An introduction to the linear representations of finite groups

A few elements of the formalism of finite group representations are recalled. As to avoid a too mathematically oriented approach the discussed items are limited to the most essential aspects of the linear and matrix representations of standard use in chemistry and physics.


INTRODUCTION
Symmetry is ubiquitous in nature and of an extremely wide variety.It may be discrete, such as the space inversion, the time inversion, the crystal isometries, . . ., or continuous, such as the euclidean isometries, the galilean invariance, the gauge invariances, . . . .It may be obvious, generally when it is of geometric nature.It may be hidden, often when it is of dynamical origin, then revealing itself indirectly. 1 It may be more or less blurred, typically as perceived in the complex systems, botanical, biological, . . . .It may be spontaneously broken, in which instance it becomes the source of a number of non-trivial phenomena, including the phase transitions, the bifurcations in the non linear processes, . . ., that gives rise to a wealth of structuration.It suffices for illustration to evoke the uncountable physical phases of matter, for instance the crystalline and mesomorphic forms or else the magnetic orders among the most familiar categories, not to mention the dynamic self-organization, the pattern formation, . . .found out in the other fields.Symmetry in fact scarcely is lowered uniformly so that the broken phase spatially builds up from different states, transforming into one another by the lost components of the symmetry, and thus is non uniform and displays defects.These in turn might interact or cross, possibly non commutatively, to organize themselves or generate further novel textures.
Symmetry gets materialized through a set of transformations of the properties of a system, which, endowed with the canonical composition law for functions, forms a group whatever the instance.Accordingly, the adequate framework within which to deal with symmetry is that of the group theory, including its ramifications into the representation theory to account for the nature of the invariances of the physical properties, the differential geometry, in particular the Morse theory, to investigate the extrema of the invariant functions of the physical properties and thus to get insights into the symmetry breaking phenomena, the algebraic topology, more specifically the homotopy theory, to feature the topological stability of defects and the formation of textures, . . . .It is clear that this is too vast a field to 1 A case in point is provided by the bound states of the non relativistic isolated hydrogen atom, which displays spectral degeneracies with respect to the principal n and orbital l quantum numbers.Whereas the l-degeneracy is an evident outcome of the symmetry group SO(3) of the rotations in the 3-dimensional space R 3 , the n-degeneracy is specific to the Kepler potentials, decreasing as the inverse of the radial distance, and emanates from the dynamical symmetry group SO (4).Considering the scattering states of the continuum in the spectrum, this metamorphoses itself into the dynamical symmetry group SO (3,1).In other words, using a more intuitive picture, the electron dynamics in a 1/r potential is equivalent to that of a free particle in the 4-dimensional space R 4 , on a sphere S 3 if it is bounded and on a double-sheeted hyperboloid H 3 if it is scattered.Another feature of the electron spectrum is the equal spacing of the energy levels when multiplied by −n 3 , which suggests duality and originates from the De Sitter spectrum generating symmetry group SO (4,1).Attempts to express the hamiltonian in terms of operators that close under commutation lead to anticipate that the largest spectrum generating symmetry group of the hydrogen atom might be the conformal group SO (4,2).
EPJ Web of Conferences describe in a few pages.The ambition of these notes is strongly limited.It is to focus on the mathematics of the linear representation of finite groups.After short recalls of basic concepts, questions of reduction and irreducibility are discussed.Next, character theory is succinctly explored.Complete reducibility of the linear representations of finite groups, the relevance and usefulness of the Schur's Lemmas, complete invariant nature of the characters with respect to intertwining and character completeness over class functions are emphasized.Construction of induced linear representations will be approached and search methods of irreducible representations will be mentioned only briefly.Of course, the discussed items are far from providing even the rough idea of all the richness of the group representations.A number of their facets are only alluded to or merely ignored, for instance concerned with the multi-valued spinor representations, the projective representations, . . ., not to mention the linear representations of continuous groups or else the non linear group actions.An extremely wide literature exists on these topics, quite often purely mathematical, including textbooks or reviews to start with.See for instance [1-5].

BASIC CONCEPTS
A representation of a group G on a mathematical object X designates an homomorphism : G → Aut(X) from the group G to the automorphism group Aut(X) of the object X: G may be any group, finite or infinite, possibly topological in which case it may be (locally) compact or non compact, n-connected, . . . .X may be any set endowed with a mathematical structure, for instance a topological space, a differentiable manifold, a module over a ring, . . . .Aut(X) is the group formed by the set of the bijective functions f : X → X that preserve the mathematical structure of X, endowed with the canonical composition law • for the functions.If X is a vector space V over a scalar field K then Aut(V) is the group GL(V, K) of the invertible linear operators on V: In this case is particularized by naming it a linear representation.V is the representation space.It is customary to call dimension of the representation the dimension d of V.Only the linear representations of the finite groups G on the vector spaces V over the field C of the complex numbers 2 are discussed in this manuscript, unless otherwise explicitly stated.With every linear representation : G → GL(V, K) is associated its kernel ker( ) and its image im( ), given as where 1 V ∈ GL(V, K) is the identity operator on the representation space V.If (g) = (h) then gh −1 ∈ ker( ).It follows that is injective if and only if (iff) ker( ) = {e}, where e is the unit element of G. by definition is surjective iff im( ) = GL(V, K).If (g, h) ∈ ker( ) 2 then (gh −1 ) = 1 V , namely gh −1 ∈ ker( ), which implies that ker( ) is a subgroup of G.It similarly is shown that im( ) is a subgroup of GL(V, K).If g ∈ G and h ∈ ker( ) then (ghg −1 ) = 1 V , namely ghg −1 ∈ ker( ), which can be defined.With w = (g)( u), this is rewritten # (g)( v # ) ( w) = v # (g) −1 ( w) .In other words # (g)( v # ) = v # • (g −1 ), which makes up another equivalent defining relation and clearly shows that # (g) does exist and is unique thanks to the existence and unicity of (g −1 ) ∀g ∈ G.Moreover, # (gh 2 , which demonstrates that # is a group homomorphism.
# is the dual representation of .All the theorems established for are valid for # , and conversely, by mere structure transport.

Canonical examples
Automorphism groups GL(V d=1 , C) of 1-dimensional vector spaces V d=1 are isomorphic to the multiplicative group C of non null complex numbers, insofar as every invertible linear operator on V d=1 is equivalent to the multiplication by a same non null scalar: if ê is the basis vector in  A linear form by definition is an application v # : V → C from a vector space V to its scalar field C such that v # (a r + b s) = a v # ( r) + b v # ( s), ∀(a, b) ∈ C 2 , ∀( r, s) ∈ V 2 .It also is called a one-form, a linear functional, a co-vector, a contravariant vector when the elements of V are called covariant vectors, . . . .This merely emphasizes the wealth of context within which the concept might be in use, such as differential geometry, measure theory, multilinear algebra, . . . .If V has the finite dimension d then V # has the same dimension d.A basis { êi # } i=1,...,d in V # is twinned in fact to any selected basis { êi } i=1,...,d in V such that êi # ( êj ) = ij , where ij is the Kronecker symbol ( ij = 1 iff i = j and ij = 0 otherwise).When V is infinite-dimensional the same construction does not end up with a basis.It leads to a family of linearly independent vectors that is not spanning.The linear forms on a finite-dimensional normed space V are bounded and therefore are continuous.

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EPJ Web of Conferences is clear that g m ∈ G ⇒ g m+1 ∈ G ∀m ∈ N whence, by increasing to infinity, m crosses integers p for which there exists strictly positive integers q < p such that g p = g q or else g p−q = e, unless G is infinite.In other words, whenever the group G is finite each of its element g is of finite order Obviously, (g) n g = (g n g ) = (e) = 1, which means that (g) (g) (g) is an n g -n gn g -th root of 1 1 1, the multiplicative unit of C .Now whatever the group G, finite or not, in the event that one has is called the trivial representation of the group G. Its significance is to reveal the full invariance of a physical property with respect to the symmetries abstracted by the elements of the group G.
Indexing with the elements x of a finite set X the basis vectors êx of a vector space V and associating each element g of a finite group G with the invertible linear operator X (g) on V that sends êx to ê (g)(x) , where : G → P X is an homomorphism of the group G into the group P X of the permutations of X, generates a linear representation X , which is called the permutation representation of the group G associated with the set X. Note that the group homomorphism : G → P X defines a representation of the group G on the set X.It is the usage in that case to state that the group G acts on the set X or else that X is a G-set.In the specific instance where the set X contains the same number n G of elements as the group G the permutation representation is isomorphic to the so-called regular representation G of the group G.One conventionally defines G by indexing the basis vectors of the vector space V with the elements h of the group G, more concisely as êh where h ∈ G, and by associating each element g of the group G with the invertible linear operator G (g) on V that transforms the basis vectors, thus G-indexed, according to the formula The regular representation G is particularized because containing each irreducible representation i of the group G with a repetition factor equal to its dimension d i .The dimension of G is the order n G of the group G.The set { G (g)(ê e ) | g ∈ G}, engendered from the single vector êe indexed with the unit element e of the group G, forms a basis of the representation space V. Conversely, given a linear representation : G → GL(V, C), if there exist a vector v in the representation space V such that the set { (g)( v) | g ∈ G} forms a basis of V then necessarily is isomorphic to G .Consider indeed the isomorphism : V → V defined by setting

Matrix representations
Let V be a vector space with dimension d over the field C. Any element (g) of the group GL(V,C) of the invertible linear operators on V is fully determined from the images (g)(ê m ) of the basis vectors êm = n y n ên then y n = m (g) nm x m .The d 2 complex coefficients (g) nm make up the entries of a d × d invertible matrix (g), called the matrix representative of the linear operator (g).Assume that (g) generically symbolizes the image of an element g of a group G by a linear representation : G → GL(V, C), so that ∀ g, h ∈ G, (gh) = (g) • (h).It follows from

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Contribution of Symmetries in Condensed Matter the equation (2.10) that (gh)(ê m ) = n ên (gh) nm and (g) This means that the mapping : G → GL(d, C) of the group G to the group GL(d, C) of d × d invertible matrices with entries in C, which to each element g in G associates the matrix representative (g) of the linear operator (g) with respect to the selected basis {ê} m=1,...,d , defines a group homomorphism.This is called a matrix representation of the group G.
The selection of another basis { f } n=1,...,d would have led to other matrix representatives (g), giving rise to another matrix representation : G → GL(d, C).Associated with the same linear representation and merely emerging from the selection of two different bases in the representation space V, the matrix representations and are said similar or equivalent.If S is the invertible matrix associated with the basis change {ê} m=1,...,d → { f } n=1,...,d , which often is called a similarity transformation, then 4 (g) = S (g) S −1 ∀g ∈ G (2.12) and are said intertwined with S. Conversely, any two finite dimensional matrix representations of a finite group intertwined with an invertible matrix are similar.As with the linear representations, a standard notation for two equivalent matrix representations is ∼ ∼ ∼ .Now, (g) could have been interpreted also as the matrix representative with respect to the initial basis vectors êm (m = 1, . . ., d) of a linear operator (g) associated with another linear representation : G → GL(V, C).The equation (2.12) then would mean that there exists an automorphism of V which is equivariant: Conversely, any automorphism of V corresponds to a change of bases.Accordingly, the isomorphism of linear representations and that of matrix representations describe the same equivalence.
As from every matrix M with entries M ij in C is built the complex conjugate M with the entries (M ) ij = (M ij ) , the transpose t M, by column-row interchange, with the entries ( t M) ij = M ji and the adjoint M † = ( t M) with the entries (M † ) ij = (M ji ) .Given a matrix representation : G → GL(d, C), by associating each element g of the group G with the complex conjugate (g), the transpose t (g) and the adjoint † (g) of (g) one respectively defines the conjugate , the transpose t and the adjoint † of the matrix representation .

Direct sums
Let : G → GL(V, C) be a linear representation.A proper subspace V 1 of the representation space V by definition is stable or invariant under the group G iff or, in terms of subsets, (g the zero-dimensional vector space { 0}.V and { 0} are trivially stable under any group G.The restriction V 1 (g) of (g) to V 1 determines an automorphism of V 1 and follows the group homomorphism rule is a linear representation of the group G on the vector space V 1 , which is called a subrepresentation of . 4 00005-p.5

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Select a basis {ê m } in V 1 and extend it to a basis {ê m } ∪ { f n } in V, which always is possible whenever V is finite-dimensional or otherwise once the axiom of choice is allowed. 5A subspace V 2 f of V is linearly spanned by the set of vectors f n .It is called a complement of the subspace V 1 in the vector space V, because any vector v in V writes uniquely as It may be emphasized that a complement of a proper subspace is a proper subspace and that V and { 0} are the complements of each other in V.One symbolically formulate the fact that two proper subspaces V 1 and V 2 of a vector space V are the complements of each other in V as In the event that not only the proper subspace V 1 but also the selected complement V 2 in V is stable under the group G, the restriction V 2 : G → GL(V 2 , C) of to the representation space V 2 makes up another subrepresentation of .Importantly, ∀g ∈ G ∀ v ∈ V, (g)( v) is fully and uniquely determined by the sum . It is customary to transcribe these properties by symbolically equating to the direct sum of V 1 and V 2 : With respect to the basis {ê m } ∪ {ê n }, built by union of the basis {ê m } in V 1 and the basis {ê n } in V 2 , the matrix representatives (g) of the linear operators (g) on V write in the block diagonal form namely as the direct sum 1 (g) ⊕ 2 (g) of the matrix representatives 1 (g) of the linear operators V 1 (g) on V 1 with respect to the basis {ê m } and of the matrix representatives 2 (g) of the linear operators V 2 (g) on V 2 with respect to the basis {ê n }.Again, now to implicitly recall the block-diagonal structure of the matrix representatives (g), it is the convention to symbolically write and, subsequently, to state that the matrix representation is the direct sum of the sub-matrix representations 1 and 2 .
As an illustration, let G : G → GL(V, C) be the regular representation of a group G on the vector space V with basis {ê g } g∈G and let V 1 be the one-dimensional subspace of V consisting in the scalar multiples of the vector g∈G êg .V 1 evidently is stable under G: , where e is the unit element of G and n G is the order of G.It easily is shown that the dimension of the subspace , it is straightforwardly inferred that the subspace V 2 is stable under G. Accordingly, the subspaces V 1 and V 2 thus constructed are effectively complement 5 The axiom of choice is not universally accepted because it leads to strange theorems, the most famous being the Banach-Tarski paradoxical decomposition.Ignoring it however also leads to disasters, for instance a vector space may have no basis or may have bases with different cardinalities.As to cure some of the inconveniences, in particular the existence of non-measurable sets of reals, the axiom of determinacy was put forward in replacement, but this still might not be all satisfactory.Under this axiom every subset of the set of reals R is Lebesgue-measurable, but, for instance, R as a vector space over the set of rationals Q has no basis.

Contribution of Symmetries in Condensed Matter
of each other and invariant under G so that G can be put into the direct sum of the subrepresentations built over these proper subspaces.Another choice of complement could have been made with the n G − 1 vectors êh∈(G−{e}) , but this is not stable under G.It suffices to observe that G (g)(ê g −1 ) = êe does not belong to this complement.

Maschke's theorem
A convenient tool to handle the direct sums of proper subspaces is the projection operator.It is recalled that given the decomposition V = V 1 ⊕ V 2 f , every vector v in V by definition writes uniquely f .The linear operator f that sends every vector v in V onto its component f .It again is clear that f • f = f , which thus makes up another equivalent definition of a projection operator f .A bijective correspondence is thus established between the projection operators f of V onto V 1 and the complements V 2 f = ker( f ) of V 1 in V. Let : G → GL(V, C) be a linear representation of a finite group G on a finite-dimensional vector space V over the field C of the complex numbers.Let V 1 be a proper subspace of the representation space V, which is invariant under the group G. Let V 2 f be an arbitrary complement of V 1 in V, not necessarily invariant under the group G. Let f be the projection operator of V onto V 1 bijectively associated to V 2 f .Let be the "average" of f over G, which is defined as: where n G is the order of the group G. is a linear operator on V, since it is a function sum of functionally composed linear operators on V. "commute" with G: by using the dummy transformation g → hg in the second equality and the identity (hg) . It finally is inferred that is a projection operator: by using the equation 2.20 in the second equality and the equation 2.21 in the third equality.Accordingly, the G-invariant subspace V 2 = ker( ) is a complement of the initially assumed G-invariant subspace V 1 : A fundamental theorem is thus proven, the so-called Maschke's Theorem, which states that whatever the linear representation

a finite group G on a finite-dimensional vector space V over the field C C C, to every invariant subspace
With the same proof arguments it is extended, for any finite group G, to any finite-dimensional vector space V over any scalar field K of any characteristic char(K) that does not divide the order n G of the group G, this merely by generalizing the average procedure in equation (2.19) to K-summation and division by n G 1 K , where 1 K is the multiplicative unit of K.It is clear that if n G ≡ 0 (mod char(K)) then this G-averaging cannot be defined since n G 1 K = 0 K , where 0 K is the additive unit of K.

Inner products
Another proof of Maschke's Theorem can be forged using inner products, inspiring generalizations to compact continuous groups G.An inner product on a vector space V over the field C designates a two-arguments application It immediately follows from the two first properties (i and ii) that the inner product is antilinear in the first argument (•): An inner product in other words is a positive definite conjugate symmetric sesquilinear form.
A sesquilinear form is the generic name for any application : V × V → C which is antilinear in the first argument and linear in the second argument.uniquely defines an antilinear application : V → V # , u → u # ≡ ( u, •).Conversely, an antilinear application from a vector space V to its dual V # uniquely determines a sesquilinear form. is non degenerate iff is injective, which means ker( is the conjugate symmetric to .If = ( = − ) then is called an hermitian form (anti-hermitian form).
A vector u is orthogonal to a vector v with respect to a sesquilinear form iff ( u, v) = 0. Let W be a subspace of V.The set then the restriction W of to W is non degenerate, which means that the restriction W of to W is injective.If, in addition, W is finitedimensional then W # is of the same dimension as W and W becomes a bijection.sends every v ∈ V to a unique linear form v # ∈ V # , since it is an application.The restriction w # of v # to W obviously is also unique.To the linear form w # finally corresponds a unique w ∈ W, because W is a bijection.In other words, to every whatever the finite-dimensional subspace W of V. Thus, to every finite-dimensional subspace W of a vector space V over the field C endowed with an inner product is associated an orthocomplement W ⊥ ⊥ ⊥ in V.
Let be a linear operator on the vector space V.The transpose of is the linear operator t on the dual space V # defined from the pointwise relation

Contribution of Symmetries in Condensed Matter
along .If is invertible then t = ( −1 ) # .Let be a non degenerate sesquilinear form.A linear operator † may be defined in V from the pointwise relation ( u, v) = ( † u, v).It is called the adjoint of with respect to .If the application : V → V # , u → u # ≡ ( u, •) is bijective, which is the case only if the vector space V is finite-dimensional, then the adjoint of always exists, given as † = −1 • t • . 6A sesquilinear form by definition is invariant with respect to a linear operator iff ( Obviously this is the case iff is invertible and † = 1 V , namely † = −1 .then is said unitary.The unitary operators are normal operators.A linear operator is normal iff it commutes with its adjoint: It is diagonalizable and its eigen-spaces are pairwise orthogonal (spectral theorem for the normal operators).Another subfamily of normal operators are the self-adoint operators: † = . 7f V is finite-dimensional and where U † (≡ t U ) is the complex conjugate row vector (u 1 , . . ., u d ) and V the column vector (v 1 , . . ., v d ).The sesquilinear matrix with the entries ij = (ê i , êj ) uniquely determines once the basis is given.is non degenerate iff Det( ) = 0.A basis { êi } i=1,...,d is orthonormal with respect to iff = I d (d × d unit matrix).Let be a linear operator and denote A the matrix representative of and A † the matrix representative of † in the { êi } i=1,...,d basis.The pointwise relation ( , since by hypothesis is non-degenerate.If in addition the chosen basis is orthonormal with respect to then It is emphasized that inner products can be defined solely on vector spaces over the field R of the real numbers, which is an ordered field, or the field C of the complex numbers, which is not ordered but makes up an ordered extension of the field R. The basic reason is that otherwise it becomes meaningless to require that a sesquilinear form be positive definite.This clearly excludes all the fields with non zero characteristic, which cannot have an ordered subfield.

Unitarity and unitarisability
A linear representation : G → GL(V, C) of a finite group G on a vector space V over the field C by definition is a unitary representation if the representation space V is endowed with an inner product which means that the linear operators (g) are unitary for every g in G. Another way telling the same thing is that the linear representation commutes with the inner product • | • . 6It is customary in physics to use the so-called bra-ket notation.The space V then is endowed with an inner product • | • (pre-Hilbert space).V is complete for the associated norm (Hilbert space), namely every Cauchy sequence in V converges within V. A vector is denoted by a ket | and a linear form by a bra |.The application of a linear operator O on a ket is described as O| .Its dual is applied on a bra | as To any ket | one may associate a bra | (Riez Theorem).The converse is true solely in finite dimension.If V is infinite-dimensional then V can be put in bijection only with the subspace of continuous linear forms in the dual V # .The "discontinuous" bra have no ket counterpart. 7A bijective correspondence exists between the self-adjoint operators H on a Hilbert space V and the families of unitary operators U( ) ∈R on V with the group property U( + ) = U( ) • U( ) and the continuity property U( → ) → U( ), to be precise U( ) = exp(i H) (Stone's theorem).When the Hilbert space is separable it suffices to assume weak measurability instead of continuity (von Newman).This bijection is useful in establishing the uniqueness of the irreducible unitary representation of the algebra of canonical commutation relations on finitely many generators (Stone-von Newman theorem).This is no more the case with infinitely many generators, concretely in quantum field theory where in general there is no unitary equivalence between canonical commutation relation representation of the free field and that of the interacting fields (Haag theorem).

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If the representation space V is finite-dimensional with dimension d then a unitary matrix representation ϒ : G → GL(d, C) of the group G is obtained by selecting in V an orthonormal basis { êi } i=1,...,d with respect to the inner product • | • .ϒ associates each element g of the group G to a unitary matrix representative ϒ(g): where for every element g in the group G.
Let W be a finite-dimensional proper subspace of the representation space V and let W ⊥ be the orhocomplement of

representation of a unitary representation is obviously unitary for the restricted inner product. Accordingly, every unitary representation of a finite group G on a vector space V over the field C that contains a finite-dimensional subspace W invariant under G can be decomposed into two unitary subrepresentations as
where W stands for the restriction of to W and W ⊥ for the restriction of to the orthocomplement W ⊥ of W in V.The two subrepresentations might in turn be decomposed into subrepresentation and so on.The process must end after a finite number of iterations if V is finite-dimensional, since by hypothesis the invariant subspace is a proper subspace so that at each step the dimension of the subrepresentation spaces to consider is decreased.It nevertheless is emphasized that no conditions is imposed on the dimension of the representation V, which thus might be infinite.So, at least as far as G is finite, the dichotomy processes might go on indefinitely and lead to infinite direct sums or even direct integrals.As a matter of fact, the construction of a meaningful direct integral often can fail, all the more as the group G is unspecified, and leads to extremely delicate and difficult problems of functional analysis.
A linear representation : G → GL(V, C) is unitarisable by definition if an inner product invariant under G can be defined in the representation space V. Assume that V possesses a basis {ê i }.Whatever the vector u = i x i ( u)ê i in V the set of complex numbers {x i ( u)} is uniquely defined.So is the product In other words, an inner product • | • in V is defined by declaring that the basis { êi } is orthonormal. 8If the group G is finite then the application In other words, • | • G is an inner product which is invariant under G.The linear representation becomes a unitary representation by endowing the representation space V with the inner product • | • G .Note that every change of inner products is equivalent to a basis change. 9 fundamental theorem is thus proven, which states that every linear representation of a finite group G on a vector space V over the field C is unitarisable and therefore isomorphic to a unitary representation.It thus can always be decomposed into subrepresentations whenever there exists a finite-dimensional proper subspace invariant under G in the representation space.The group average displayed in the equation (2.26) is the so-called Weyl's Trick.It already was employed in a disguised manner for a projection operator in the equation (2.19).It can be extended to linear representations of topological groups, 10 provided the summation over the group elements can be generalized to an appropriate integration. 11ne finally may wonder whether the unitarity concept is worth extending to invariance with respect to hermitian forms not necessarily positive definite, to deal with linear representations on vector spaces 9 A basis { f i } orthonormal with respect to • | • G can even always be built, using for instance the Gram-Schmidt procedure: Of course, the change from the basis {ê i } to the basis { f i } describes nothing but a similarity transformation. 10A topological group by definition is a set G endowed with a group structure and a topological structure such that the group operation G op : (g, h) → gh −1 is a continuous function, to be precise the inverse image of any open set of G by this function is an open set of the topological product space G × G.A topological space is separated iff for any pair of distinct points there exists disjoint neighborhoods (Hausdorff).It is quasi-compact iff a finite cover can be extracted from every open cover (Borel-Lebesgue).It is compact iff it is separated and quasi-compact.It is locally compact iff every point possesses a compact neighborhood.It is simply connected iff every loop is homotopic to the null loop.A loop is a continuous function : [0, 1] → G such that (0) = (1).A loop at a point g is null iff im( ) = {g}.A loop is homotopic to a loop iff there exists a continuous function : [0, 1] × [0, 1] → G such that (0, ) = (1, ) ∀ and ( , 0) = ( ), ( , 1) = ( ) ∀ .A topological group is m-connected iff at every point it shows m homotopy classes of loops.Its representations then might be m-valued, but for each multiply-connected group there exists a simply connected group, the universal cover, that is homomorphic to it.A few examples: A field is topological iff its additive and multiplicative groups are topological.A vector space on a topological field endowed with a topological structure such that the vector addition and the scalar multiplication are continuous is topological.A continuous representation of a topological group G on a topological vector space V over the field C is a linear representation 11 If G is a locally compact topological group then there always exist a measure dg and only one carried by G and enjoying the properties i-G F(g)dg = G F(gh)dg for every h in G and every continuous function F on G (invariance of dg under right translation) and ii-G dg = 1 (mass normalization).If G is compact then dg is also invariant under left translation: G F(g)dg = G F(hg)dg, in which case dg is called the bi-invariant or Haar measure of G.If the group G is finite of order n G , the measure dg is obtained by assigning to each g in G a mass equal to 1/n G .If G is the group SO(2) of the planar rotations and if every g ∈ SO(2) is represented in the form g ≡ exp(i ) ( taken modulo 2 ) the invariant measure is d /2 .As a matter of fact, the concrete construction of the Haar measure generally is far from being obvious, except possibly for groups of geometric nature (O(n, K), SO(n, K), U (n, K), . ..).An efficient method can be worked out for a Lie group G of dimension n represented by unitary matrices U = exp(iH) of order N .The hermitian matrix H belongs to the associated Lie algebra G and can be parametrized as H(x) = p x p X p with x q = Tr(HX q ), by means of the generators X p chosen such that X p , X q = iC pqr X r and Tr(X p X q ) = pq .As from the invariant metric Tr(dU † dU) = −Tr U −1 dUU −1 dU = pq (x)dx p dx q 00005-p.11EPJ Web of Conferences over fields with non zero characteristic.A more generalized approach might even be considered, since sesquilinear forms might be defined on any module over a ring for an unspecified antiautomorphism (in place of the conjugate complex involution).The drawback is that the crucial result according to which every proper subspace possesses an orthocomplement then would be lost.Isotropic subspaces, the vectors of which are all orthogonal to at least one of their own non null vectors, might exist, that thus might not necessarily have a complement.

Irreducibility and reduction
A linear representation of a group is said irreducible if its representation space contains no proper invariant subspace under the action of the group and reducible otherwise.A reducible representation is not necessarily decomposable into subrepresentations, since this requires that to the identified invariant subspace is associated an invariant complement.A linear representation then might be reducible but indecomposable.A linear representation is said completely reducible if it is decomposable down to irreducible components.
Let : G → GL(V, K) and : G → GL(W, K) be two linear representations intertwined with the isomorphism : V → W. Assume that there exists a G-invariant subspace V 1 in V and denote W 1 its image by in W. W 1 obviously is a subspace of W, which is G-invariant: and the dimensions of V i and W i (i = 1, 2) are the same.W 2 of course is also G-invariant.This means that every linear representation isomorphic to a decomposable linear representation is itself decomposable.Assume now that there is no G-invariant subspace V 1 in V then obviously there can be no invariant subspace in W, otherwise its image by −1 would be a G-invariant subspace in V in contradiction with the hypothesis.Accordingly, every linear representation isomorphic to an irreducible linear representation is itself irreducible.It similarly is shown that every linear representation isomorphic to a reducible but indecomposable linear representation is itself reducible but indecomposable and every and the identity d(e ) is diagonalized by the same unitary matrix as the n × n real antisymmetric matrix M(x) = −M † (x) with the entries M pq (x) = r x r C rqp .It follows that if ±i j ( j ∈ R + ) denotes the eigenvalues of the matrix M(x) then The eigenvalue problem needed to evaluate the Haar measure are differences of eigenvalues ν i of H(x).
Assume that : G → GL(V, C) is a linear representation of a compact group G and assume that the representation space is endowed with an inner product We thus have demonstrated that every linear representation of a compact group is unitary.Using similar arguments as with the unitary representation of finite groups it then is shown that every finite-dimensional linear representation of a compact group is completely reducible.As a matter of fact, as far as only the finite-dimensional representations on the vector spaces over the field C are considered, almost all the theorems that are proved for finite groups are safely extended to compact groups, be it that at some places a sum must be replaced by an integral.

00005-p.12
Contribution of Symmetries in Condensed Matter linear representation isomorphic to a completely reducible linear representation is itself completely reducible.It is the usage also to call irreducible (resp.reducible and decomposable, reducible but indecomposable, completely reducible) the matrix representation obtained from an irreducible (resp.reducible and decomposable, reducible but indecomposable, completely reducible) linear representation by selecting a basis in the representation space.
Complete Reducibility Theorems may be formulated for certain families of linear representations.Among the most important for the physics of the finite groups of symmetry is the one which states that every linear representation of a finite group on a finite-dimensional vector space over the field of complex numbers is completely reducible.As to prove it one proceeds by induction on the dimension d of the representation space V. Assume that the statement holds for all the representations of dimension smaller than d, and let be a linear representation of dimension d.If V is irreducible, then there is nothing to prove.Otherwise, there exists a proper subspace V 1 , therefore of dimension d 1 < d, invariant under G.According to the Maschke's Theorem, V 1 has in V a complement V 2 , therefore of dimension d 2 < d, which is also invariant under G. Accordingly, = 1 ⊕ 2 , where i (i = 1, 2) is the restriction of to V i (i = 1, 2).Now, by the induction hypothesis the subrepresentation i (i = 1, 2) is completely reducible, since d i < d (i = 1, 2).So the same is true of , which ends the proof.Note that although the mathematical induction might suggest that the theorem might be true for infinite countable dimension, the corresponding extension would make up an abuse at this step for the Maschke's Theorem is demonstrated only for finite-dimensional V.
The theorem is straightforwardly extended to the linear representation of the finite groups on the finite-dimensional vector spaces over the fields whose characteristic does not divide the order of the group, from the corresponding extension of the Maschke's Theorem.Using the Weyl's Trick the theorem also is extended to the linear representation of the compact groups on the finite-dimensional vector spaces over the field C. Note, meanwhile, that the finite groups are compact, for the discrete topology.It happens that finally the infinite-dimensional case does not cause excessively more troubles for compact groups.

It indeed is shown that every continuous representation of a compact group on a Hilbert space V, be it infinite-dimensional, is isomorphic to the Hilbert sum of finite-dimensional unitary representations and the set of G-finite vectors is dense in V. A Hilbert sum of unitary representations
: < ∞} on the Hilbert sum of the representation spaces V , that coincides with on each sector.⊕ V is the Hilbert space with inner product (( u ), ( v )) = u , v and contains ⊕ V as a dense subspace with V ⊥V ∀ = .A set of G-finite vectors is the set of all vectors v f in in V such that the dimension of the vector space spanned by { (g)( v f in ), g ∈ G} is finite.It follows in particular that the irreducible unitary representations of the compact groups are all finite dimensional.A proof is provided first by showing that there always exists a finite-dimensional G-invariant (closed) subspace in V, for instance the eigenspace of any non zero eigenvalue of a G-averaged compact operator on V, and next, using the Zorn's Lemma, by establishing that the set ⊕ V , partially ordered by inclusion, necessarily shows a maximal element.As a result ⊕ V cannot be different from V, otherwise there would exist V ∈ ( ⊕ V ) ⊥ in violation of the maximality.Note that the "Zorn's Lemma" is equivalent to the axiom of choice (see footnote 5).Non compact groups do show infinite-dimensional representations which are more delicate to handle or else linear representations that cannot be isomorphic to unitary representations or reducible representations that are indecomposable. 1212 Although to some extent either exotic or pathological for what might concern physical systems the counterexamples to the complete reducibility of the linear representations are not that uncommon, even with finite groups, and it always is instructive to have scrutinized at least one.Consider for instance the matrix representation of the cyclic group C p of order p and generator s on the linear group of the 2 × 2 invertible matrices with entries in the field Z/qZ of characteristic char(Z/qZ) = q.At first it is observed that if q does not divide p then cannot be a group homomorphism and 00005-p.13

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Now, let : G → GL(V, C) be a completely reducible linear representation of a finite group G. Choose an initial G-invariant subspace, find its complement and perform a first decomposition into two sub-representations, then proceed similarly on each of these and so on until getting only irreducible sub-representations.Grouping isomorphic irreducible summands, one most generally would write where k is isomorphic to the direct sum of n k copies of an irreducible linear representation k : G → GL(V k , C), these by construction being non-isomorphic for different k's.A symbolic manner transcribing all this is where V ⊕n k k is isomorphic to the subspace X k of V spanned by the different G-invariant subspaces of V associated with each copy of k and n k defines the multiplicity of the irreducible component k contained in .It is customary to call = k k the canonical decomposition of , or else the decomposition of into isotypical components k .An irreducible matrix representation k : G → GL(d k , C) is associated with the irreducible linear representation k : G → GL(V k , C) as soon as a basis is selected in the representation space V k .With every isomorphism of V that transforms a given copy of V k in V to another copy of V k in V is associated two distinct bases in one-to-one correspondence and two isomorphic irreducible matrix representations.A basis of X k ∼ = V ⊕n k k thus may be built from different isomorphisms in V sending an initial copy of V k in V to the different copies of V k in V.With respect to this basis the linear representation k is associated to a matrix representation k : G → GL(n k d k , C) isomorphic to the direct sum of n k copies of the irreducible matrix representations k : G → GL(d k , C).A basis in V is obtained from the union of the bases built on each subspace X k , since V is the direct sum of the X k ∼ = V ⊕n k k .The matrix representation : G → GL(d, C) associated with the linear representation : G → GL(V, C) with respect to this basis in V is given as the direct sum = 1 ⊕ 2 ⊕ . . .⊕ s = k k .It again is standard to write and customary to call = k k the canonical decomposition of , or else the decomposition of into isotypical components k .A similar procedure may be replicated to get canonical decompositions of linear representations of compact groups, possibly by using Hilbert sums of representations.Note that at this stage it is not sure whether the canonical decomposition is unique, so deserves its name, and whether the n k are unambiguously defined.
therefore cannot be a matrix representation associated with a linear representation.Next ker( ) = {e}, that is to say is injective, iff q = p.Now assuming that either q divides p or equals p, the one-dimensional space spanned by the (1, 0) vector is invariant under C p , but it has no invariant complement: the representation is reducible but indecomposable.In a different context, if l is a prime then the set Z l = inv.lim.Z/l n Z of l-adic integers makes up a compact topological group, which has the continuous reducible but indecomposable representation on a 2-dimensional vector space over the field Q l of l-adic numbers.This example tells that "compact group" and "continuous representation" are not enough conditions.The basis field must be C. Substituting the additive group R for Z l and the automorphism group GL(2, C) for GL(2, Q l ) a third example of continuous representation is obtained, which again is reducible but indecomposable.It also is not unitarizable.In this case the failure of complete reducibility is to ascribe to the fact that R is not compact.It is only locally compact, because it is not bounded.The compact subsets of R n (C n ) are the closed and bounded subsets of R n (C n ).

Schur's lemmas
It is clear that there exists a number of ways to decompose reducible linear representations down to irreducible components, so that to proceed further it is necessary to get deeper insights into their isomorphisms.As a matter of fact, the irreducible linear (or matrix) representations are special in their intertwining.This is formulated in the Schur's Lemmas: The irreducibility of 1 and 2 leaves ker( ) = { 0 1 } or V 1 and im( ) = V 2 or { 0 2 } as the only options. is non zero iff ker( ) = { 0 1 }, which means that is injective, and im( ) = V 2 , which means that is surjective, that is to say iff is an isomorphism.As a consequence, 1 , which is only a vector space, End G (V i ) (i = 1, 2), endowed with the canonical composition law • for the functions, shows the structure of a division algebra, with unit i (i = 1, 2) and composition inverse for each of its non zero elements.Now, select a non zero 1 in End G (V 1 ) and pick up another arbitrary ∈ End G (V 1 ).Obviously . It is implicitly assumed that the representation space V 1 is finite-dimensional.Accordingly, as the field C is algebraically closed, there always exists for • −1 1 an eigenvalue ∈ C: ker( which ends the proof of Schur 1. Schur's Lemma are straightforwardly generalized to finite-dimensional irreducible representations of compact groups, using the same proof arguments.With infinite-dimensional representations discrete eigenvalues might not necessarily exist and one has to resort to the spectral theorem for normal bounded operators, which states that for any in End G (V) there exists a projection valued measure such that = spec( ) d and that the only bounded endomorphisms of V commuting with are the ones commuting with the self-adjoint projection (B) for each Borel subset B of the spectrum spec( ).Whatever the case, Schur 1 obviously implies that Schur's Lemma may be extended to scalar fields K other than the field C of complex numbers under the weaker formulation:

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which is inferred solely from the G-invariance of the subspaces ker( ) and im( ) for any in Hom G (V 1 , V 2 ) and the irreducibility of 1 and 2 .It also is clear that any non zero in Hom G (V 1 , V 2 ) is an isomorphism and therefore Hom , endowed with the canonical composition law • for the functions, shows the structure of a division algebra over the field K.This leads to three possibilities: is to say if K is not algebraically closed but its closure is a finite extension, then by virtue of the (1,2,4,8)-Theorem on the real division algebras and since it implicitly is clear that End G (V i , K) (i = 1, 2) is associative but not necessarily commutative, d End G (V i ,K) (i=1,2) may take the values 1, 2, 4 and the division algebra End G (V i , K) (i = 1, 2) may be isomorphic to either R 1 V , C 1 V or Q 1 V , where Q stands for the field of quaternions. 13iii-if K is neither algebraically closed nor real closed then d End G (V i ,K) (i=1,2) is the square of an integer.
The transcription of the Schurs Lemmas into the language of complex matrix representations of finite groups is easily inferred as: d, C) is an irreducible complex matrix representation of dimension d of a finite group G then every d × d matrix A commuting with is a multiple of the d × d identity matrix 1 d : Schur 2 -No intertwining may exist between two irreducible complex matrix representation of a finite group G except if these are associated with isomorphic representation spaces: Schur's Lemmas have a number of impacting outcomes.Schur 1 for instance implies that every irreducible complex representation The irreducibility of then implies that the representation space V itself is 1-dimensional.This easily is generalized to compact groups using similar arguments, 14 but fails with scalar fields K that are not algebraically closed.A simple illustration is provided by the real representations : C 3 = s| s 3 = e → GL(V, R) of the cyclic group C 3 .If is irreducible then it either is isomorphic to the 1-dimensional trivial representation or to the 2-dimensional representation that associates the generator s of C 3 to the 2-dimensional geometric rotation by an angle 2 /3 in a plane.The matrix representative of this rotation with respect to any selected basis in V has complex eigenvalues.It thus 13 The (1,2,4,8)-Theorem can be given different equivalent formulations.It in particular states that, up to isomorphism, the only division algebra over a real closed field are the 1-dimensional real algebra R, the 2-dimensional complex algebra C, the 4-dimensional quaternion algebra Q and the 8-dimensional octonion algebra O.At each increase of the algebra dimension an essential property is lost: a nonidentical involution must be introduced to get C, commutativity is lost with Q then associativity is lost with O, but these algebra still are alternative.Algebras of higher dimension are constructed using the dimension-doubling Cayley-Dickson process: (x 1 , x 2 )(y 1 , y 2 ) = (x 1 y 1 − y 2 x 2 , x 1 y 2 + y 1 x 2 ), (x 1 , x 2 ) = (x 1 , −x 2 ).According to this, the next in the list is the 16-dimensional sedenion algebra S, which is no more alternative nor a division algebra, but retains the property of power associativity.The (1,2,4,8)-Theorem encompasses the weaker previous Frobenius', Hurwitz's and Zorn's Theorems on the real division algebras, but unlike these is not proved algebraically.It actually emerges as a corollary to a theorem of topological nature: the existence of an arbitrary division algebra of dimension n over the reals implies parallelizability of the sphere S n−1 but according to the Bott-Milnor-Kervaire Theorem spheres are parallelizable only in dimensions n = 1, 2, 4, 8 (a manifold is parallelizable iff the tangent space at each point stay isomorphic to its transform induced by any parallel transport along a curve).There exists a variety of other avatars of the (1,2,4,8)-Theorem, in Topology (Hopf bundles over the spheres S n , . ..), in Geometry (construction of exceptional Lie algebra, . ..), in Number Theory (a sum of n squares of integers times another sum of n squares of integers is a sum of n squares of integers iff n = 1, 2, 4, 8, . ..), . . . . 14A number of way exists to establish that all the irreducible representations of a compact group are 1-dimensional iff G is abelian.One may use for instance the fact that the commutator group C G = {ghg −1 h −1 | g, h ∈ G} = {e} iff G is abelian and that this acts trivially on 1-dimensional representations.

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Contribution of Symmetries in Condensed Matter cannot be diagonalized with only real entries in the diagonals.As a matter of fact, it can be shown that the irreducible representations : G → GL(V, K) of an abelian group G are 1-dimensional over the endomorphism ring End G (V, K), which makes up an extension field of the field K.
Schur 2 allows demonstrating that the canonical decomposition of completely reducible linear representations is unique.Let = k k and = k k be canonical decompositions of two linear representations : G → GL(V, C) and : G → GL(U, C).Any in Hom G (U, V) maps the representation space Z k ∼ = U ⊕m k k of k to the representation space X k ∼ = V ⊕n k k of k , because every restriction kq of from a copy of U k to a copy of V q intertwines with two irreducible representations so is null as soon as k = q by virtue of Schur 2. In the more intuitive language of matrix representations, if = k k and = k ϒ k are two canonical decompositions and if and are intertwined with a matrix S then this cannot contain a non null off-diagonal block S k,q =k with which the isotypical components k of and ϒ q =k of would be intertwined.It follows, by taking for an irreducible representation k : G → GL(V k , C), that every sub-representation of which is isomorphic to an irreducible representation k is contained in k , which gives an intrinsic description of k as isomorphic to the direct sum of all the copies of k contained in .Accordingly, the canonical decomposition does not depend on the manner it might be performed, which proves its uniqueness.
Another consequence of the Schur's Lemmas, of utmost practical relevance for irreducible matrix representations, is the so-called Orthogonality Theorem.Whatever the two irreducible representations k : G → GL(V k , C) and q : G → GL(V q , C) of a finite group G and the linear application from V q to V k , the average of over the group G, which is defined as 15 In other words, ∈ Hom G (V q , V k ).It then follows from the Schur's Lemmas that k ∼ q ⇔ = 1 V k ∼ =Vq and k q ⇔ = 0. = Tr [ ] /Tr 1 V k , since and Tr 1 V k = d k , where d k is the dimension of k .Now, selecting a basis in V k and a basis in V q , the linear representations k and q and the linear operators and get associated respectively with matrix representations k and q and d k × d q (k lines − q columns) matrices T and S. In terms of matrix elements of the corresponding matrices the equation (2.31) writes: which comes out as a linear form with respect to the variables T lm .If k q , that is to say if k = q, then this form vanishes for all systems of values of the T lm .Its coefficients therefore are null, whence g∈G k jl (g) q mn (g −1 ) = 0 for arbitrary j , l, m, n.
00005-p.17 q mn (g −1 ) = 0 otherwise.All the possibilities are summarized under the compact formula: where kq stands for a generalized Kronecker symbol, defined as kq = 1 if k ∼ q and kq = 0 if k q .jn (resp.lm ) is the standard Kronecker symbol jn = 1 (resp.lm = 1) iff j = n (resp.l = m) and 0 otherwise.If the matrix representations are unitary then q mn (g −1 ) = (( q (g)) −1 ) mn = (( q (g)) † ) mn = q nm (g) , which leads to the alternative formula: The theorem can be proved also by directly using any pair of irreducible matrix representations k and q and applying the Schur's Lemmas to the matrix A = g∈G k (g) q (g −1 ), where is a d k × d q matrix with entries all null except at line l and column m where it is set to lm = 1.The theorem is straightforwardly extended to the finite-dimensional linear representations of compact groups G on the vector spaces over the field C. It suffices in the proof to replace every normalized sum 1 n G g∈G . . .over a finite group G by the corresponding integration G . . .dg using the Haar measure dg of the compact group G.It also is extended to every ground field K whose characteristic char(K) that does not divide the order n G of the group G, except only that 1 n G g∈G k nm (g) q mn (g −1 ) can fail to give 1 d k if K is not algebraically closed.This can be determined from the Galois Theory of the centre of the division algebra End G (V, K).

CHARACTER THEORY
What now one needs are effective methods for reducing a linear representation and constructing the irreducible components of its representation space, to allow discerning the invariances of a physical quantity with respect to a symmetry group.It is obvious from the considerations of the previous sections that, quite quickly, this might become cumbersome.Invariants over the isomorphism classes of the linear representations should be of the greatest help, at the condition that these also allow distinguishing between non isomorphic linear representations.
Whatever the finite dimensional linear representation : G → GL(V, C) of a compact group G the linear operators (g) for every element g in the group G are diagonalizable, since is unitarisable and unitary operators are diagonalisable with pairwise orthogonal eigenspaces (cf.spectral theorem for normal operators).It is recalled that finite groups are compact, for the discrete topology.As a matter of fact, with finite groups it even may be asserted that all the eigenvalues of (g) are roots of unity, since every element g ∈ G necessarily is of finite order, that is to say ∃n g : g n g = e so that (g) ng = 1 V .Numerical invariants may be deduced from the symmetric functions of these eigenvalues, more precisely from the coefficients n (g) of the characteristic polynomial Det[ (g) − , where d is the dimension of the representation space V.Among the most familiar are the coefficient d (g) = Det[ (g)] of the constant term and the coefficient 16 and Tr[ • (g) 17 whatever the invertible linear operator on V. Thus, Det[ (g)] and Tr[ (g)] show the required invariance over every isomorphism class of linear representations.Now, it follows from the multiplicativity of the Determinant that Det[ (g) • (h)] = Det[ (g)]Det[ (h)], which means that the application g ∈ G → Det[ (g)] ∈ C makes up a 1-dimensional representation of G.It thus turns out that the Determinant invariant is often unable to distinguish between different classes of isomorphism when, by contrast, the Trace invariant, which is not multiplicative, can.So this is the searched invariant.It actually will be shown below that the complex-valued function on G defined as is a complete invariant, in the sense that it uniquely determines the linear representation : G → GL(V, C) up to isomorphism.defines the character of the linear representation .

Elementary properties
Let : G → GL(V, C) be a d-dimensional linear representation of a finite (or even continuous compact) group G and let : G → GL(d, C) be the matrix representation associated to with respect to the basis vectors êm (m = 1, . . ., d) selected in the representation space V.It follows from the definition of the trace of a linear operator that g n g = e (unit element of G), otherwise the successive powers of g would generate an infinite group.It follows that (g n g ) = (g) n g = 1 d .It then is directly clear that (g) is diagonisable.Let 1 (g), . . ., d (g) be the g-dependent eigenvalues of (g).Obviously, i (g) n g = 1, which means that i (g) is a root of unity, ∃ i (g) : i (g) = e j i (g) with j Note that by the theorem of Lagrange the order n g of g divides the order n G of the group G.So the eigenvalues 1 (g), . . ., d (g) of (g) are roots of unity of orders dividing the order n G of the group G.More generally, every linear representation of a compact group and à fortiori of a finite group is unitarisable.An inner product thus may be defined in the representation space V so that (g −1 ) = (g) −1 = (g) † ∀g ∈ G.In terms of matrix representations with respect to

Orthogonality theorem
Getting back to the equation 2.34 and setting j = l and n = m then summing over all j and all n and finally using the identity jn ( jn ) 2 = jn ( jn ) = d k , one ends up at where k and q are the characters of the irreducible representations k : G → GL(V k , C) and q : G → GL(V q , C). kq is a generalized Kronecker symbol, defined as kq = 1 if k ∼ q and kq = 0 if k q .The notation | is used to emphasize that the quantity 1 n G g∈G ( (g)) (g) does define an inner product in the vector space C [G] of complex-valued functions on G, being obviously linear with respect to , conjugate symmetric and positive definite ( 3) makes up the First Orthogonality Theorem for the Characters and has far-reaching consequences.
Consider a decomposition = 1 ⊕ . . .⊕ s of a linear representation : G → GL(V, C) with character into the irreducible representations k : G → GL(V k , C) with characters k .It results from the additivity property of the characters that = 1 + . . .+ s and from the linearity of the inner product that q | = q | 1 + . . .+ q | s .According to the First Orthogonality Theorem for the Characters, It follows that q | determines the number of k isomorphic to q contained in the decomposition of .As previously transcribed in the equation (2.29), this number is nothing but the multiplicity n q of q in the expansion of the representation over its irreducible components k : The multiplicity of the trivial representation in this expansion for instance is g∈G (g).Obviously n q = q | does not depend on the chosen decomposition, which means that the decomposition of a finite-dimensional linear representation of a finite group into irreducible representations is unique.This in turn immediately implies that every two completely reducible linear representations with the same character are necessarily isomorphic, for they contain each given irreducible representation the same number of times.Characters thus are in one-to-one correspondence with isomorphic classes of linear representations, which is the essence of the Theorem of Complete Invariance of the Characters.

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Given that every decomposition of a linear representation uniquely writes ∼ k n k k every character uniquely writes = k n k k .Computing the square norm of and taking account of the First Orthogonality Theorem for the Characters one gets k n 2 k is equal to 1 only if one of the n k 's is equal to 1 and the others to 0, that is if is isomorphic to one of the irreducible representation k , whence if is the character of a representation then | | | is the sum of squares of integers and | | | = 1 iff is irreducible.We obtain thus a very convenient irreducibility criterion.

Dimensional closure
Consider the regular representation G of a finite group G (cf. Section 2.1).G by definition transcribes the left action of the group G on the representation space V G spanned by basis vectors êh indexed with the group elements h ∈ G by permuting these as G (g)(ê h ) = êgh ∀g ∈ G ∀h ∈ G.It is clear by the group properties that gh = h ⇔ g = e, where e is the unit element of G.It follows that G (g)(ê h ) = êh ⇔ g = e.This means that the diagonal elements of the matrix representatives G (g) of the linear operators G (g) with respect to the basis {ê h } h∈G are all null for g = e and all equal to 1 for g = e.The character G of of the regular representation G then is given by the formula: where n G is the order of G.One finds that So G is far from being irreducible.If q stands for the character of an irreducible representation q : G → GL(V q , C) with dimension d q of the group G then one also computes Note that the span V G of {ê h } h∈G is isomorphic to the vector space C [G] of complex valued functions on the group G.As to build an isomorphism it suffices to match the basis vector êh in G with the function h : G → C, g → gh .Under this isomorphism the elements g in G act on the left on C [G] by sending the function to the function G (g)( ) such that G (g)( )(h) = (g −1 h).As a matter 00005-p.21EPJ Web of Conferences of fact, this is the way to generalize the concept of regular representations to the compact groups.The representation space V G then is isomorphic to the Hilbert space L 2 (G, C) of the square integrable functions on the group G and G (g) for each g ∈ G operates on this space by sending every It again is shown that the number of times each irreducible linear representation k is contained in the regular representation G is equal to the dimension d k of that irreducible representation, but now no dimensional closure prevails since the group G is not finite.The regular representation G then is infinite-dimensional.

Class functions
Owing to the invariance Tr[ • • −1 ] = Tr [ ] of the Trace of any pair ( , ) of invertible linear operators on any vector space, the character of every linear representation : G → GL(V, C) is conjugation-invariant: It is recalled that two elements g and h of a group G are conjugate iff there exists another element t in the group G such that h = tgt −1 .Conjugacy is an equivalence relation that partitions the group G into conjugacy classes C i .A complex valued function on G is called a class function iff (tgt −1 ) = (g) ∀g ∈ G ∀t ∈ G, that is to say iff it is constant over each conjugacy class C i .It is clear from the equation (3.10) that every character of a linear representation : G → GL(V, C) of a finite group G is a class function.
The set of the class functions on a group G, endowed with addition and scalar multiplication makes up a subspace C [C G ] of the vector space C [G] of the complex valued functions on G. Whatever the linear representation : G → GL(V, C) of a finite group G and whatever the complex valued function ∈ C [G], we always may define a linear operator on V as: is a class function iff commutes with the group G through any linear representation : 19 It follows that if is a class function and is isomorphic to an irreducible representation k : G → GL(V k , C) of the group G with character k then, by Schur 1, ∃ ∈ C :

8). can be determined by computing Tr
. 20 As a partial conclusion, we write Note that the last deduction is obvious if we take for the regular representation G . 20

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where n G is the order of G and d k the dimension of V k .Now, assume that the class function is orthogonal to the character k of every irreducible representation k then, by virtue of the equation (3.13), = g∈G (g) (g) is zero so long as is irreducible and by the decomposition into irreducible representations we conclude that is always zero.Applying this to the regular representation G and computing the image under of the basis vector êe indexed with unit element e of G, we obtain but (ê e ) = 0, since is zero, therefore (g) ∀g ∈ G, whence is the null function on G.In short It is on the other hand clear from the equation (3.3) that the characters k of the irreducible representations of the group G make up an orthonormal system in the space of the class functions

In other words the characters of the irreducible representations of a finite group G form an orthonormal basis for the space of the complex class functions
which is the expression of the Theorem of Character Completeness over the Class Functions.Again this is straightforwardly generalized to the compact groups G by using the Haar integration for summation over G and considering the Hilbert space L 2 (C G , C) of the square integrable class functions on G.With the other ground fields K the application of Schur 1 on the linear operator will involve the division algebra End(V k , K).Given a finite group G the first stage to construct its Character Table is to find its conjugacy classes.A series of properties of conjugate elements exist that ease this search.A few of them are:

As an immediate consequence, the number of irreducible representations of a finite group G up to isomorphism is equal to the number
The unit element e of every group always forms a conjugacy class {e} by its own.In an abelian group every element form a conjugacy class by its own.The orders of the elements of the same conjugacy class C i are all equal, since obviously g n g i i = e and ∃t ∈ G : i is conjugate to g −1 i so that all the inverses of the elements of a given conjugacy class C i belong to a same conjugacy class C −1 i .If g i and g −1 i are conjugate then we a have a single conjugacy class, C i = C −1 i , which is said ambivalent, otherwise we have two distinct conjugacy classes C i = C −1 i , which are said inverse of each other.If n C i stands for the number of elements in each conjugacy class C i then, inherently to the partition of the group G into conjugacy classes, we have the class equation i n C i = n G where n G is the order of the group G.The elements of the conjugacy class C i of any given element g i of the group G are in bijective correspondence with the cosets of the normalizer N G (g defines the index in G of N G (g i ).Conjugating g i with any element s j t of the coset s j N G (g i ) we get (s j t)g i (s j t) −1 = s j tg i (t −1 s −1 j ) = s j g i s −1 j .On the other hand, if (s j t)g i (s j t) −1 = (s k r)g i (s k r) −1 then (s k r) −1 s j t g i (s j t) −1 s k r = g i so (s k r) −1 s j t = h ∈ N G (g i ) or else s j = s k (rht −1 ) which means It then is inferred that the conjugation of g i by the elements of distinct cosets leads to distinct conjugates.Thus each conjugate of g i by an element of the coset s j N G (g i ) can be uniquely labelled by this coset as g j i .It follows that n C i is the index [G : N G (g i )] in G of the normalizer of the representative g i of the conjugacy class C i , but by the Lagrange Theorem [G : N G (g i )] = n G /n N G (g i ) .Therefore n C i is a divisor of n G .It is recalled more generally that the normalizer N G (S) of a subset S of elements of a group G is defined as N G (S) = {t ∈ G | tSt −1 = S}.A related concept is the centralizer C G (S) of the subset S, which is defined as C G (S) = {t ∈ G | tS = St}.It goes without saying that, obviously, the normalizer N G (g i ) of a single element g i of the group G is identical to the centralizer C G (g i ) = {t ∈ G | tg i = g i t} of that element g i in the group G.The intersection Z(G) = ∩ g∈G C G (g) defines the Center of G. Z(G) is an abelian subgroup of G and contains all the elements of the group G that form a class by their own. . . .The second stage to construct the Character Table of a finite group G is to get the list of the character k of its irreducible linear representations k .In the case of small enough groups the already established theorems may be enough to find them all.We recall the elementary property k (e) = d k , the equations where n C i is the number of elements in the conjugacy class C i and n G the order of the group G.This makes up a "Row-by-Row Orthogonality Theorem" for the Character Table .The second orthonormality EPJ Web of Conferences sums of subgroups, involving the concept of induced representation, making use of conjugacy class multiplication, exploiting arithmetic properties of the characters, . . . .A few of these theorems and methods will be approached in the following but only sketchily.

Projectors and exchangers
As to fully discern the effects of a symmetry group in the concrete instances it actually is inevitable to have to explicitly determine the invariant subspaces of the linear representations.One then is sent back to the discomforts of the arbitrariness associated with the intertwinings of the representations and of the consequent lack in general of a natural decomposition of a completely reducible linear representation : G → GL(V, C) of a group G into the irreducible representations k : G → GL(V k , C).This clearly prompts us to formulate a standard method, although not unique, of reduction.
An exception is the coarse-grained canonical decomposition = k k of the linear representation into isotypical components k : G → GL(X k , C), these being isomorphic to the direct sum of n k copies of the irreducible representations As proved from Schur 2 the canonical decomposition is unique, which implies that the isotypical components k can be unambiguously determined.k for each k is nothing but the restriction of to the representation space X k and only a little intuition is necessary to find out that each subspace X k of the representation space V is fully identified by the linear operator on V given by the formula It indeed is inferred from the equation (3.13) that the restriction of P k on every subspace V k s of V that is isomorphic to the representation space V k of the irreducible representation k is the identity operator 1 V k s ∼ =Vk and the zero operator on any other subspace of V. A linear operator the restriction of which on a family of spaces is the identity (resp.zero) operator is the identity (resp.zero) operator on the direct sum space of the family, symbolically s 1 V k s = 1 s V k s (resp.s 0 V k s = 0 s V k s ).It follows that P k is the identity operator on the representation space X k = s V k s of the isotypical component k and the zero operator everywhere else in the representation space V, that is to say P k P k P k is the projector of V = q X q V = q X q V = q X q onto X k k k .
Consequently, to formulate a method for a standard reduction of any linear representation of a group G, it suffices to do so for each of its isotypical components k .Choose, in that purpose, a basis {ê n } n=1,...,d k in the representation space V k of each irreducible representation k of G and denote k : G → GL(d k , C) the matrix representation associated with k with respect to the selected basis in each V k .We are free to define for each k the linear operators on the representation space V of .As from the orthogonality theorem for the matrix representations, to be precise from the equation (2.34), it immediately is inferred that ∀(n, m) Q k mn is null on every subspace V s q =k and therefore on every subspace X q =k = s V s q =k of V.One similarly establishes, focussing solely at X k , that if {ê s n } n=1,...,d k in V k s stands for an isomorphic replica of {ê n } n=1,...,d k in V k then One also deduces that Q k mn defines an isomorphism of X m k to X n k and is null elsewhere in the space V, that is to say . . the direct sum of which gives back X k .It then is clear that the restrictions of to these G-invariant subspaces can be taken as the k -copy components of the searched standard decomposition of the isotypical component k .One may proceed systematically in the concrete cases, by selecting an arbitrary basis in the representation space V of and projects each vector of this basis onto the spaces X m k by using the projectors Q k mm then applies the exchangers Q k mn to get the bases of all the standard G-invariant subspaces.Generalization to the fields K whose characteristic char(K) does not divide the order n G of the group G is straightforward as well as to the compact groups G.In the latter case the projectors and exchangers are built by replacing the summation 1 n G g by the Haar integration:

MISCELLANEA
A few additional topics are more succinctly discussed in this section, in order to only catch a glimpse of the wealth of the topic.Constructions of new linear representations of groups from existing representations through tensor products of the representation spaces or through groups products are described.The concept of induced representation is approached with a qualitative discussion of a few essential theorems.A method of systematic search of the irreducible representations of finite groups is mentioned.The section ends with a very short description of group representations on more general mathematical objects than vector spaces.

Tensor product
A vector space V over a field K is the tensor product V 1 ⊗ V 2 of two vector spaces V 1 and V 2 over the field K iff it is endowed with an application ( The tensor product of vector spaces is associative and distributive with respect to the direct sum, to be precise U ⊗ Natural is to mean that no choice of basis is requested to produce the property.Let i (i = 1, 2) be a linear operator on the vector space V i (i = 1, 2).The tensor product 1 ⊗ 2 of the linear operators 1 and 2 is the linear operator on the tensor product vector space

whose entries are given in terms of the entries of the matrices A
which is checked by observing that the application of (A 1 ⊗ A 2 ) to the basis vector ê1 j ⊗ ê2 l contains the basis vector ê1 i ⊗ ê2 k with the awaited coefficient A 1 ij A 2 kl .An interesting property is Now let 1 : G → GL(V 1 , C) and 2 : G → GL(V 2 , C) be two linear representations of the group G.The tensor product = 1 ⊗ 2 of the linear representations 1 and 2 is the linear representation : G → GL(V, C) that associates to each g in G the linear operator (g) on the tensor product vector space is uniquely defined up to isomorphism.The matrix representative (g) of the linear operator (g) for each g in G with respect to the basis {ê 1 n 1 ⊗ ê2 n 2 } n 1 =1,...,d 1 ,n 2 =1,...,d 2 is the tensor product 1 (g) ⊗ 2 (g) of the matrix representatives i (g) (i = 1, 2) of the linear operators i (g) (i = 1, 2) with respect to the bases One says that the matrix representation is the tensor product of the matrix representations 1 and 2 , symbolically = 1 ⊗ 2 .Generalization to multiple tensor product is obvious.Consider then a linear representation : G → GL(V, C) of the group G.The ν-th ν-th ν-th tensor power of the vector space V is the vector space V ⊗ν = V ⊗ . . .⊗ V (ν times) and the ν-th tensor power of the linear representation is the linear representation ⊗ν : G → GL(V ⊗ν , C) that associates to each g in G the linear operator ⊗ν (g) = (g) ⊗ . . .⊗ (g) (ν times) on V ⊗ν .If {ê n } n=1,...,d is a basis of V then a basis in V ⊗ν is obtained from the collection of vectors ên1 ⊗ . . .⊗ ênν where the indices n1, . . ., nν range over {1, . . ., d} ν : the dimension of ⊗ν is d ν .Applying ⊗ν (g) before or after any permutation : ên1 ⊗ . . .⊗ ênν → ê (n1) ⊗ . . .⊗ ê (nν) of factors leads to the same result.This means that the action of the group S ν of permutations commutes with ⊗ν .S ν thus must preserves the canonical decomposition of ⊗ν .So every S ν -isotypical component of ⊗ν makes up a sub-representation of G.Among these it is customary to discern the ν-th symmetric power Sym ν : G → GL(Sym ν V, C) associated with the trivial representation of S ν and the ν-th alternate power Alt ν : G → GL(Alt ν V, C) associated with the sign representation of S ν , which is defined by declaring that every transposition produces a multiplication by −1.Define the linear operators where N ( ) is the number of transposition under which decomposes.One easily shows that + is a projector of V ⊗ν onto Sym ν V and − a projector of V ⊗ν onto Alt ν V.The vectors + (ê n1 ⊗ . . .⊗ ênν ) (1 ≤ n 1 ≤ . . .nν ≤ d) make up a basis of Sym ν V and the vectors The characters of the tensor products of linear representations are elementarily determined: . Denoting 1 , . . ., d the eigenvalues of (g), one indeed computes 2 H by g s / ∈ H.The concept of induced representations provide powerful tools to demonstrate a variety of important theorems.We only mention among them the Artins'Theorem, which allows stating that each character of a group G is a linear combination with rational coefficients of characters of representations induced from cyclic subgroups of G. Induction is also extremely efficient in the determination of the irreducible representations from representations of its subgroups.Note finally that the notion of induced representations extends with the same definition to the compact groups G so long as H is a closed subgroup of finite index.With infinite index the notion may be defined through the Hilbert space of square integrable functions on the group.

Searching irreducibles
An essential problem of representation analysis is whether algorithmic procedures might be forged that would allow finding out the invariant subspaces of any linear representation and the invariant complements.A general method to determine the Character Table of any finite group can be given.In that purpose let us consider back the conjugacy classes of a group.
We may define the "product" of two conjugacy classes C i and C j formally as the set C i C j = {g i g j | g i ∈ C i , g j ∈ C j }.If g ∈ C i C j then any conjugate to g is also the product of an element of C i by an element of C j , merely because hg i g j h −1 = hg i h −1 hg j h −1 .In other words, if an element of the conjugacy class C l appears a given number C(C i C j C l ) of times in the set C i C j then every other element of the same conjugacy class C l will appear the same number C(C i C j C l ) of times in the set C i C j .This means that the conjugacy class product C i C j expands onto conjugacy classes C l as where the class multiplication coefficients are strictly positive integers: C(C i C j C l ) ∈ N − {0}.C i C j = C j C i , since g i g j = g j (g −1 j g i g j ), so that C(C i C j C l ) = C(C j C i C l ).The expansion in the equation 4.16 contains the conjugacy class C l = {e}, where e is the unit of the group G, iff the two conjugacy classes C i and C j are inverse of each, merely because g i g j = e ⇔ g i = g −1 j , and whenever this is so the conjugacy class e will appear n C i times in the conjugacy class product of C i with itself if it is ambivalent and with its inverse if this is distinct from it.In other words, Summing the linear operators k (g) over a class C i the linear operator k i = g i ∈C i k (g i ) is defined on the representation space V. k i belongs to End G (V ) 21 so, by Schur 1, ∃ i ∈ C : k i = i 1 V k (cf.Section 2.8), which in terms of characters is transcribed into n C i k i = i k (e).As from the 21 (h)

00005-p.33
EPJ Web of Conferences equation 4.16 it is inferred that k i • k j = g i ∈C i k (g i ) • g j ∈C j k (g j ) = g i ∈C i g j ∈C j k (g i g If N C is the number of the conjugacy classes of the group G then this makes up a system of N 2 C equations over the N C variables k i (i = 1, N C ).This is the starting point of a variety of algorithms to determine the Character Tables of the finite groups.Consult [6] for further details.The computations of irreducible representations are harder, as emphasized in [7].
Arithmetic properties of the characters are also extremely useful.Note that since every element of a finite group has finite order, the character values always are sums of eigenvalues that are roots of the multiplicative unit, that is to say roots of a polynomial with coefficients in the set of integers Z.This defines algebraic integers.It then follows, for instance, from the equation (3.13) that the dimensions d k of the irreducible representations k : G → GL(V k , C) are all divisors of the order n G of the group G, since the set of algebraic integers is closed under addition and multiplication and since algebraic integers given as rationals are in fact integers.

Group actions
Let : G → Aut(X) be a representation of a group G on a mathematical object X.One always may define a function : G × X → X that canonically maps each couple (g, x) ∈ G × X into (g, x) = (g)(x) ∈ X.It is straightforward to show that preserves the law of G, namely (gh, x) = (g, (h, x)) ∀g, h ∈ G ∀x ∈ X, since is an homomorphism, and that the unit e of G is neutral for , namely (e, x) = x ∀x ∈ X, because (e) necessarily is the identity of Aut(X).In other words, is nothing but an action of the group G on the mathematical object X.Conversely, given an action : G × X → X one always may define a function : G → Aut(X) that canonically maps each g ∈ G into the isomorphism (g) : x → (g, x) of X.It is not more difficult to demonstrate that the properties of an action imply that is a group homorphism.Accordingly, it is equivalent to define a representation of a group G on a mathematical object X or an action of this group G on that object X.It then is tempting to state that a representation is identical to an action, but that would make up a mathematical abuse.
Using either of the two concepts of action or of representation, symmetry can be defined in a very wide context.A subset Y of X is said invariant under a subgroup S of G if { (g, x) | (g, x) ∈ S × Y} ⊆ Y.The elements of S then are called the symmetries of Y.
A group action : G × X → X is said isomorphic to a group action : G × Y → Y, symbolically ∼ ∼ ∼ , if they are intertwined with an isomorphism, namely if there exists an isomorphism : X → Y which is equivariant: ( (g, x)) = (g, (x)) ∀(g, x) ∈ G × X.Of course, if : G → Aut(X) and : G → Aut(Y) are the representations canonically associated with and then The set Orb (x) = { (g, x) | g ∈ G} by definition is the orbit of x ∈ X. Writing xR y for y ∈ Orb (x) one gets an equivalence relation, which partition the set X into orbits.The quotient set defines the orbit space X | G.If : G × X → X is an action of a finite group G on a manifold then X | G is an orbifold with the singularities on the fixed points of in X.Interest in the orbifolds strongly raised in the context of the geometrization conjecture, formulated by Thurston then proved by Perelman, as essential pieces of manifold decompositions.An action is transitive if Orb (x) = X.
The set Stab (x) = {g ∈ G | (g, x) = x} by definition is the stabilizer of x ∈ X.It forms a subgroup of G, whatever x in X.It is also called a little group.One easily establishes that 00005-p.34

Contribution of Symmetries in Condensed Matter
Stab ( (g, x)) = g Stab (x)g −1 . 22It follows that the collection {Stab ( (g, x)) | g ∈ G} of the stabilizers of the elements of an orbit Orb (x) forms a conjugacy class of subgroups of G.If Stab (x) = G then Orb (x) = x and x is termed a fixed point.If Stab (x) = {e} then Orb (x) is termed a principal orbit.An action is effective if all its orbits are principal: Stab (x) = {e} ∀x ∈ X, which means that every element of G other than the unit e of G acts by changing every element of X.
The function x : G/Stab (x) → Orb (x), from the set of the left cosets of the stabilizer Stab (x) in G to the orbit Orb (x) is well defined and bijective.It then is inferred that: i-if G is finite then the number of elements of any orbit with the same conjugacy class of stabilizers as Orb (x) is n Orb (x) = n G /n Stab (x) , denoting n E the number of elements in a set E. ii-if is an infinitely differentiable action of a Lie group then any orbit with the same conjugacy class of stabilizers as the orbit Orb A stratum by definition is the union of the orbits with the same conjugacy class of stabilizers.An example is the set of the fixed points of the action.Another is the union of the principal orbits, which consists in the points that are changed under any element of G other than the unit e of G.If is an infinitely differentiable action of a compact group G on a real manifold X then every real valued function invariant with respect to G possesses extrema on each stratum corresponding to maximal little groups, namely proper little group not contained in any other proper little group, and all real valued function invariant with respect to G have in common orbits of extrema, which precisely are those critical in their stratum (consult [8]).

CONCLUSION
It is hoped that this little trip to the mathematical lands of linear representations of groups was not boring in spite of the many digression made with respect to the initial scope of the lecture and that, instead, was rather pleasant and enjoyable by providing an abstract glimpse of the basics on which the theory is founded.The reported literature provides more details.Clearly, it by no way is exhaustive and emanates only from the author's own arbitrary taste.
Any homomorphism : G → C : G → C : G → C thus makes up a linear representation of dimension d = 1 d = 1 d = 1 of the group G.An evocative example is : GL(d, C) → C , M → Det(M), where GL(d, C) designates the group of d × d non singular matrices with entries in C and Det(M) the determinant of a matrix M. ker( ) = SL(d, C) consists in the d × d matrices with determinant 1, which thus is a normal subgroup of GL(d, C).Since im( ) = C we have GL(d, C)/SL(d, C) ∼ = C .GL(d, C) is called the general linear group of order d over C and SL(d, C) the special linear group of order d over C. It

3
.26) can always be defined.It is straightforwardly shown that i-• | • G is linear in the second argument because • | • is linear in the second argument and (g) is a linear operator on V for every g ∈ G, ii-• | • G inherits from • | • the conjugate symmetry property, and iii-• | • G is positive definite Contribution of Symmetries in Condensed Matter because the sum of strictly positive numbers is strictly positive.It further is found out that

EPJ
Web of Conferences with = Tr [ ] /d k = (1/d k ) lm lm T lm , equating the coefficients of the T lm , gives 1 n G g∈G k jl (g) q mn (g −1 ) = 1 d k if l = m and j = n

2 )
It is the usage to also call the character of the matrix representation .The trace of a product of matrices being invariant by cyclic permutation, we have ∀g ∈ G Tr[S (g) S −1 ] = Tr[ (g)], whatever the invertible matrix S. Of course, this is nothing but the transposition to the matrix representations of the group G of the invariance of the character over an isomorphism class.concretely is independent of any choice of basis vectors in the representation space V. • (e) = d (e) = d (e) = d, where e is the unit element of G. (e) = Tr [ (e)] = Tr [I d ] = d i=1 1 = d, where I d is the d × d unit matrix.• (g −1 ) = (g) (g −1 ) = (g) (g −1 ) = (g) and | (g) | ≤ d ∀g | (g) | ≤ d ∀g | (g) | ≤ d ∀g in every finite group G G G. ∀g ∈ G ∃n g ∈ N : 16  A Determinant most generally designates every alternating d-linear form F: End(M, A) → A on the module End(M, A) of the endomorphisms on a free module M of dimension d over a commutative ring A. F is unique up to the image F(1 M ) of the identity endomorphism 1 M .One standardly put F( )/F(1 M ) = Det[ ].It results from the functorial properties of the exterior algebra on the module M that Det is multiplicative:Det[ • ] = Det[ ]Det[ ] ∀( , ) ∈ End(M, A) 2 .As an obvious consequence, the image by Det of any composition of endomorphisms i is invariant by any permutation of these:Det[ i i ] = Det[ i (i) ].17A Trace most generally designates every linear form F: End(M, A) → A on the module End(M, A), of the endomorphisms on a free module M of dimension d over a commutative ring A, enjoying the property F( • ) = F( • ) ∀( , ) ∈ End(M, A) 2 .F is unique up to the image F(1 M ) of the identity endomorphism 1 M .One standardly put F( )/F(1 M ) = Tr[ ]/d.Obviously, by substituting • for and so on, the property F( • ) = F( • ) implies that the Trace of any composition of endomorphisms is invariant under cyclic permutation, whence Tr[ • • −1 ] = Tr[ ] for invertible linear operators on a vector space.Note that Det[e ] = e Tr[ ] .
where i stands for the character of i .Evident from the property Tr [A ⊕ B] = Tr [A] + Tr [B] for any pair of matrices A and B.

7 )
that is to say the number of times each irreducible linear representationk k k is contained in the regular representation G G G is equal to the dimension d k d k d k of that irreducible representation.The equation (3.7) implies that G (g) = k d k k (g) for all g in G.Taking g = e leads to the dimensional closure identity e) = n G and k (e) = d k .This identity is useful in the determination of the irreducible representations of a group G, to check in particular that all of these have been found out.If g = e then, since G (g = e) = 0, n C of conjugacy classes of G. Indeed, if C 1 , . . ., C n C are the distinct conjugacy classes of G then every class function ∈ C [C G ] is fully determined by its values C i ∈ C on each conjugacy class C i .It therefore has n C degrees of freedom.This merely means that the dimension of C [C G ] is n C , but, by the Character Completeness over the Class Function, this is equal to the number of irreducible representations of G.This is still true of compact groups, but without any interest since there then are infinitely many classes and infinitely many irreducible representations in the group G. Completeness means that every class function ∈ C [C G ] on a group G is the linear combination = k k | k of the characters k of the irreducible representations k of the group G.With the class function g that takes the value 1 for every element of the class C g = {h ∈ G | ∃ t ∈ G, h = tgt −1 } and 0 elsewhere, we compute k | g = n Cg n G ( k (g)) where n C g is the number of elements in the class C g and n G the order of the group G.It follows, by definition of g , that

k d 2 k
= n G and k d k k (g = e) = 0 inferred from the regular representation G = k d k k , the equality d C[C G ] = n C between the total number of the k and that of the conjugacy classes C i and, of course, the orthonormality of the k .Denoting k i the value of the character k of an irreducible representation k : G → GL(V k , C) over a conjugacy class C i , the first orthonormality equation (3.3) re-writes: i one gets the symmetric square Sym 2 and the alternate square Alt 2 .Note that ⊗ = Sym 2 ⊕ Alt 2 .The dimension of Sym 2 is d Sym 2 = d(d + 1)/2 and the dimension of Alt 2 is d Alt 2 = d(d − 1)/2.The matrix representation associated with Sym 2 with respect to the symmetrized basis {ê n1 ⊗ ên2 + ên2 ⊗ ên1 } 1≤n1≤n2≤d defines the symmetric square matrix representation [ ⊗ ] and the matrix representation associated with Alt 2 with respect to the antisymmetrized basis {ê n1 ⊗ ên2 − ên2 ⊗ ên1 } 1≤n1<n2≤d defines the antisymmetric square matrix representation { ⊗ }.Of course ⊗ = [ ⊗ ] ⊕ { ⊗ }.
where i stands for the character of i .Evident from the property Tr [A ⊗ B] = Tr [A] Tr [B] for any pair of matrices A and B. • The character of the symmetric square [ (x) is a manifold of dimension d Orb (x) = d G − d Stab (x) .If d Orb (x) = d G − d Stab (x) =0 then the orbit is finite and its cardinal is the quotient of the number of connected components of G over the number of connected components of S.

Orthogonality Theorem for the Characters. 3.5 Character tables
which, it is recalled, is equal to the number of classes n C in G. Equation 3.16 makes up the Second Character Orthogonality, Complete Invariance and Completeness over the Class Functions offer the great advantage to allow globally handling all the irreducible linear representations of a finite group G up to isomorphism by means of the so-called Character Table.This is a square matrix with rows labelled by the isomorphism classes of irreducible representations, columns labelled by the conjugacy classes of the group and entries given by the values of the character for each isomorphism class of irreducible representation and for each conjugacy class.Every linear representation of the group can be characterized from this table by determining the multiplicities of its irreducible components from the 00005-p.23 EPJ Web of Conferences inner product with the rows of the table and even its decomposition into isotypical components from projection operators on the representation space built over the irreducible characters as discussed in the Section 3.6.
If the operators i (i = 1, 2) are diagonalizable then so is A 1 ⊗ A 2 and if {ê i n i } n i =1,...,d i (i = 1, 2) are the eigenbasis of i ) .A matrix representation s H of the subgroup H s = g s Hg −1 s ∩ H of H is defined for each g s in S by putting s H (h) = H (g −1 s hg s ) for h ∈ H.One then shows that H↑G is irreducible iff H is irreducible and s H and H↓H s are disjoints ∀g s / ∈ H, that is have no common irreducible component.If H is a normal subgroup of G then H s = H and H↑G is irreducible iff H is irreducible and not equivalent to any of its conjugate s