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um of subrepresentations then the corresponding character is the sum of the characters of those subrepresentations if a character of the finite group g is restricted to a subgroup h then the result is also a character of h every character value χ g is a sum of n m th roots of unity where n is the degree that is the dimension of the associated vector space of the representation with character χ and m is the order of g in particular when f c every such character value is an algebraic integer if f c and χ is irreducible then g c g x χ x χ 1 displaystyle g c_ g x frac chi x chi 1 is an algebraic integer for all x in g if f is algebraically closed and char f does not divide the order of g then the number of irreducible characters of g is equal to the number of conjugacy classes of g furthermore in this case the degrees of the irreducible characters are divisors of the order of g and they even divide g z g if f c arithmetic properties edit let ρ and σ be representations of g then the following identities hold χ ρ σ χ ρ χ σ displaystyle chi _ rho oplus sigma chi _ rho chi _ sigma χ ρ σ χ ρ χ σ displaystyle chi _ rho otimes sigma chi _ rho cdot chi _ sigma χ ρ χ ρ displaystyle chi _ rho overline chi _ rho χ a l t 2 ρ g 1 2 χ ρ g 2 χ ρ g 2 displaystyle chi _ scriptscriptstyle rm alt 2 rho g tfrac 1 2 chi _ rho g 2 chi _ rho g 2 χ s y m 2 ρ g 1 2 χ ρ g 2 χ ρ g 2 displaystyle chi _ scriptscriptstyle rm sym 2 rho g tfrac 1 2 chi _ rho g 2 chi _ rho g 2 where ρ σ is the direct sum ρ σ is the tensor product ρ denotes the conjugate transpose of ρ and alt 2 is the alternating product alt 2 ρ ρ ρ and sym 2 is the symmetric square which is determined by ρ ρ ρ ρ sym 2 ρ displaystyle rho otimes rho rho wedge rho oplus textrm sym 2 rho character tables edit further information character table the irreducible complex characters of a finite group form a character table which encodes much useful information about the group g in a compact form each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of g the columns are labelled by representatives of the conjugacy classes of g it is customary to label the first row by the character of the trivial representation which is the trivial action of g on a 1 dimensional vector space by ρ g 1 displaystyle rho g 1 for all g g displaystyle g in g each entry in the first row is therefore 1 similarly it is customary to label the first column by the identity therefore the first column contains the degree of each irreducible character here is the character table of c 3 u u 3 1 displaystyle c_ 3 langle u mid u 3 1 rangle the cyclic group with three elements and generator u 1 u u 2 1 1 1 1 χ 1 1 ω ω 2 χ 2 1 ω 2 ω where ω is a primitive third root of unity the character table is always square because the number of irreducible representations is equal to the number of conjugacy classes 2 orthogonality relations edit main article schur orthogonality relations the space of complex valued class functions of a finite group g has a natural inner product α β 1 g g g α g β g displaystyle langle alpha beta rangle frac 1 mathopen vert g mathclose vert sum _ g in g alpha g overline beta g where β g is the complex conjugate of β g with respect to this inner product the irreducible characters form an orthonormal basis for the space of class functions and this yields the orthogonality relation for the rows of the character table χ i χ j 0 if i j 1 if i j displaystyle langle chi _ i chi _ j rangle begin cases 0 mbox if i neq j 1 mbox if i j end cases for g h in g applying the same inner product to the columns of the character table yields χ i χ i g χ i h c g g if g h are conjugate 0 otherwise displaystyle sum _ chi _ i chi _ i g overline chi _ i h begin cases mathopen vert c_ g g mathclose vert mbox if g h mbox are conjugate 0 mbox otherwise end cases where the sum is over all of the irreducible characters χ i of g and the symbol c g g denotes the order of the centralizer of g note that since g and h are conjugate iff they are in the same column of the character table this implies that the columns of the character table are orthogonal the orthogonality relations can aid many computations including decomposing an unknown character as a linear combination of irreducible characters constructing the complete character table when only some of the irreducible characters are known finding the orders of the centralizers of representatives of the conjugacy classes of a group finding the order of the group character table properties edit certain properties of the group g can be deduced from its character table the order of g is given by the sum of the squares of the entries of the first column the degrees of the irreducible characters more generally the sum of the squares of the absolute values of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class all normal subgroups of g and thus whether or not g is simple can be recognised from its character table the kernel of a character χ is the set of elements g in g for which χ g χ 1 this is a normal subgroup of g each normal subgroup of g is the intersection of the kernels of some of the irreducible characters of g the commutator subgroup of g is the intersection of the kernels of the linear characters of g if g is finite then since the character table is square and has as many rows as conjugacy classes it follows that g is abelian iff each conjugacy class is a singleton iff the character table of g is g g displaystyle g times g iff each irreducible character is linear it follows using some results of richard brauer from modular representation theory that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table an observation of graham higman the character table does not in general determine the group up to isomorphism for example the quaternion group q and the dihedral group of 8 elements d 4 have the same character table brauer asked whether the character table together with the knowledge of how the powers of elements of its conjugacy classes are distributed determines a finite group up to isomorphism in 1964 this was answered in the negative by e c dade the linear representations of g are themselves a group under the tensor product since the tensor product of 1 dimensional vector spaces is again 1 dimensional that is if ρ 1 g v 1 displaystyle rho _ 1 g to v_ 1 and ρ 2 g v 2 displaystyle rho _ 2 g to v_ 2 are linear representations then ρ 1 ρ 2 g ρ 1 g ρ 2 g displaystyle rho _ 1 otimes rho _ 2 g rho _ 1 g otimes rho _ 2 g defines a new linear representation this gives rise to a group of linear characters called the character group under the operation χ 1 χ 2 g χ 1 g χ 2 g displaystyle chi _ 1 chi _ 2 g chi _ 1 g chi _ 2 g this group is connected to dirichlet characters and fourier analysis induced characters and frobenius reciprocity edit main articles induced character and frobenius reciprocity the characters discussed in this section are assumed to be complex valued let h be a subgroup of the finite group g given a character χ of g let χ h denote its restriction to h let θ be a character of h ferdinand georg frobenius showed how to construct a character of g from θ using what is now known as frobenius reciprocity since the irreducible characters of g form an orthonormal basis for the space of complex valued class functions of g there is a unique class function θ g of g with the property that θ g χ g θ χ h h displaystyle langle theta g chi rangle _ g langle theta chi _ h rangle _ h for each irreducible character χ of g the leftmost inner product is for class functions of g and the rightmost inner product is for class functions of h since the restriction of a character of g to the subgroup h is again a character of h this definition makes it clear that θ g is a non negative integer combination of irreducible characters of g so is indeed a character of g it is known as the character of g induced from θ the defining formula of frobenius reciprocity can be extended to general complex valued class functions given a matrix representation ρ of h frobenius later gave an explicit way to construct a matrix representation of g known as the representation induced from ρ and written analogously as ρ g this led to an alternative description of the induced character θ g this induced character vanishes on all elements of g which are not conjugate to any element of h since the induced character is a class function of g it is only now necessary to describe its values on elements of h if one writes g as a disjoint union of right cosets of h say g h t 1 h t n displaystyle g ht_ 1 cup ldots cup ht_ n then given an element h of h we have θ g h i t i h t i 1 h θ t i h t i 1 displaystyle theta g h sum _ i t_ i ht_ i 1 in h theta t_ i ht_ i 1 because θ is a class function of h this value does not depend on the particular choice of coset representatives this alternative description of the induced character sometimes allows explicit computation from relatively little information about the embedding of h in g and is often useful for calculation of particular character tables when θ is the trivial character of h the induced character obtained is known as the permutation character of g on the cosets of h the general technique of character induction and later refinements found numerous applications in finite group theory and elsewhere in mathematics in the hands of mathematicians such as emil artin richard brauer walter feit and michio suzuki as well as frobenius himself mackey decomposition edit the mackey decomposition was defined and explored by george mackey in the context of lie groups but is a powerful tool in the character theory and representation theory of finite groups its basic form concerns the way a character or module induced from a subgroup h of a finite group g behaves on restriction back to a possibly different subgroup k of g and makes use of the decomposition of g into h k double cosets if g t t h t k textstyle g bigcup _ t in t htk is a disjoint union and θ is a complex class function of h then mackey s formula states that θ g k t t θ t t 1 h t k k displaystyle theta g _ k sum _ t in t bigl theta t _ t 1 ht cap k bigr k where θ t is the class function of t 1 ht defined by θ t t 1 ht θ h for all h in h there is a similar formula for the restriction of an induced module to a subgroup which holds for representations over any ring and has applications in a wide variety of algebraic and topological contexts mackey decomposition in conjunction with frobenius reciprocity yields a well known and useful formula for the inner product of two class functions θ and ψ induced from respective subgroups h and k whose utility lies in the fact that it only depends on how conjugates of h and k intersect each other the formula with its derivation is θ g ψ g θ g k ψ t t θ t t 1 h t k k ψ t t θ t t 1 h t k ψ t 1 h t k displaystyle begin aligned langle theta g psi g rangle langle theta g _ k psi rangle sum _ t in t bigl langle bigl theta t _ t 1 ht cap k bigr k psi bigr rangle sum _ t in t bigl langle theta t _ t 1 ht cap k psi _ t 1 ht cap k bigr rangle end aligned where t is a full set of h k double coset representatives as before this formula is often used when θ and ψ are linear characters in which case all the inner products appearing in the right hand sum are either 1 or 0 depending on whether or not the linear characters θ t and ψ have the same restriction to t 1 ht k if θ and ψ are both trivial characters then the inner product simplifies to t twisted dimension edit one may interpret the character of a representation as the twisted dimension of a vector space 3 treating the character as a function of the elements of the group χ g its value at the identity is the dimension of the space since χ 1 tr ρ 1 tr i v dim v accordingly one can view the other values of the character as twisted dimensions clarification needed one can find analogs or generalizations of statements about dimensions to statements about characters or representations a sophisticated example of this occurs in the theory of monstrous moonshine the j invariant is the graded dimension of an infinite dimensional graded representation of the monster group and replacing the dimension with the character gives the mckay thompson series for each element of the monster group 3 characters of lie groups and lie algebras edit see also weyl character formula and algebraic character if g displaystyle g is a lie group and ρ displaystyle rho a finite dimensional representation of g displaystyle g the character χ ρ displaystyle chi _ rho of ρ displaystyle rho is defined precisely as for any group as χ ρ g tr ρ g displaystyle chi _ rho g operatorname tr rho g meanwhile if g displaystyle mathfrak g is a lie algebra and ρ displaystyle rho a finite dimensional representation of g displaystyle mathfrak g we can define the character χ ρ displaystyle chi _ rho by χ ρ x tr e ρ x displaystyle chi _ rho x operatorname tr e rho x the character will satisfy χ ρ ad g x χ ρ x displaystyle chi _ rho operatorname ad _ g x chi _ rho x for all g displaystyle g in the associated lie group g displaystyle g and all x g displaystyle x in mathfrak g if we have a lie group representation and an associated lie algebra representation the character χ ρ displaystyle chi _ rho of the lie algebra representation is related to the character x ρ displaystyle mathrm x _ rho of the group representation by the formula χ ρ x x ρ e x displaystyle chi _ rho x mathrm x _ rho e x suppose now that g displaystyle mathfrak g is a complex semisimple lie algebra with cartan subalgebra h displaystyle mathfrak h the value of the character χ ρ displaystyle chi _ rho of an irreducible representation ρ displaystyle rho of g displaystyle mathfrak g is determined by its values on h displaystyle mathfrak h the restriction of the character to h displaystyle mathfrak h can easily be computed in terms of the weight spaces as follows χ ρ h λ m λ e λ h h h displaystyle chi _ rho h sum _ lambda m_ lambda e lambda h quad h in mathfrak h where the sum is over all weights λ displaystyle lambda of ρ displaystyle rho and where m λ displaystyle m_ lambda is the multiplicity of λ displaystyle lambda 4 the restriction to h displaystyle mathfrak h of the character can be computed more explicitly by the weyl character formula see also edit irreducible representation applications in theoretical physics and chemistry association schemes a combinatorial generalization of group character theory clifford theory introduced by a h clifford in 1937 yields information about the restriction of a complex irreducible character of a finite group g to a normal subgroup n frobenius formula real element a group element g such that χ g is a real number for all characters χ references edit nicolas bourbaki algèbre spr...
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