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Extra resources for Representation Theory of Finite Groups: An Introductory Approach (Universitext)
1. allow χ be a personality of a finite staff G. Then χ(g) is an algebraic integer for all g ∈ G. facts. allow be a illustration with personality χ and permit n be the order of G. Then gn = 1 and so ϕgn = I. Corollary 4. 1. 10 then signifies that ϕg is diagonalizable with eigenvalues λ1, …, λm, that are nth-roots of team spirit. specifically, the eigenvalues of ϕg are algebraic integers. considering and the algebraic integers shape a hoop, we finish that χ(g) is an algebraic integer. Remark 6. 2. 2. discover that the evidence of Corollary 6. 2. 1 indicates that χϕ(g) is a sum of mnth-roots of solidarity. we will use this truth later. Our subsequent target is to teach that the measure of an irreducible illustration divides the order of the crowd. to do that we have to conjure up a few extra algebraic integers. Theorem 6. 2. three. permit ϕ be an irreducible illustration of a finite crew G of measure d. allow g ∈ G and allow h be the scale of the conjugacy type of g. Then hχ ϕ (g)∕d is an algebraic integer. evidence. allow C1, …, Cs be the conjugacy periods of G. Set hi = | Ci | and allow χi be the price of χϕ at the classification Ci. we wish to express that hiχi ∕ d is an algebraic integer for every i. contemplate the operator declare. . evidence (of claim). We first express that ϕgTiϕg − 1 = Ti for all g ∈ G. certainly, in view that Ci is closed less than conjugation and conjugation through g is a permutation. by way of Schur’s lemma, Ti = λI a few . Then on account that I is the identification operator on a d-dimensional vector area, it follows that and so , setting up the declare. we want yet one more declare, which says that the Ti “behave” like algebraic integers in that they fulfill a formulation like in Lemma 6. 1. five. declare. for a few . facts (of claim). regimen calculation indicates the place is the variety of how you can write g = xy with x ∈ Ci and y ∈ Cj. We declare that aijg relies in basic terms at the conjugacy classification of g. feel that this can be certainly the case and allow aijk be the worth of aijg with g ∈ Ck. Then proving the declare. So allow us to cost that aijg relies in basic terms at the conjugacy classification of g. enable so aijg = | Xg | . permit g′ be conjugate to g. We convey that | Xg | = | Xg′ | . think that g′ = kgk − 1 and outline a bijection ψ: Xg→Xg′ via observe that kxk − 1 ∈ Ci, kyk − 1 ∈ Cj and , and so ψ(x, y) ∈ Xg′. obviously, ψ has inverse τ : Xg′→Xg given via τ(x′, y′) = (k − 1x′k, k − 1y′k) so ψ is a bijection and hence | Xg | = | Xg′ | . We now entire the evidence of the theory. Substituting the formulation for the Ti from the 1st declare into the formulation from the second one declare yields and so hiχi ∕ d is an algebraic integer by means of Lemma 6. 1. five. Theorem 6. 2. four (Dimension theorem). enable ϕ be an irreducible illustration of G of measure d. Then d divides |G|. facts. the 1st orthogonality family (Theorem 4. three. nine) supply and so (6. 1) enable C1, …, Cs be the conjugacy sessions of G and allow χi be the worth of χϕ on Ci. allow hi = | Ci | . Then from (6. 1) we receive (6. 2) yet hiχi ∕ d is an algebraic integer by means of Theorem 6. 2. three, while is an algebraic integer by way of Corollary 6. 2. 1 and the closure of algebraic integers below complicated conjugation. because the algebraic integers shape a hoop, it follows from (6.