By David M. R. Jackson
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Additional resources for Notes on the representation theory of finite groups [Lecture notes]
Example text
THE ORTHOGONALITY OF THE CHARACTERS = = 1 |G| 1 |G| tr (X (x)) tr X x−1 x∈G n m [X (x)]i,i X x−1 j,j . i=1 j=1 x∈G Let Pi,j be the n × m matrix with (k, l) -element equal to δi,k δl,j . Let Si,j = 1 |G| X (x) Pi,j X x−1 . x∈G Then Si,j i,j = = = 1 |G| 1 |G| 1 |G| [X (x)]i,k Pi,j k,l X x−1 l,j x∈G k,l [X (x)]i,k δi,k δl,j X x−1 l,j x∈G k,l [X (x)]i,i X x−1 j,j x∈G so n m Si,j χ, χ = i,j . 3, 0n×m 1 i,j n tr P Si,j = In if X X , if X ∼ X . But tr Pi,j = 0 unless i = j, in which case tr Pi,j = 1. Thus Si,j i,j 0 0 = 1 n if X X , if i = j, if i = j and X ∼ X .
Am,l xθ11 · · · xθrr ¯ r,l =il l=1 a1,l +···a il la xla1,l · · · xr r,l a1,l , . . , am,l n = xθ11 · · · xθrr xl1 + · · · + xln il = xθ11 · · · xθrr pα = [mθ ] pα , l=1 which completes the proof. To obtain the irreducible characters, we will use the following facts about the ring Λ of symmetric functions in the ground indeterminates x1 , x2 , . . For α ∈ Nn , let α aα = det xi j n×n . 1. FROBENIUS’S CONSTRUCTION Let δ = (n − 1, n − 2, . . , 1, 0) . Then (xj − xi ) , aδ = 1≤ij≤n the Vandermonde determinant.
I=1 j=1 x∈G Let Pi,j be the n × m matrix with (k, l) -element equal to δi,k δl,j . Let Si,j = 1 |G| X (x) Pi,j X x−1 . x∈G Then Si,j i,j = = = 1 |G| 1 |G| 1 |G| [X (x)]i,k Pi,j k,l X x−1 l,j x∈G k,l [X (x)]i,k δi,k δl,j X x−1 l,j x∈G k,l [X (x)]i,i X x−1 j,j x∈G so n m Si,j χ, χ = i,j . 3, 0n×m 1 i,j n tr P Si,j = In if X X , if X ∼ X . But tr Pi,j = 0 unless i = j, in which case tr Pi,j = 1. Thus Si,j i,j 0 0 = 1 n if X X , if i = j, if i = j and X ∼ X . 8 , whence χ, χ = 1 if X ∼ X , 0 if X X , and the result follows.