Download e-book for iPad: Algebraic Combinatorics I: Association Schemes by Eiichi Bannai

By Eiichi Bannai

ISBN-10: 0805304908

ISBN-13: 9780805304909

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Example text

A 1 j,1 j,n ! j=1 ways. Thus r |Cα ∩ Sθ | = A≥0;(I,II) j=1 1aj,1 θj ! n a j,1 ! · · · aj,n ! Then, after some routine simplification, n φθ (x) il ! r j=1 aj,l ! = A≥0;(I,II) l=1 n = A≥0;(I) l=1 il a1,l , . . , am,l n = xθ11 · · · xθrr A≥0 l=1 n = il la xla1,l · · · xr r,l a1,l , . . , 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 , .

THE CHARACTERS OF THE SYMMETRIC GROUP n Proof: Let φθ (x) = 1 ↑S Sθ . 7, for x ∈ Cα , φθ (x) = |Sn | |Sθ | hα 1= z∈Cα ∩Sθ z (α) | C α ∩ Sθ | . θ! We now determine |Cα ∩ Sθ | . ,θ1 +···+θj } . Then A = [aj,k ]r×n , where r = l (θ) , satisfies the two conditions I) II) n k=1 kaj,k = θj , for j = 1, . . , r, r j=1 aj,k = ik for k = 1, . . , n. In the sums that arise the attachment of the symbols I and II to the summation conditions indicates which of these conditions is to be applied. The cycles of π can be selected in r θj !

RegG so regG = ⊕ki=1 (deg ρi ) ρi. Then χregG = (x) = i=1 (deg ρi ) χ , so χ k ρi. regG (deg ρ ) χ (x) for all x ∈ G. 8(2), |G| = i=0 (deg ρi ) . The result follows immediately. 38 CHAPTER 5. 1. 1 The centre ZCG Let G = {x1 , . . , xg } be a finite group and let CG = {a1 x1 + · · · + ag xg : a1 , . . , ag ∈ C} . g g Let x, y ∈ CG, so x = i=1 ai xi and y = j=1 bj xj . The product xy is defined g to be xy = i,j=1 ai aj (xi xj ) . Then CG is called the group algebra of G over C. Its dimension is g, and {x1 , .

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Algebraic Combinatorics I: Association Schemes by Eiichi Bannai


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