Hecke algebras at roots of unity.
Now de ne the set JS (n;; d) to be the subset of JS (n) comprising those partitions with n-core and n-weight d (see 13, 7] for de nitions of n-core and n-weight). Then de ne the generating seriesn; (z )=
Xd 0
#JS (n;; d)z d:
Theorem 5.3 (i) If 2 JS (n) then the n-core of is a rectangular partition= (kl )such that k+ l n (it is assumed here that if either k= 0 or l= 0 then (kl ) means the empty partition). (ii) If k 6= 0 6= l then?s k+?l n;(kl ) (z )= z b k?l 0 (z ); where s= min(k; l). (iii) !n;; (z )= n?1 X k=0
b
k+ 0 (z ) k 0
? (n? 1):
Proof: In the tensor product V ( i ) V ( 0 ), all highest weights are of the form k+?l? e with k? l= i mod n (for later convenience we use?l and not l here). We take 0 k; l< n here and can also assume that k?l mod n whereupon k+ l n, and l= 0 only if k= 0. By de nition, the multiplicity of V ( k+?l? e ) is given by the coe cient of z e in b ik;+ 0?l (z ) and hence, by Corollary 2.3, by the number of partitions 2 JS (n) for which wt ( )= k+?l? i? e . We claim that the n-core of such a is the rectangular partition= (kl ), for which we calculate wt ( )= k+?l? k?l? s= k+?l? i? s, where s= min(k; l) is the multiplicity of the colour charge 0 in . This follows from the fact that b for every string of weights;?;? 2;:::; of the sln -module V ( 0 ) (where+ is not a weight of V ( 0 )), those partitions having these weights have the same n-core, and this n-core has weight . Then since= (kl ) w
ith k+ l n is manifestly an n-core, it follows that it is the n-core of, thence proving part (i). Part (ii) follows immediately since in the case k 6= 0 (so that l 6= 0), the partitions k+ enumerated by the branching function b k?l?l are precisely those elements 2 JS (n) 0 having weight wt ( )= k+?l? k?l? e, for some e, and hence n-core (kl ). 0 Finally, for k= 0 and arbitrary l, each partition counted by b?+ 0?l has empty n-core, l and hence contributes to n;;. However, the empty partition occurs for each l, hence an adjustment of n? 1 is needed after summing over all l. No other partition is repeated since, 0 as indicated by Theorem 2.2, the b?+ 0?l to which it contributes is uniquely determined by l?l mod n= ( 1? a1 ) mod n. (The summation over?l is replaced by one over k to give the nal result).