On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$
On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$
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DOI：
10.1215/kjm/1250518452
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发表时间：
1996
影响因子：
--
通讯作者：
S. Ariki
中科院分区：
文献类型：
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作者：
S. Ariki
已知该代数是无 A 的。如果我们将其专门化为 vi = v,, q= q，其中 vi e C, qe Cx，则该代数用 i(c e 表示。我们在此注意到，整数环上的该代数的研究推测与一般线性群 [BM] 的分块代数的模表示论有关。X e 的模表示论的构建块之一是 v1,•••, vm 的情况是 q2 1 的幂，并且在本文中我们将 u n 设为 Xe 模范畴的 Grothendieck 群。本文的目的是证明，如果 q 不是单位根，则 u 的分级对偶是 g (A .) （分别为 g (A 1 ) ）的最高权模。 （分别是原 r 次单位根），并且不可约模的对偶基与规范基一致，该证明在很大程度上取决于 Lusztig 的仿射 Hecke 代数和量子群理论以及 G inzburg 的仿射 Hecke 代数理论。对于 m =1，我们的结果验证了 [LLT] 的猜想。该算法实际上计算 A 型赫克代数的分解数。这里我们注意到，Grojnowski [G r] 对 A 型赫克代数的分解数有一个公告，但我们在这里看到的是，我们可以避免单位根处的结果来计算解 -
This algebra is known to be A -free. If we specialize it to v i = v,, q= q, where vi e C, qe Cx, this algebra is denoted by i(c. W e note here that the study of this algebra over a ring of integers is conjecturely related to th e m odular representation theory fo r the block algebras of the general linear group [BM]. One of the building blocks for the modular representation theory of X e is the case that v1,•••, vm a r e powers of q2 1 , a n d w e consider this case in this paper. Let u n b e the Grothendieck group of the category of Xe-modules. W e set u--= un . T he purpose of th is paper is to show tha t the graded dual o f u is a highest weight module of g (A .) (resp. g (A 1 ) ) if q is no t roo t of unity (resp. a prim itive r-th root of un ity ), and the dual basis of irreducible modules coincides with canonical basis. T he proof heavily depends on Lusztig 's theory of affine Hecke algebras a n d quantum groups, and G inzburg 's theory of affine Hecke algebras. For m =1, our result verifies a conjecture of [LLT]. Hence their conjectura l a lgorithm ac tua lly com putes t h e decom position num bers of the H ecke algebra of type A. W e note here that there is an announcement of Grojnowski [G r] on the decomposition numbers of the Hecke algebra of type A , bu t what we see here is that we can avoid the result at roots of unity to compute the de-