Research on the representations of cyclotomic Hecke algebras and finite algebraic groups of classical type

分圆Hecke代数和经典型有限代数群的表示研究

• 批准号: 12640016
• 负责人: ARIKI Susumu
• 金额: $ 0.9万
• 依托单位: Tokyo University of Mercantile Marine
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640016/
• 关键词: finite representation type; Hecke algebra; decomposition numbers; Fock space; crystal basis; canonical basis; Uno's conjecture; 巡回ヘッケ環; モジュラー表現

We have determined when a Hecke algebra has finite representation type. In classical types, the case where the Hecke algebra has type B is essential, and the result in this case is based on the theory which I explained in my book published in 2000, in the project term (Exceptional types were settled by Hyoue Miyachi)More precisely, using Morita equivalence theorem of Dipper and Mathas the proof is reduced to the case where a two-parameter Hecke algebra of type B has 1 and q^f as the parameters. Then we obtain certain decomposition numbers by the computation of some canonical basis elements. We combine this with Specht module theory and the theory of Artinian algebras. As a result we have a necessary and sufficient condition n<min (e, 2f+4, 2e-2f+4) where q is a primitive eth root of unity. Results for Hecke algebros of classical type are obtained as a Corollary of this result. Related to this project, I also published papers one on the result that λ : Kleshchev<->D^λ*0 the other on combinatorios and crystal.Moreover, I've completed the English translation of the book mentioned above. This will be published by the American Mathematical Society.

我们已经确定了Hecke代数何时具有有限表示类型。在经典类型中，Hecke代数具有B型的情况是本质的，而在这种情况下的结果是基于我在2000年出版的书中解释的理论，在项目项(例外类型是由Hyoue Miyachi解决的)中，利用Dipper和Mathas的Morita等价定理，将证明简化为B型的两参数Hecke代数具有1和q^f作为参数的情况。然后通过对一些标准基元的计算，得到了一定的分解数。我们将其与Speht模理论和Artin代数理论相结合。作为结果，我们得到了一个充要条件n&lt；min(e，2f+4，2e-2f+4)，其中q是本原单位根。作为这一结果的推论，得到了经典Hecke代数的结果。与此项目相关，我还发表了一篇关于λ：Kleshchev&lt；-&gt；D^λ*0的论文，另一篇关于组合体和晶体的论文。此外，我还完成了上面提到的这本书的英译。这份报告将由美国数学学会出版。

项目成果

S. Ariki: "Robinson-Schensted correspondence and left cells"Adv. Stud. Pure Math.. 28. 1-20 (2000)

S. Ariki：“Robinson-Schensted 通信和左细胞”Adv。

S.Ariki: "Some remarkes on A_1^<(1)> soliton cellular automata"J. Math. Sci. Univ. Tokyo. 8. 143-156 (2001)

S.Ariki：“关于 A_1^<(1)> 孤子元胞自动机的一些评论”J。

S.Ariki: "Some reworks on A^<(1)>_1 solution cellular automata"J. Math. Sci. Univ. Tokyo. 8. 143-156 (2001)

S.Ariki：“对 A^<(1)>_1 解元胞自动机的一些修改”J.

S.Ariki, A.Mathas: "The number of simple modules of the Hecke algebias of type G(r.l.n)"Math. Zeit. 233. 601-623 (2000)

S.Ariki，A.Mathas：“G(r.l.n) 型 Hecke 代数的简单模数”数学。

S.Ariki: "On the classification of simple modules for cyclotomic Hecke algebars of type G(m, l, n)"Osaka J. Math.. 38. 827-837 (2001)

S.Ariki：“关于 G(m, l, n) 型分圆 Hecke 代数的简单模的分类”Osaka J. Math.. 38. 827-837 (2001)