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代数何时具有有限的表示类型。 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 theory 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参数。然后，我们通过计算某些规范基础元素来获得某些分解数。我们将其与Speccht模块理论和Artinian代数理论相结合。结果，我们有一个必要且充分的条件n <min（e，2f+4，2e-2f+4），其中q是统一的原始元根。获得经典类型的Hecke代数的结果是作为该结果的必然的。与这个项目相关，我还发表了一篇论文，其中一篇结果是λ：kleshchev <-> d^λ*0另一个在Combinatorios和Crystal上。此外，我已经完成了上述书的英文翻译。这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 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: "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, 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)