Representation theory of algebraic groups, Hecke algebras and complex reflection groups

代数群的表示论、赫克代数和复反射群

• 批准号: 13440014
• 负责人: SHOJI Toshiaki
• 金额: $ 5.31万
• 依托单位: Nagoya University (2003)Tokyo University of Science (2001-2002)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440014/
• 关键词: Luszfig's conjecture; special linear group; irreducible character; almost character; character sheaf; Shintani descent; symmetric space; 代数群; Green関数; Hall-Littlewood関数; Macdonald関数; 複素鏡映群; 一般Green関数; graded Hecke algebra; ヘッケ環; 表現論; Ariki

(1) I have proved the Luszfig's conjecture concerning the characters of finite special linear group SLn(〓_q) over finite field 〓_q. This conjecture provides as a general algorithm of computing irreducible characters of reductive group G(〓_q) over finite field, which is stated as 「almost characters and characteristic functions of character sheaves coincides with up to scalar multiple」. This conjecture we proved by the head investigator in the case where the center of G is connected. SLn is an example of disconnected enter. We can determine the scalars also, and we obtain a complete algorithm of computing irreducible characters.(2) By the joint work with K.Sorlin, I have determined the deconpositon of the presentation G(〓_q^2) -with G(〓_q^2)/G(〓_q) for G=SLn.

(1)证明了有限特殊线性群SLn（〓_q）在有限域〓_q上的性质的Luszfig猜想。该猜想提供了在有限域上计算可约群G（〓_q）不可约字符的一般算法，即“字符束的几乎字符和特征函数最多重合于标量倍”。在G中心连通的情况下，这个猜想被首席调查员证明了。SLn是一个断开连接输入的例子。并给出了计算不可约字符的完整算法。(2)通过与K.Sorlin的共同工作，我确定了G（〓_q^2）表示与G（〓_q^2）/G（〓_q）在G=SLn下的分解。

项目成果

Susumu Ariki: "Uno's conjecture on representation types of Hecke algbra"Algebraic Combinatorics and Quantum groups. 1-9 (2003)

Susumu Ariki：“Uno 关于 Hecke 代数表示类型的猜想”代数组合和量子群。

Kiyotaka Ohkura: "On certain bases for Ariki-Koike algebras arising from canonical bases for U_v(sl_m)"SUT J. of Mathematics. 38. 145-173 (2002)

Kiyotaka Ohkura：“关于由 U_v(sl_m) 规范基产生的 Ariki-Koike 代数的某些基”SUT J. of Mathematics。

Susumu Ariki: "Representations of Quantum algebras and combinatorics of Young tableaux"AMS, University lecture series. 155 (2002)

Susumu Ariki：“Young tableaux 的量子代数和组合学的表示”AMS，大学讲座系列。

T.Shoji: "Green functions attached to limit symbols"Adv.Studies in Pure Math.. (発表予定).

T.Shoji：“附加到极限符号的绿色函数”Adv.Studies in Pure Math..（待提交）。

T.Shoji: "On Green functions associated to Complex reflection groups"Sugaku Exposition. (発表予定).

T.Shoji：“论与复杂反射群相关的绿色函数”Sugaku Exposition（待提交）。