Representation theory of algebraic groups, Hecke algebras and canpex refiecion groups

代数群、Hecke 代数和 Canpex 折射群的表示论

基本信息

批准号：
17340003
负责人：
SHOJI Toshiaki
金额：
$ 6.39万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340003/
关键词：
algebraic groups Hecke algebras complex reflection groups cyclotomic q-Scqur algebras Fock space cellular aloebras finite reductive groups finite symmetric space Ariki-Koike algebra cyclotomic q-Schur algerbra 分解定数積公式フォック空間有限C

项目摘要

We have studied on the following three themes.1. Determination of scalars involved in Lusztig's conjecture concerning irreducible characters of finite reductive groups_G (F_q), and the determination of unipotent elements with certain good properties related to the algorithm of computing irreducible characters. In particular concerning the second problem, in the case where G (F_q) = SL_n(F_q), Sp_{2n}(F_q), SO_{2n+1} (F_q), SO {2n} (F_q) we have determined a class of good unipotent elements. In the case of Classical groups, this results holds also for the case where the characteristic is equal to 2. By this result, an algorithm computing Green functions of classical groups of even characteristic was established, which had involved certain ambiguity before.2. It is known that Green functions of finite reductive groups is a polynomial in q. We have proved a formula for the values of Green functions obtained by substituting a root of unity for q. This type of formula was known in the case … More where G(F_q) = GL_n(F_q) by a combinatorial method. In this study, we have proved it in the general case by making use of induction theorem for Springer representations due to Lusztig.3. We have studied the modular representation theory of Ariki-Koike algebras which are Hecke algebras associated to the complex reflection groups G(r,1,n), and the cyclotomic q-Schur algebras related to the Ariki-Koike algebras. We have constructed various subalgebras of cyclotomic q-Schur algebras and their quotients, and proved a product formulas for certain type of decomposition numbers, by comparing the decomposition numbers of them.On the other hand, thanks to Yvonne's conjecture, it is conjectured that the decomposition numbers of cyclotomic q-Schur algebras are obtained from the transition matrix between standard bases and canonical bases of higher level Fock space. Based on this conjecture", we have proved a product formula for the Fock space which conjecturally corresponds to the original product formula Less

我们研究了以下三个主题1。确定Lusztig涉及的标量有关有限的降低组_G（F_Q）的不可还原字符的概念，以及确定具有与计算不可减少字符算法相关的某些良好属性的单位元素的确定。特别是关于第二个问题，在g（f_q）= sl_n（f_q），sp_ {2n}（f_q）的情况下，so_ {2n+1}（f_q），所以{2n}（f_q）我们确定了一类好的不透明元素。在经典组的情况下，对于字符等于2的情况，此结果也存在。通过此结果，建立了偶数性格的经典群体的算法计算绿色函数，该功能涉及某些歧义。2。众所周知，有限减少组的绿色功能是Q中的多项式。我们为通过将Unity的根替换为Q获得的绿色函数值提供了一个公式。这种类型的公式在这种情况下是知道的……更多的是通过组合方法G（f_q）= GL_N（f_q）。在这项研究中，我们在一般情况下通过使用诱导理论来为springer代表造成的诱导理论提供了它。3。我们研究了与复杂反射组G（R，1，N）和与Ariki-Koike代数相关的环形Q-Schur代数相关的Ariki-Koike代数的模块化表示理论。 We have constructed various subalgebras of cyclotomic q-Schur algebras and their quotients, and provided a product formulas for certain type of decomposition numbers, by comparing the decomposition numbers of them.On the other hand, thank you to Yvonne's conjecture, it is conjectured that the decomposition numbers of cyclotomic q-Schur algebras are obtained from the transition matrix Between standard bases and canonical bases更高级别的fock空间。基于这个概念”，我们为Fock空间提供了一个产品公式，该公式可能与原始产品公式相对应