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Study on representation theory of the classical groups and their related combinatorics

经典群表示论及其相关组合学研究

基本信息

批准号：
18540046
负责人：
KOIKE Kazuhiko
金额：
$ 2.57万
依托单位：
Aoyama Gakuin University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540046/
关键词：
algebra combinatoics representation theor classical groups Weyl groups 組み合せ論対称式ヘッケ環

项目摘要

For the classical Lie groups, the famous formulas by Herman Weyl, which say that the denominators of their irreducible characters are completely decomposed into the product of the linear factors, are regarded as the determinants whose components are the irreducible characters of rank one Lie groups of the same type. From this view point, Koike and Ito generalize these formulas and give some formulas that the determinants, whose components (the row indices are parameterized by the Young diagrams included in some fixed rectangular and the column indices are parameterized by the subsets of the variables) are given by the irreducible characters corresponding to the Young diagrams of the row indices in their variables prescribed by the column indices, are completely decomposed into the products of the linear factors. The resulting formulas turn out to be the powers of the original denominator formulas. These formulas play an crucial role in the result by Aomoto-Ito, in which they construct the fundamental system of solutions of holonmic q-difference system of Jackson integrals of type BC_n. Ito also gives explicit connection formulas of the solutions of holonmic q-difference system of Jackson integrals of type BC_n. Taniguchi gives an elementary construction for the invariant polynomials of type F_4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D_4, based on the fact that Weyl group of type D(-4) is included in a normal subgroup of Weyl group of type F_4

对于经典李群，把Herman Weyl关于其不可约特征的分母完全分解为线性因子的乘积的著名公式，作为其组成部分为同类型的1阶李群的不可约特征的行列式。从这个观点出发，Koike和Ito推广了这些公式，并给出了一些公式，这些公式的组成部分（行指标由固定矩形中的杨氏图参数化，列指标由变量的子集参数化）由列指标规定的变量的行指标的杨氏图对应的不可约特征给出。完全分解成线性因子的乘积。得到的公式是原来的分母公式的幂。这些公式在Aomoto-Ito的结果中起着至关重要的作用，在Aomoto-Ito的结果中，它们构造了BC_n型Jackson积分的全q差分系统的基本解系统。给出了Jackson积分的全q差分系统解的显式连接公式。Taniguchi根据D（-4）型Weyl群包含在F_4型Weyl群的正规子群中，利用D_4型Weyl群作用下的二阶不变多项式，给出了F_4型Weyl群在Weyl群作用下的初等构造