Combinatorial Studies of Demazure Modules

Demazure 模块的组合研究

基本信息

批准号：
09640034
负责人：
OKADO Masato
金额：
$ 1.98万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640034/
关键词：
affine Lie Algebra guantum group combinatorics アフィンリ一環

项目摘要

The affine Lie algebra is of great importance in mathematical physics. We investigate representation theory of affine Lie algebras (or quantum affine algebras) from the combinatorial viewpoint.1. Demazure modules and pathsFor a quantum affine algebra there exist perfect crystals, and the crystal base of an integrable repre-sentation is described by the semi-infinite tensor product of perfect crystals (path). On the other hand, there also exists a Demazure module, which can be thought of as a finite truncation of the integrable representation. We presented a criterion for the Demazure module to have the tensor product structure in general setting. We also checked that this criterion is satisfied for almost all known perfect crystals.2. Fermionic formulaAn expression without minus sign of the affine Lie algebra character is sometimes called fermionic formula. We proved such fermionic formulae for the string and branching function when the affine Lie algebra is of type A and the highest weight of the representation is lALAMBDA_0. We also conjectured fermionic formulae of one dimensional configuration sums of classically restricted paths for an arbitrary untwisted affine Lie algebra. It is an important open problem to prove them.3. Solvable lattice modelsKuniba et al. considered a solvable vertex model of type A and performed the spectral decomposition of its corner transfer matrix. They also derived an integral equation for the transverse, longitudinal correlation length for the XXZ model at q a root of unity via quantum transfer matrix method.4. Nearly-integrable systemsOgawa considered an equation with perturbation effect from the KdV equation. To understand theoretically the selectivity of wave numbers, he used the eigenfunctions of periodic traveling wave solutions for the KdV equation, and determined the spectra of the wave numbers by perturbative calculation.

仿射李代数在数学物理中非常重要。我们从组合的角度研究了仿射李代数（或量子仿射代数）的表示论。 1． Demazure 模块和路径对于量子仿射代数，存在完美晶体，并且可积表示的晶体基础由完美晶体的半无限张量积（路径）来描述。另一方面，还存在 Demazure 模块，它可以被认为是可积表示的有限截断。我们提出了 Demazure 模块在一般设置下具有张量积结构的标准。我们还检查了几乎所有已知的完美晶体都满足这个标准。2．费米子公式仿射李代数特征没有负号的表达式有时称为费米子公式。当仿射李代数为A型且表示的最高权重为lALAMBDA_0时，我们证明了弦函数和分支函数的费米子公式。我们还猜想了任意非扭曲仿射李代数的经典限制路径的一维构型和的费米子公式。证明它们是一个重要的悬而未决的问题。 3．可解晶格模型Kuniba 等人。考虑 A 型可解顶点模型并对其角转移矩阵进行谱分解。他们还通过量子转移矩阵法推导了单位根q处XXZ模型横向、纵向相关长度的积分方程。 4．近可积系统小川从 KdV 方程考虑了一个具有扰动效应的方程。为了从理论上理解波数的选择性，他利用KdV方程的周期行波解的本征函数，并通过微扰计算确定了波数的谱。