A Q-polynomial structure for the Attenuated Space poset Aq(N,M)

IF 0.9 2区 数学 Q2 MATHEMATICS
Paul Terwilliger
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引用次数: 0

Abstract

The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset Aq(N,M). The poset Aq(N,M) is briefly described as follows. Start with an (N+M)-dimensional vector space H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of Aq(N,M) consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q-polynomial structure involves two matrices A,AMatX(C) with the following entries. For y,zX the matrix A has (y,z)-entry 1 (if y covers z); qdimy (if z covers y); and 0 (if neither of y,z covers the other). The matrix A is diagonal, with (y,y)-entry qdimy for all yX. By construction, A has N+1 eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with 2N+1 eigenspaces. We show that A acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that A,A satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of MatX(C) generated by A,A. We show that A,A act on each irreducible T-module as a Leonard pair.

衰减空间正集 Aq(N,M) 的 Q 多项式结构
本文的目的是展示衰减空间正集 Aq(N,M) 的 Q 多项式结构。正集 Aq(N,M) 简述如下。从有限域上具有 q 个元素的 (N+M) 维向量空间 H 开始。Aq(N,M)的顶点集 X 由与 h 有零交集的 H 子空间组成。Q 多项式结构涉及两个矩阵 A,A⁎∈MatX(C),其条目如下。对于 y,z∈X,矩阵 A 有 (y,z) 项 1(如果 y 覆盖了 z);qdimy(如果 z 覆盖了 y);0(如果 y,z 都没有覆盖另一个)。矩阵 A⁎ 是对角线,对于所有 y∈X 都有 (y,y) 项 q-dimy。根据构造,A⁎ 有 N+1 个特征空间。根据构造,A 以(分块)三对角方式作用于这些特征空间。我们证明 A 是可对角的,有 2N+1 个特征空间。我们证明 A⁎ 以(块)三对角方式作用于这些特征空间。利用这一作用,我们证明 A 是 Q 多项式。我们证明 A、A⁎ 满足一对称为三对角关系的关系。我们考虑由 A,A⁎ 生成的 MatX(C) 子代数 T。我们证明,A,A⁎ 作为伦纳德对作用于每个不可还原 T 模块。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.90
自引率
9.10%
发文量
94
审稿时长
12 months
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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