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

中文翻译：

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衰减空间偏序集 Aq(N,M) 的 Q 多项式结构

本文的目标是显示衰减空间偏序集的多项式结构。该偏序集简要描述如下。从包含元素的有限域上的 维向量空间开始。修复 的 维子空间。的顶点集由 与 的交集为零的子空间组成。上的偏序就是包含关系。多项式结构涉及两个具有以下条目的矩阵。对于矩阵有 -entry 1 （如果包含 ）；（如果覆盖）；和 0（如果两者都不覆盖另一个）。该矩阵是对角矩阵，所有 都有 -entry 。通过构造，具有特征空间。通过构造，以（块）三对角方式作用于这些特征空间。我们证明它是可对角化的，具有特征空间。我们证明了以（块）三对角方式作用于这些特征空间。使用这个动作，我们证明它是-多项式。我们证明满足一对称为三对角关系的关系。我们考虑由 生成的子代数。我们展示了对每个不可约模的作用作为伦纳德对。