{"title":"A Q-polynomial structure for the Attenuated Space poset Aq(N,M)","authors":"Paul Terwilliger","doi":"10.1016/j.jcta.2024.105872","DOIUrl":null,"url":null,"abstract":"<div><p>The goal of this article is to display a <em>Q</em>-polynomial structure for the Attenuated Space poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>. The poset <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> is briefly described as follows. Start with an <span><math><mo>(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>)</mo></math></span>-dimensional vector space <em>H</em> over a finite field with <em>q</em> elements. Fix an <em>M</em>-dimensional subspace <em>h</em> of <em>H</em>. The vertex set <em>X</em> of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> consists of the subspaces of <em>H</em> that have zero intersection with <em>h</em>. The partial order on <em>X</em> is the inclusion relation. The <em>Q</em>-polynomial structure involves two matrices <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> with the following entries. For <span><math><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></math></span> the matrix <em>A</em> has <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></math></span>-entry 1 (if <em>y</em> covers <em>z</em>); <span><math><msup><mrow><mi>q</mi></mrow><mrow><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> (if <em>z</em> covers <em>y</em>); and 0 (if neither of <span><math><mi>y</mi><mo>,</mo><mi>z</mi></math></span> covers the other). The matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is diagonal, with <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mrow><mi>dim</mi></mrow><mspace></mspace><mi>y</mi></mrow></msup></math></span> for all <span><math><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. By construction, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. By construction, <em>A</em> acts on these eigenspaces in a (block) tridiagonal fashion. We show that <em>A</em> is diagonalizable, with <span><math><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn></math></span> eigenspaces. We show that <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that <em>A</em> is <em>Q</em>-polynomial. We show that <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra <em>T</em> of <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> generated by <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. We show that <span><math><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> act on each irreducible <em>T</em>-module as a Leonard pair.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000116","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset . The poset is briefly described as follows. Start with an -dimensional vector space H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of 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 with the following entries. For the matrix A has -entry 1 (if y covers z); (if z covers y); and 0 (if neither of covers the other). The matrix is diagonal, with -entry for all . By construction, has eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with eigenspaces. We show that acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of generated by . We show that act on each irreducible T-module as a Leonard pair.
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.