Projective Geometries, Grassmann Graphs, and a Generalization of the Askey-Wilson Relations
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| Publicado en: | ProQuest Dissertations and Theses (2025) |
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| Acceso en línea: | Citation/Abstract Full Text - PDF |
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| Resumen: | This thesis is about projective geometries, Grassmann graphs, and a generalization of the Askey-Wilson relations. The thesis contains three main results. The projective geometry and its corresponding Grassmann graphs are defined as follows. Let Fq denote a finite field with q elements. Fix an integer n ≥ 1 Let V denote an n-dimensional vector space over Fq. Let the set P consist of the subspaces of V. The set P, together with the inclusion partial order, is a poset called a projective geometry. For 1 ≤ k ≤ n - 1 the Grassmann graph Jq(n,k) is defined as follows. The vertex set X of Jq(n,k) consists of the k-dimensional subspaces of V. Two vertices x, y are adjacent whenever x∩y has dimension k-1. Our first main result concerns how to use Jq(n,k) to potentially recover P. Pick distinct x, y ∈ Х. The geometry P contains the elements x, y, x∩y, x+y. DefineBxy = {z∈ X | ∂(z,x) = 1, ∂(z,y)= ∂ (x, y) + 1}Cxy = {z∈ X | ∂(z,x) = 1, ∂(z,y) = ∂(x, y) - 1},where ∂ is the path-length distance function of Jq(n,k). We consider a Euclidean space E of dimension (qn - 1) / (q - 1) - 1 We turn E into a Euclidean representation of Jq(n,k) associated with the second largest eigenvalue of the adjacency matrix. We represent the elements x, y, x∩y, x+y and the sets Bxy, Cxy as vectors in E. We write the vector representations of x∩y, x+y as linear combinations of the vector representations of x, y, Bxy, Cxy: this is our first main result. For our second main result, we consider the stabilizer Stab(x, y) of x, y in GL(V). We find the orbits of the Stab(x, y)-action on the local graph of x. As we will see, there are five orbits. These orbits form an equitable partition. We compute the corresponding structure constants; this is our second main result. In our third main result, we display two matrices A, A* ∈ MatP(C) that satisfy a generalization of the Askey-Wilson relations. We define a matrix A ∈ MatP(C) as follows. For u,v ∈ P, the (u, v)-entry of A is 1 if each of u, v covers u∩v, and 0 otherwise. Fix y ∈ P with dim y = k. We define a diagonal matrix A* ∈ MatP(C) as follows. For u ∈ P. the (u, u)-entry of A* is qdim(u∩y). We show thatA2 A* - (q + q-1)A A* A+A* A2 - Y (AA* +A* A)- P A* = ΩA+G,A*2 A - (q + q-1) A* AA* +AA*2 = Y A*2 + ΩA* +G*,where Y, P. Ω, G, G* are matrices in MatP (C) that commute with each of A, A*. We give precise formulas for Y, P, Ω, G, G*. |
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| ISBN: | 9798314852842 |
| Fuente: | ProQuest Dissertations & Theses Global |