Projective Geometries, Grassmann Graphs, and a Generalization of the Askey-Wilson Relations

Abstract

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 F_q denote a finite field with q elements. Fix an integer n ≥ 1 Let V denote an n-dimensional vector space over F_q. 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 J_q(n,k) is defined as follows. The vertex set X of J_q(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 J_q(n,k) to potentially recover P. Pick distinct x, y ∈ Х. The geometry P contains the elements x, y, x∩y, x+y. Define

B_xy = {z∈ X | ∂(z,x) = 1, ∂(z,y)= ∂ (x, y) + 1}

C_xy = {z∈ X | ∂(z,x) = 1, ∂(z,y) = ∂(x, y) - 1},

where ∂ is the path-length distance function of J_q(n,k). We consider a Euclidean space E of dimension (qⁿ - 1) / (q - 1) - 1 We turn E into a Euclidean representation of J_q(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 B_xy, C_xy as vectors in E. We write the vector representations of x∩y, x+y as linear combinations of the vector representations of x, y, B_xy, C_xy: 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* ∈ Mat_P(C) that satisfy a generalization of the Askey-Wilson relations. We define a matrix A ∈ Mat_P(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* ∈ Mat_P(C) as follows. For u ∈ P. the (u, u)-entry of A* is q^dim(u∩y). We show that

A² 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 Mat_P (C) that commute with each of A, A*. We give precise formulas for Y, P, Ω, G, G*.