An operator-theory construction on geometric lattices

TL;DR

This paper introduces an operator-theoretic framework for finite geometric lattices, linking lattice structures to orthogonal polynomial systems via a novel nonassociative product and explicit combinatorial formulas.

Contribution

It presents a new construction that associates geometric lattices with orthogonal polynomials, extending to various lattice types without symmetry constraints.

Findings

Centered Krawtchouk Jacobi matrix for Boolean lattices

q-deformations for projective geometries

Explicit combinatorial formulas for Jacobi coefficients

Abstract

We introduce a canonical operator-theoretic construction associated to a finite geometric lattice, in which a simple nonassociative ``diamond product'' on the lattice basis gives rise to a family of creation operators indexed by atoms and a corresponding self-adjoint Hamiltonian on $\mathbb R[L]$. A key structural feature is that the Hamiltonian changes rank by at most one, so that its compression to the rank-radial subspace is a Jacobi matrix. In this way, geometric lattices give rise in a direct and uniform manner to finite orthogonal polynomial systems. The Jacobi coefficients admit explicit combinatorial formulas. For Boolean lattices one obtains the centered Krawtchouk Jacobi matrix, while for projective geometries one obtains natural $q$-deformations consistent with the $q$-Hahn family. The construction applies to arbitrary geometric lattices and requires no symmetry assumptions.