A tale of two $q$-deformations : connecting dual polar spaces and weighted hypercubes

TL;DR

This paper introduces two $q$-analog hypercube graphs, explores their connection via a graph quotient, and links algebraic structures like $U_q(rak{su}(2))$ and dual $q$-Krawtchouk polynomials to these graphs.

Contribution

It establishes a novel relationship between dual polar spaces and weighted hypercubes through $q$-deformations and algebraic tools.

Findings

Introduction of two $q$-analog hypercube graphs

Connection established via a graph quotient

Relation to $U_q(rak{su}(2))$ and dual $q$-Krawtchouk polynomials

Abstract

Two $q$-analogs of the hypercube graph are introduced and shown to be related through a graph quotient. The roles of the subspace lattice graph, of a twisted primitive elements of $U_q(\mathfrak{su}(2))$ and of the dual $q$-Krawtchouk polynomials are elaborated upon. This paper is dedicated to Tom Koornwinder.