2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$

TL;DR

This paper explores the structure of certain bipartite distance-regular graphs using quantum group theory, specifically constructing modules for $U^ extprime_q( ext{so}_6)$ via an $S_3$-symmetric approach.

Contribution

It introduces a novel $S_3$-symmetric framework to realize modules of the quantum group $U^ extprime_q( ext{so}_6)$ from bipartite distance-regular graphs.

Findings

Constructed a subspace $ extLambda$ of $V^{ ensor 3}$ with dimension $inom{D+3}{3}$.

Developed six linear maps turning $ extLambda$ into an irreducible module for $U^ extprime_q( ext{so}_6)$.

Established a connection between graph symmetry and quantum group representations.

Abstract

We consider a 2-homogeneous bipartite distance-regular graph $\Gamma$ with diameter $D \geq 3$. We assume that $\Gamma$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $\Gamma$. This $Q$-polynomial ordering is described using a nonzero parameter $q \in \mathbb C$ that is not a root of unity. We investigate $\Gamma$ using an $S_3$-symmetric approach. In this approach one considers $V^{\otimes 3} = V \otimes V \otimes V$ where $V$ is the standard module of $\Gamma$. We construct a subspace $\Lambda$ of $V^{\otimes 3}$ that has dimension $\binom{D+3}{3}$, together with six linear maps from $\Lambda$ to $\Lambda$. Using these maps we turn $\Lambda$ into an irreducible module for the nonstandard quantum group $U^\prime_q(\mathfrak{so}_6)$ introduced by Gavrilik and Klimyk in 1991.