The Terwilliger algebra of the twisted Grassmann graph: the thin case

TL;DR

This paper characterizes all irreducible modules of the Terwilliger algebra associated with a specific orbit of the twisted Grassmann graph, focusing on the case where the algebra is thin, revealing detailed algebraic structure.

Contribution

It explicitly determines all irreducible Terwilliger algebra modules for the thin case of the twisted Grassmann graph, expanding understanding of its algebraic properties.

Findings

All irreducible T(x)-modules are classified for the thin case.

The structure of the Terwilliger algebra in this setting is fully described.

The results deepen the understanding of the algebraic symmetry of the twisted Grassmann graph.

Abstract

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.