The Terwilliger algebra of the halved cube

TL;DR

This paper studies the Terwilliger algebra of the halved cube graph, decomposing its standard module into irreducible components, which advances understanding of its algebraic structure.

Contribution

It provides a decomposition of the standard module of the Terwilliger algebra for the halved cube, revealing its irreducible modules and their structure.

Findings

Decomposition of the standard module into irreducible modules.

Explicit description of the Terwilliger algebra's structure.

Insights into the algebraic symmetry of the halved cube.

Abstract

Let $D\geq 3$ denote an integer. For any $x\in \mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $\mathbb F_2^D$ consisting of all $x\in \mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved cube $\frac{1}{2}H(D,2)$ is a finite simple connected graph with vertex set $X$ and $x,y\in X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $x\in X$. The Terwilliger algebra $\mathcal T=\mathcal T(x)$ of $\frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${\rm Mat}_X(\mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) \qquad \hbox{for all $y\in X$}. $$ In this paper we decompose the standard $\mathcal T$-module into a direct sum of irreducible $\mathcal T$-modules.