The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$

TL;DR

This paper explores the algebraic structure of symplectic dual polar graphs by connecting their Terwilliger algebra to $U_q(sl_2)$, revealing their irreducible components and submodule multiplicities.

Contribution

It establishes a novel link between the Terwilliger algebra of symplectic dual polar graphs and quantum algebra $U_q(sl_2)$, providing explicit module decompositions.

Findings

Identifies the action of the adjacency matrix on eigenspaces as a weighted subspace lattice.

Determines the irreducible components of the standard module.

Calculates the multiplicities of submodules.

Abstract

The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces of an abelian automorphism subgroup is shown to act as the adjacency matrix of a weighted subspace lattice. The connection between the latter and $U_q(sl_2)$ is used to find the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.