Terwilliger Algebra of the Ordered Hamming Scheme

TL;DR

This paper investigates the structure of the Terwilliger algebra for the ordered Hamming scheme, extending known results from Hamming schemes and wreath products to this more general setting.

Contribution

It generalizes the description of the Terwilliger algebra from Hamming schemes and wreath products to the ordered Hamming scheme.

Findings

The Terwilliger algebra of the ordered Hamming scheme is characterized.

Extension of previous results to a broader class of schemes.

Provides a structural decomposition of the algebra.

Abstract

This paper delves into the Terwilliger algebra associated with the ordered Hamming scheme, which extends from the wreath product of one-class association schemes and was initially introduced by Delsarte as a natural expansion of the Hamming schemes. Levstein, Maldonado and Penazzi have shown that the Terwilliger algebra of the Hamming scheme of length $n$ is the $n$-fold symmetric tensor algebra of that of the one-class association scheme. Furthermore, Bhattacharyya, Song and Tanaka have established that the Terwilliger algebra of the wreath product of a one-class association scheme is a direct sum of the ``primary'' subalgebra and commutative subalgebras. This paper extends these findings to encompass both conclusions.