The Terwilliger algebra of digraphs I -- Hamming digraph $H^*(d,3)$
Tsuyoshi Miezaki, Hiroshi Suzuki, Keisuke Uchida
TL;DR
The paper determines the Terwilliger algebra of the digraph H^*(d,3) by relating it to the Lie algebra sl_3(C) and identifying the Terwilliger algebra with the symmetric tensor algebra $\mathrm{Sym}^{(d)}(\mathrm{Mat}_3(\mathbb{C}))$. It develops a detailed Lie-algebraic framework using matrices derived from the digraph, proves an sl_3(C) structure with sl_2 subalgebras, and classifies irreducible modules via highest-weight theory, yielding a complete decomposition $\mathcal{T}(H^*(d,3)) \cong \bigoplus_{i\in\Lambda} \mathrm{Mat}_i(\mathbb{C})$. A constructive PBW-type basis and explicit weight/multiplicity formulas are given, with a first-principles second proof based on representation theory. The results reveal a deep link between Hamming-type digraph Terwilliger algebras and classical Lie-algebra representations, and they identify the principal module and its multiplicities, providing a foundation for further study of weakly distance-regular digraphs of Hamming type. The work has potential implications for combinatorial representation theory and algebraic graph theory, especially in understanding symmetry-encoded decompositions of digraph algebras.
Abstract
In the present paper, we define the Terwilliger algebra of digraphs. Then, we determine the irreducible modules of the Terwilliger algebra of a Hamming digraph $H^*(d,3)$. As is well known, the representation of the Terwilliger algebra of a binary Hamming graph $H(d,2)$ is closely related to that of the Lie algebra $\mathit{sl}_2(\mathbb{C})$. We show that in the case of $H^*(d,3)$, it is related to that of the Lie algebra $\mathit{sl}_3(\mathbb{C})$. We also identify the Terwilliger algebra of $H^*(d,3)$ as the $d$ symmetric tensor algebra of ${\rm Mat}_3(\mathbb{C})$.