Tridiagonal pairs, alternating elements, and distance-regular graphs

Abstract

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation with two generators $W_0$, $W_1$ and two relations called the $q$-Serre relations. The algebra $U_q^{+}$ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted $\{W_{-k}{\}}_{k\in \mathbb{N}}$, $\{W_{k+1}{\}}_{k\in \mathbb{N}}$, $\{G_{k+1}{\}}_{k\in \mathbb{N}}$, $\{{\tilde{G}}_{k+1}{\}}_{k\in \mathbb{N}}$. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps $A,A^{\ast }$ on a nonzero, finite-dimensional vector space $V$, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let $A$, $A^{\ast }$ denote a tridiagonal pair on $V$. Associated with this pair are six well-known direct sum decompositions of $V$; these are the eigenspace decompositions of $A$ and $A^{\ast }$, along with four decompositions of $V$ that are often called split. In our main results, we assume that $A$, $A^{\ast }$ has $q$-Serre type. Under this assumption $A$, $A^{\ast }$ satisfy the $q$-Serre relations, and $V$ becomes an irreducible $U_q^{+}$-module on which $W_0=A$ and $W_1=A^{\ast }$. We describe how the alternating elements of $U_q^{+}$ act on the above six decompositions of $V$. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of $A$ and $A^{\ast }$ all have dimension one. In the second case $A$ and $A^{\ast }$ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.

arXiv comment39 pages, 9 diagrams

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Cited by in corpus (2)

• The nucleus of a $Q$-polynomial distance-regular graph
• An equitable partition for the distance-regular graph of the bilinear forms