Distance-regular graphs and the q-tetrahedron algebra

金沢大学理工研究域数物科学系Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b ≠ 1, α = b - 1. The condition on α implies that Γ is formally self-dual. For b = q2 we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra {squared times}q on the standard module of Γ. We describe four algebra homomorphisms into {squared times}q from the quantum affine algebra Uq (over(s l, ̂)2); using these we pull back the above {squared times}q-action to obtain four actions of Uq (over(s l, ̂)2) on the standard module of Γ. © 2008 Elsevier Ltd. All rights reserved

Towards a classification of the tridiagonal pairs

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \in End(V)$ that satisfy (i)--(iv) below: (i) Each of $A,A^*$ is diagonalizable. (ii) There exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. (iii) There exists an ordering $\{V^*_i\}_{i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1}=0$ and $V^*_{\delta+1}=0$. (iv) There is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\em tridiagonal pair} on $V$. Let $E^*_0$ denote the element of $End(V)$ such that $(E^*_0-I)V^*_0=0$ and $E^*_0V^*_i=0$ for $1 \leq i \leq d$. Let $D$ (resp. $D^*$) denote the $K$-subalgebra of $End(V)$ generated by $A$ (resp. $A^*$). In this paper we prove that the span of $E^*_0 D D^*DE^*_0$ equals the span of $E^*_0D E^*_0DE^*_0$, and that the elements of $E^*_0 D E^*_0$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.Comment: 18 page

Tridiagonal pairs of Krawtchouk type

Let $K$ denote an algebraically closed field with characteristic 0 and let $V$ denote a vector space over $K$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^*$ has Krawtchouk type whenever the sequence $\lbrace d-2i\rbrace_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^*$. Assume $A,A^*$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $$ on $V$ such that $=$ and $=$ for $u,v\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^*$; (ii) $-A,-A^*$; (iii) $A^*,A$; (iv) $-A^*,-A$. We give a number of related results and conjectures.Comment: 20 page

A duality between pairs of split decompositions for a Q-polynomial distance-regular graph

AbstractLet Γ denote a Q-polynomial distance-regular graph with diameter D≥3 and standard module V. Recently, Ito and Terwilliger introduced four direct sum decompositions of V; we call these the (μ,ν)-split decompositions of V, where μ,ν∈{↓,↑}. In this paper we show that the (↓,↓)-split decomposition and the (↑,↑)-split decomposition are dual with respect to the standard Hermitian form on V. We also show that the (↓,↑)-split decomposition and the (↑,↓)-split decomposition are dual with respect to the standard Hermitian form on V