The Attenuated Space Poset ${\mathcal{A}}_q(N,M)$

Abstract

In this paper, we study the incidence algebra $T$ of the attenuated space poset ${\mathcal{A}}_q(N,M)$. We consider the following topics. We consider some generators of $T$: the raising matrix $R$, the lowering matrix $L$, and a certain diagonal matrix $K$. We describe some relations among $R,L,K$. We put these relations in an attractive form using a certain matrix $S$ in $T$. We characterize the center $Z(T)$. Using $Z(T)$, we relate $T$ to the quantum group $U_{\tau }({\mathfrak{sl}}_2)$ with ${\tau }^2=q$. We consider two elements $A,A^{\ast }$ in $T$ of a certain form. We find necessary and sufficient conditions for $A,A^{\ast }$ to satisfy the tridiagonal relations. Let $W$ denote an irreducible $T$-module. We find necessary and sufficient conditions for the above $A,A^{\ast }$ to act on $W$ as a Leonard pair.