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

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 $U_τ({\mathfrak{sl}}_2)$ with $τ^2=q$. We consider two elements $A, A^*$ in $T$ of a certain form. We find necessary and sufficient conditions for $A, A^*$ to satisfy the tridiagonal relations. Let $W$ denote an irreducible $T$-module. We find necessary and sufficient conditions for the above $A, A^*$ to act on $W$ as a Leonard pair.