The Incidence Algebra of a Uniform Poset

Abstract

Let P,≤ denote a finite graded poset of rank N ≥ 2, with fibers P ₀, P ₁, ... , P _N. Let the matrices L _i, R _i, E ^*_i (0 ≤ i ≤ N) have rows and columns indexed by P, and entries

$$ \begin{gathered} {{({{L}_{i}})}_{{xy}}} = 1\;if\;x \in {{P}_{{i - 1}}},\;y \in {{P}_{i}},\;x \leqslant y,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N), \hfill \\ {{({{R}_{i}})}_{{xy}}} = 1\;if\;x \in {{P}_{{i + 1}}},\;y \in {{P}_{i}},\;y \leqslant x,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N - 1), \hfill \\ {{(E_{i}^{*})}_{{xy}}} = 1\;if\;x,y \in {{P}_{i}},\;\;x = y,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N), \hfill \\ \end{gathered} $$

and L ₀ = R _N = 0. The incidence algebra of P is the real matrix algebra generated by L _i, R _i, E ^*_i (0 ≤ i ≤ N). P is uniform if there exists real numbers e ⁺_i , e ^-_i , f_i, (1 ≤ i ≤ N) (satisfying a certain condition) such that

$$ e_{i}^{ - }{{R}_{{i - 2}}}{{L}_{{i - 1}}}{{L}_{i}} + {{L}_{i}}{{R}_{{i - 1}}}{{L}_{i}} + e_{i}^{ + }{{L}_{i}}{{L}_{{i + 1}}}{{R}_{i}} = {{f}_{i}}{{L}_{i}}\quad (1 \leqslant i \leqslant N)\;({{R}_{{ - 1}}} = {{L}_{{N + 1}}} = 0). $$