# The Incidence Algebra of a Uniform Poset

• Paul Terwilliger
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)

## 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 ≤ iN). P is uniform if there exists real numbers e i + , e i - , fi, (1 ≤ iN) (satisfying a certain condition) such that
$$e_{i}^{ - }{{R}_{{i - 2}}}{{L}_{{i - 1}}}{{L}_{i}} + {{L}_{i}}{{R}_{{i - 1}}}{{L}_{i}} + e_{i}^{ + }{{L}_{i}}{{L}_{{i + 1}}}{{R}_{i}} = {{f}_{i}}{{L}_{i}}\quad (1 \leqslant i \leqslant N)\;({{R}_{{ - 1}}} = {{L}_{{N + 1}}} = 0).$$

## Keyword

Graded poset Partial geometry Partial geometric lattice Association scheme

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