The Incidence Algebra of a Uniform Poset
The Incidence Algebra of a Uniform Poset
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DOI：
10.1007/978-1-4613-8994-1_15
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发表时间：
1990
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影响因子：
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通讯作者：
Paul M. Terwilliger
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作者：
Paul M. Terwilliger
设P,≤ 表示rankN≥ 2 的有限分级偏序集，其中纤维P0,P1,...,PN。令矩阵Li,Ri,Ei*(0 ≤ i ≤N) 的行和列由P 索引，条目$$ \begin{gathered} {{({{L}_{i}})}_{{xy}}} = 1\;if\;x \in {{P}_{{i - 1}}},\;y \in {{P}_{i}},\;x \leqslant y,\quad 和\quad 0\;否则\;(1 \leqslant i \leqslant N), \hfill \\ {{({{R}_{i}})}_{{xy}}} = 1\;如果\;x \in {{P}_{{i + 1}}},\;y \in {{P}_{i}},\;y \leqslant x,\quad且\quad 0\;否则\;(1 \leqslant i \leqslant N - 1), \hfill \\ {{(E_{i}^{*})}_{{xy}}} = 1\;如果\;x,y \in {{P}_{i}},\;\;x = y,\quad 且\quad 0\;否则\;(1 \leqslant i \leqslant N), \hfill \\ \end{gathered} $$ andL0=RN= 0. 的重合代数Pi是由Li,Ri,Ei*(0 ≤i≤N)生成的实矩阵代数。Pisuniformif存在实数ei+, ei-, fi, (1 ≤i≤N) (满足一定条件)使得$$ e_{i}^{ - }{{R}_{{i - 2}}}{{L}_{{i - 1}}}{{L}_{i}} + {{L}_{i}}{{R}_{{i - 1}}}{{L}_{i}} + e_{i}^{ + }{{L}_{i}}{{L}_{{i + 1}}}{{R}_{i}} = {{f}_{i}}{{L}_{i}}\quad (1 \leqslant i \leqslant N)\;({{R}_{{ - 1}}} = {{L}_{{N + 1}}} = 0)。 $$
LetP,≤ denote a finite graded poset of rankN≥ 2, with fibersP0,P1,... , PN. Let the matricesLi,Ri,Ei*(0 ≤ i ≤N) have rows and columns indexed byP, and entries $$ \begin{gathered} {{({{L}_{i}})}_{{xy}}} = 1\;if\;x \in {{P}_{{i - 1}}},\;y \in {{P}_{i}},\;x \leqslant y,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N), \hfill \\ {{({{R}_{i}})}_{{xy}}} = 1\;if\;x \in {{P}_{{i + 1}}},\;y \in {{P}_{i}},\;y \leqslant x,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N - 1), \hfill \\ {{(E_{i}^{*})}_{{xy}}} = 1\;if\;x,y \in {{P}_{i}},\;\;x = y,\quad and\quad 0\;otherwise\;(1 \leqslant i \leqslant N), \hfill \\ \end{gathered} $$ andL0=RN= 0. Theincidence algebraofPis the real matrix algebra generated byLi,Ri,Ei*(0 ≤i≤N).Pisuniformif there exists real numbers ei+, ei-, fi, (1 ≤i≤N) (satisfying a certain condition) such that $$ e_{i}^{ - }{{R}_{{i - 2}}}{{L}_{{i - 1}}}{{L}_{i}} + {{L}_{i}}{{R}_{{i - 1}}}{{L}_{i}} + e_{i}^{ + }{{L}_{i}}{{L}_{{i + 1}}}{{R}_{i}} = {{f}_{i}}{{L}_{i}}\quad (1 \leqslant i \leqslant N)\;({{R}_{{ - 1}}} = {{L}_{{N + 1}}} = 0). $$