On the homology of distributive lattices
For a finite poset \(P\) the set of order-ideals \(J(P)=L\) ordered by containment is a distributive lattice with rank \(| P|\) and rank function \(\rho(I) =| I|\). If \(S\subseteq [n-1]= \{1,\dots, n-1\}\), then \(J(P)_S= \{I\in J(P) \mid| I|\in S\cup \{0,n\}\}\). Given a finite poset \(Q=J(P)_S\) with \(\widehat 0\) and \(\widehat 1\), let \(C_r(Q)\), \(r=-1,0, \dots\), denote the complex vector-space with basis the chains \(\widehat 0<a_0< \cdots <a_r< \widehat 1\), \((C_{-1 }(Q)\) has basis \(\widehat 0< \widehat 1)\). If \(\partial_r: C_r(Q)\to C_{r-1} (Q)\) with \(\partial_r (\widehat 0<\cdots <\widehat 1)= \sum^r_{i=0} (-1)^i (\widehat 0<\cdots <a_{i-1}< a_{i+1}< \cdots \widehat 1)\), let \(\widetilde H_r(Q)=\ker \partial_r/ \text{Im} \partial_{r+1}\), \(\widetilde H_*(Q)= \oplus_r\widetilde H_r(Q)\). (If a finite group \(G\) acts on \(P\), then it acts naturally on \(J(P)\), by \(g\cdot I=\{g \cdot a\mid a\in I\}\), on \(J(P)_S\), on \(C_r(J(P)_S)\), on \(\widetilde H_r(J(P)_S)\) and on \(\widetilde H_*(J(P)_S)\). If the character of the last action of \(G\) is \(\beta(S,g)\), then the author shows how the generating function \(F^g(\underline x)=\sum (\prod x_{f(a)})\) where the sum runs over all \(f\in {\mathcal O} (P)\) with \(g\cdot f=f\), where \({\mathcal O}(P)\) is the set of order-preserving maps \(f:P\to\mathbb{Z}_+\) and where for \(g\in G\), \(g\cdot f(a) =f(g^{-1} \cdot a)\), can be expanded as \(F^g(\underline x)= \sum_{T\subseteq [n-1]}\beta (T,g) F_T(\underline x)\), the \(F_T(\underline x)\) being Gessel's quasisymmetric functions defined as: \(F_T(\underline x)= \sum (\prod_{1\leq i\leq n} x_{a_i})\), where the sum runs over the set \(\{a_1\leq \cdots\leq a_n\) and \(i\in T\) implies \(a_i<a_{i+1}\}\). Using this generating function and the relationship shown above, the author is able to compute explicitly the \(\widetilde H_*(Q)\) for \(Q=J(P)_S\) in various interesting cases, including \(L=[a+1] \times \cdots \times [a+1]\) \((b\) times) with \(G=S_b\) acting on \(L\) by permuting coordinates, leading for \(S\subseteq [ab]\) to nicely formulated generalizations for known results, such as L. Solomon's decomposition of the \(S_b\)-modules of \(\widetilde H_* (({\mathcal B}_b)_S)\).
scientific article; zbMATH DE number 1199925