Rook placements and generalized partition varieties
The paper generalizes the concept of partition varieties introduced in a preceding paper by the same author. For this purpose, \(\gamma\)-compatible partitions \(\lambda\) are introduced, where \(\gamma\) is a composition of some integers. It is shown that a certain quotient space associated to such composition \(\gamma\) and partition \(\lambda\) is a projective variety. Moreover, this generalized partition variety is a CW-complex which can be described combinatorially in terms of \(\gamma\)-compatible rook placements of the Ferrers board of \(\lambda\). Finally, the Poincaré polynomial \(P_{\lambda,\gamma}(q)\) for the cohomology equals \(RL_{\lambda,\gamma}(q^2)\), where \(RL_{\lambda,\gamma}\) is the \(\gamma\)-rook length polynomial. The combinatorial description of generalized partition varieties applies to the homology and cohomology of flag manifolds and Grassmann manifolds.
scientific article; zbMATH DE number 1109958
scientific article