ROOK PLACEMENTS AND CLASSIFICATION OF PARTITION VARIETIES B\ Mλ

Abstract

Let M be the set of m by n complex matrices of rank m. Let λ = (λ1,…,λm) be a partition with λi ≥ λi + 1 for 1 ≤ i ≤ m - 1 and λ1 = n. A Ferrers board Fλ is a right justified subarray in a m by n matrix with the length of the ith row being λi, and define Mλ={a ∈ M|ai,j = 0 if (i,j) ∉ Fλ}. Let B be the Borel subgroup of the general linear group GLm (C) consisting of upper triangular matrices. Define B\Mλ={Ba|a∈ Mλ}. The quotient space B\Mλ is a projective variety called a partition variety associated to λ. In this note, we classify partition varieties B\Mλ according to their homology and cohomology groups (up to isomorphisms).