Equivariant cohomology of rank varieties associated with vexillary permutations (Q5928273)

In the space of complex \(m\times\ell\) matrices a rank variety \(\Sigma(r)\) is defined by equations of the form \(\text{rank} A_{\mu, \lambda}= r_{\mu, \lambda}\), where \(A_{\mu,\lambda}\) denotes the submatrix of \(A\) weakly northwest of position \((\mu,\lambda)\) and \(r\) is a rank condition, i.e., a non-negative integer valued function on some set \(\gamma(r)\subset [1,m]\times [1,\ell]\) of matrix positions. Particularly important rank varieties are those associated with `linear' rank conditions, that is to say with functions \(r\) defined on a sequence \(\gamma(r)\) of matrix positions each of which lies weakly southwest of the previous one. These varieties are closely related to vexillary permutations. The torus \(T^\ell \times T^m\) acts on rank varieties and their unions in the classical way via left and right matrix multiplication. With regard to the corresponding equivariant cohomology, it is proved that if a smooth manifold is a union of rank varieties, its equivariant cohomology is torsion free, of finite type and vanishing in odd dimensions. In the particular case where \(r\) is a linear rank condition, the following results are established: (i) \(\Sigma(r)\) is a smooth manifold; there is indeed a parabolic subgroup \(P\) of \(\text{GL} (m+\ell)\) which acts on the Grassmannian \(G_\ell (\mathbb{C}^{m+ \ell})\) such that \(\Sigma(r)\) is diffeomorphic to the intersection of a \(p\)-orbit with an open cell centered at some \(T^{m+ \ell}\)-fixed ppoint of the Grassmannian. (ii) The space \(\Sigma^\leq(r)\) which one obtains by requiring the matrices \(A_{\mu, \lambda}\) to have rank at most \(r_{\mu,\lambda}\) admits a topologically intrinsic stratification, whose strata are precisely the various \(\Sigma(\rho)\) as \(\rho\) varies among the linear rank conditions defined on a suitable subset of \(\gamma(r)\) and satisfying \(\rho\leq r\). (iii) The equivariant Poincaré series of \(\Sigma(r)\) is computed via a method which yields simple explicit formulas.