On the geometry of varieties of invertible symmetric and skew-symmetric matrices

Abstract

Let Sym(n; F) and Sk(n; F) denote the algebraic varieties of n n invertible symmetric and skew-symmetric matrices over a eld F, respectively. We rst show how the homotopy type of Sym(n; R) and the homology groups of Sk(n; R) can be determined using an alternative method to Iwasawa decomposition. Then, using recent results of Dimca and Lehrer, the weight polynomials of Sym(n; C) and Sk(n; C) are calculated. 1. Introduction and notations. Let F be a eld, GL(n; F) the set of n n invertible matrices over F, Sym(n; F) the set of nn invertible symmetric matrices over F and Sk(n; F) the set of n n invertible skew-symmetric matrices over F. Note that in the case of the invertible skew-symmetric matrices n has to be even. In this paper we rst determine the toplogy of the real varieties Sym(n; R) and Sk(n; R). More precisely, we show that Sym(n; R) has the homotopy type of a Grassmannian by constructing a homotopy equivalence and compute the Betti numbers of Sk(n; R) by bering this variety over the sphere and using the associated Leray spectral sequence. The Serre-Poincar e polynomial (weight polynomial) of a complex algebraic variety X is dened to be Wc(X;t )= X