On the geometry of varieties of invertible symmetric and skew-symmetric matrices
type: Item
instance of: scholarly article
title: On the geometry of varieties of invertible symmetric and skew-symmetric matrices
zbMATH Open document ID: 0885.14008
author: Oliver Jones
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
full work available at URL: v180no1.html
review text: Let Sym\((n,K)\) and Sk\((n, K)\) denote the sets of \(n\times n\) invertible symmetric and skew-symmetric matrices over a field \(K\) respectively. Let GL\((n, K)\) be the set of \(n\times n\) invertible matrices over \(K\) and define an action GL\((n, K)\times\text{Sym}(n, K)\to\text{Sym}(n, K)\) by \(g\cdot A= gA^t g\). An action of GL\((n, K)\) on Sk\((n, K)\) is defined in an analogous way. The first results presented in this paper are that the homotopy type of Sym\((n,\mathbb{R})\) can be determined by fibering the orbits over the Grassmann variety and then showing that the fibres are contractible and that the Betti numbers of the orbits of Sk\((n, \mathbb{R})\) can be determined by fibering them over the sphere and using the associated Wang exact sequence. We also show how these results can be derived using Iwasawa decompositon and theorems of Borel and Ehresmann. Deligne showed that any complex algebraic variety \(X\) has a canonical mixed Hodge structure, i.e. a finite increasing filtration \(W\) on \(H^i (X; \mathbb{Q})\), called the weight filtration, and a finite decreasing filtration \(F\) on \(H^i (X; \mathbb{C})\), called the Hodge filtration such that the filtration induced by \(F\) on \(\text{Gr}^W_k H^i (X; \mathbb{C})\) is a Hodge structure of weight \(k\). Using the weight filtration an invariant for the variety can be defined known as the weight polynomial. The next result is that the weight polynomial of Sk\((n, \mathbb{C})\) is determined by applying a theorem of Dimca and Lehrer to the fibration \[ \text{Sp}(n, \mathbb{C}) \hookrightarrow\text{GL}(n, \mathbb{C})\to\text{Sk}(n, \mathbb{C}). \] Let Sym\((i, n, K)\) denote the variety of \(n\times n\) symmetric matrices over \(K\) with rank \(i\) and \(G(k, n, K)\) the Grassmannian variety of \(k\)-dimensional subspaces of \(K^n\). An inductive formula for the weight polynomial of Sym\((n, \mathbb{C})\) is found by using the fibration \[ \text{Sym}(i, \mathbb{C}) \hookrightarrow\text{Sym}(i, n, …
reviewed by: Oliver Jones
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14D10
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 950200
zbMATH Keywords: number of rational points
zbMATH Keywords: homotopy type
zbMATH Keywords: Grassmann variety
zbMATH Keywords: Betti numbers
zbMATH Keywords: complex algebraic variety
zbMATH Keywords: mixed Hodge structure
zbMATH Keywords: weight filtration
zbMATH Keywords: Hodge filtration
zbMATH Keywords: invariant
zbMATH Keywords: weight polynomial
zbMATH Keywords: Euler characteristic
MaRDI profile type: Publication
OpenAlex ID: W2122875942
DOI: 10.2140/PJM.1997.180.89
description: scientific article; zbMATH DE number 950200