The equivariant Euler characteristic of real Coxeter toric varieties

Anthony Henderson and Gus Lehrer

Abstract

Let W be a crystallographic Weyl group, and let T_W be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of W, and its weight lattice. The real locus T_W(R) is a smooth, connected, compact manifold with a W-action. We give a formula for the Euler characteristic of T_W(R) as a generalised character of W. In type A_{n-1} for n odd, one obtains a generalised character of Sym_n whose degree is (up to sign) the nth Euler number.

Keywords: real toric variety, Euler characteristic, symmetric group.

AMS Subject Classification: Primary 14P25; secondary 20C30.