Combinatorial Models for the Variety of Complete Quadrics

Abstract

We develop several combinatorial models that are useful in the study of the $S{L}_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations parameterize the fixed points of the action of a maximal torus $T$ of $S{L}_n$, while $\hat{I}¼$-involutions parameterize the orbits of a Borel subgroup of $S{L}_n$. Using these combinatorial objects, we characterize the $T$-stable curves and surfaces on $\mathcal{X}$, compute the $T$-equivariant $K$-theory of $\mathcal{X}$, and describe a Białynicki-Birula cell decomposition for $\mathcal{X}$. Furthermore, we give a computational characterization of the Bruhat order on Borel orbits in $\mathcal{X}$.

Completely rewritten. Comments welcome!