Combinatorial Models for the Variety of Complete Quadrics

TL;DR

This paper introduces combinatorial models like barred permutations and μ-involutions to analyze the geometry and equivariant K-theory of the variety of complete quadrics under group actions.

Contribution

It develops new combinatorial tools to study the fixed points, orbits, and cell decompositions of the complete quadrics variety, advancing understanding of its geometric and algebraic structure.

Findings

Parameterization of fixed points by barred permutations

Characterization of T-stable curves and surfaces

Computation of equivariant K-theory and orbit orderings

Abstract

We develop several combinatorial models that are useful in the study of the $SL_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations parameterize the fixed points of the action of a maximal torus $T$ of $SL_n$, while $\mu$-involutions parameterize the orbits of a Borel subgroup of $SL_n$. 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 Bia{\l}ynicki-Birula cell decomposition for $\mathcal{X}$. Furthermore, we give a computational characterization of the Bruhat order on Borel orbits in $\mathcal{X}$.