Characteristic classes of Borel orbits of square-zero upper-triangular matrices

TL;DR

This paper studies the geometry and topology of Borel orbits of square-zero upper-triangular matrices, providing formulas for their characteristic classes using combinatorial and algebraic tools, and relating them to Schubert polynomials.

Contribution

It introduces a new combinatorial construction for computing characteristic classes of these orbits, extending known results to a broader class of matrix varieties.

Findings

Derived explicit formulas for cohomological and K-theoretic classes of Borel orbits.

Connected the classes to double Schubert polynomials and trigonometric weight functions.

Provided a Bott-Samelson type resolution for orbit closures.

Abstract

Anna Melnikov provided a parametrization of Borel orbits in the affine variety of square-zero $n \times n$ matrices by the set of involutions in the symmetric group. A related combinatorics leads to a construction a Bott-Samelson type resolution of the orbit closures. This allows to compute cohomological and K-theoretic invariants of the orbits: fundamental classes, Chern-Schwartz-MacPherson classes and motivic Chern classes in torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig operations. The case of square-zero upper-triangular matrices is reach enough to include information about cohomological and K-theoretic classes of the double Borel orbits in $Hom(\mathbb C^k,\mathbb C^m)$ for $k+m=n$. We recall the relation with double Schubert polynomials and show analogous interpretation of Rim\'anyi-Tarasov-Varchenko trigonometric weight function.