An equivariant basis for the cohomology of Springer fibers

Abstract

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for GL n (C) using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.

The "Young" and "Reverse" Dichotomy of Polynomials

“…In a similar vein, Young analogues of pre-existing reverse bases of QSym n were applied in the study of q-analogues of combinatorial Hopf algebras [Li15] and skew variants of quasisymmetric bases [MN15] to take advantage of classical combinatorics in Sym n concerning Schur functions and Young tableaux. This type of relabelling is also used in [PR21] (there called "shifting") to simplify arguments relating to the equivariant cohomology of Springer fibers for GL n (C). We are motivated by the utility of the flip-and-reverse perspective to explore and develop further Young analogues of bases of Poly n and establish structural results.…”