Cyclic sieving of finite Grassmannians and flag varieties

TL;DR

This paper demonstrates instances of the cyclic sieving phenomenon in finite Grassmannians and flag varieties under torus actions, using polynomials derived from symmetric group coset weights.

Contribution

It establishes new cases of cyclic sieving for geometric structures acted upon by finite groups, linking combinatorics and algebraic geometry.

Findings

Proves cyclic sieving for finite Grassmannians.

Extends cyclic sieving to partial flag varieties.

Uses weights of symmetric group coset representatives.

Abstract

In this paper we prove instances of the cyclic sieving phenomenon for finite Grassmannians and partial flag varieties, which carry the action of various tori in the finite general linear group GL_n(F_q). The polynomials involved are sums of certain weights of the minimal length parabolic coset representatives of the symmetric group S_n, where the weight of a coset representative can be written as a product over its inversions.