Wronskians, cyclic group actions, and ribbon tableaux

TL;DR

This paper explores the fixed points of the Wronski map under cyclic and dihedral group actions, linking algebraic geometry with combinatorics through ribbon tableaux and providing new proofs of cyclic sieving phenomena.

Contribution

It computes the number of fixed points in fibers of the Wronski map under group actions using ribbon tableaux and relates these counts to cyclic sieving, offering new combinatorial proofs.

Findings

Number of fixed points expressed via r-ribbon tableaux

Connection between fixed points and promotion-invariant tableaux

New proof of Rhoades' cyclic sieving theorem

Abstract

The Wronski map is a finite, PGL_2(C)-equivariant morphism from the Grassmannian Gr(d,n) to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If C_r < PGL_2(C) is a cyclic subgroup of order r, how may C_r-fixed points are in the in a fibre of the Wronski map over a C_r-fixed point in the base? In this paper, we compute a general answer in terms of r-ribbon tableaux. When r=2, this computation gives the number of real points in the fibre of the Wronski map over a real polynomial with purely imaginary roots. More generally, we can compute the number of real points in certain intersections of Schubert varieties. When r divides d(n-d) our main result says that the generic number of C_r-fixed points in the fibre is the number of standard r-ribbon tableaux rectangular shape (n-d)^d. Computing by a different method, we show that the answer in this case is also given by the number of of standard Young tableaux of shape (n-d)^d that are invariant under N/r iterations of jeu de taquin promotion. Together, these two results give a new proof of Rhoades' cyclic sieving theorem for promotion on rectangular tableaux. We prove analogous results for dihedral group actions.