Wronskians, cyclic group actions, and ribbon tableaux

Abstract:

The Wronski map is a finite, $\mathrm {PGL}_2(\mathbb {C})$-equivariant morphism from the Grassmannian $\mathrm {Gr}(d,n)$ to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If $C_r \subset \mathrm {PGL}_2(\mathbb {C})$ is a cyclic subgroup of order $r$, how may $C_r$-fixed points are in the 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 of 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 standard Young tableaux of shape $(n{-}d)^d$ that are invariant under $\frac {N}{r}$ 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.