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Wronskians, cyclic group actions, and ribbon tableaux

Author: Kevin Purbhoo
Journal: Trans. Amer. Math. Soc. 365 (2013), 1977-2030
MSC (2010): Primary 14N10; Secondary 05E10, 14P05
Published electronically: October 24, 2012
MathSciNet review: 3009651
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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.

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Additional Information

Kevin Purbhoo
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

Received by editor(s): April 20, 2011
Received by editor(s) in revised form: July 26, 2011
Published electronically: October 24, 2012
Additional Notes: This research was partially supported by an NSERC discovery grant.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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