Wronskians, cyclic group actions, and ribbon tableaux
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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
- Email: kpurbhoo@math.uwaterloo.ca
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 1977-2030
- MSC (2010): Primary 14N10; Secondary 05E10, 14P05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05676-5
- MathSciNet review: 3009651