Jeu de taquin and a monodromy problem for Wronskians of polynomials

TL;DR

This paper explores the monodromy of the Wronskian map from Grassmannians to projective space, revealing connections to combinatorial jeu de taquin and providing new geometric insights into classical algebraic results.

Contribution

It establishes a link between the monodromy of Wronskians and jeu de taquin, offering new geometric interpretations and proofs of key combinatorial rules.

Findings

Monodromy is encoded by jeu de taquin when roots are real.

Provides geometric proofs of the Littlewood-Richardson rule.

Connects algebraic geometry with combinatorial jeu de taquin theory.

Abstract

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schutzenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.