390 research outputs found
Jeu de taquin and a monodromy problem for Wronskians of polynomials
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.Comment: 37 pages, 3 examples containing figures; detailed example of main
theorem added, corrections and clarifications made to some proofs, other
minor revision
The Wronski map and shifted tableau theory
The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a
number of interesting consequences. Among them is a well-behaved correspondence
between certain points on a Grassmannian - those sent by the Wronski map to
polynomials with only real roots - and (dual equivalence classes of) Young
tableaux.
In this paper, we restrict this correspondence to the orthogonal Grassmannian
OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and
only if the corresponding tableau has a certain type of symmetry. From this we
recover much of the theory of shifted tableaux for Schubert calculus on
OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule
for OG(n,2n+1).Comment: 11 pages, color figures, identical to v1 but metadata correcte
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