Universal Pl\"ucker coordinates for the Wronski map and positivity in real Schubert calculus
Steven N. Karp, Kevin Purbhoo

TL;DR
This paper provides explicit formulas for solutions to the inverse Wronski problem using symmetric group algebra operators, confirming positivity conjectures in real Schubert calculus.
Contribution
It introduces a novel approach to solving the inverse Wronski problem via Grassmann-Plücker coordinates as symmetric group algebra operators, extending previous work and verifying key positivity conjectures.
Findings
Explicit formulas for Grassmann-Plücker coordinates of solutions
Operators are positive semidefinite for real nonnegative parameters
Verifies several conjectures in real Schubert calculus
Abstract
Given a -dimensional vector space of polynomials, its Wronskian is the polynomial whose zeros are the points of such that contains a nonzero polynomial with a zero of order at least at . Equivalently, is a solution to the Schubert problem defined by osculating planes to the moment curve at . The inverse Wronski problem involves finding all with a given Wronskian . We solve this problem by providing explicit formulas for the Grassmann-Pl\"ucker coordinates of the general solution , as commuting operators in the group algebra of the symmetric group. The Pl\"ucker coordinates of individual solutions over are obtained by restricting to an eigenspace and replacing each operator by its eigenvalue. This…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Mathematical functions and polynomials · Advanced Combinatorial Mathematics
