Séminaire Lotharingien de Combinatoire, 91B.20 (2024), 12 pp.
Steven N. Karp and Kevin Purbhoo
Universal Plücker Coordinates for the Wronski Map and Positivity in Real Schubert Calculus
Abstract.
Given a d-dimensional vector space V ⊂ C[u], its
Wronskian is the polynomial (u+z1)
... (u+zn) whose zeros
-zi are the points of C such that V contains a nonzero
polynomial with a zero of order at least d at
-zi. Equivalently,
V is a solution to the Schubert problem defined by osculating planes
to the moment curve at z1,
..., zn. The inverse Wronski
problem involves finding all V with a given Wronskian. We solve
this problem by providing explicit formulas for the
Grassmann-Plücker coordinates of the general solution V, as
commuting operators in the group algebra C[Sn]
of the symmetric group. The Plücker coordinates of individual
solutions over C are obtained by restricting to an
eigenspace and replacing each operator by its eigenvalue. This
generalizes work of Mukhin, Tarasov, and Varchenko (2013) and of
Purbhoo (2022), which give formulas in C[Sn]
for the differential equation satisfied by V. Moreover,
if z1,
..., zn are real and nonnegative, then our operators are positive
semidefinite, implying that the Plücker coordinates of V are all
real and nonnegative. This verifies several outstanding conjectures in
real Schubert calculus, including the positivity conjectures of Mukhin
and Tarasov (2017) and of Karp (2021), the disconjugacy conjecture of
Eremenko (2015), and the divisor form of the secant conjecture of
Sottile (2003). The proofs involve the representation theory of
Sn, symmetric functions, and τ-functions of the KP
hierarchy.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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