The partial derivative of ratios of Schur polynomials and applications to symplectic quotients

2025/04/28 by Herbig, Hans-Christian, Herden, Daniel, Kolehmainen, Harper +1
#53D20 #Combinatorics (math.CO) #FOS: Mathematics #Primary 05E05 #Secondary 05A20 #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2504.19466

Abstract

We show that a ratio of Schur polynomials s_λ/s_ρ associated to partitions λ and ρ such λ\subsetneqρ has a negative partial derivative at any point where all variables are positive. This is accomplished by establishing an injective map between sets of pairs of skew semistandard Young tableaux that preserves the product of the corresponding monomials. We use this result and the description of the first Laurent coefficient of the Hilbert series of the graded algebra of regular functions on a linear symplectic quotient by the circle to demonstrate that many such symplectic quotients are not graded regularly diffeomorphic. In addition, we give an upper bound for this Laurent coefficient in terms of the largest two weights of the circle representation and demonstrate that all but finitely many circle symplectic quotients of each dimension are not graded regularly diffeomorphic to linear symplectic quotients by SU₂.

Citations

• Chevalley operations on TNN Grassmannians
• The diagonal derivative of a skew Schur polynomial
• When is a symplectic quotient an orbifold?
• The Hilbert series of a linear symplectic circle quotient
• On Orbifold Criteria for Symplectic Toric Quotients
• Schur positivity and Schur log-concavity
• Stratified Symplectic Spaces and Reduction

Related

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