Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hans-Christian Herbig, Daniel Herden, Harper Kolehmainen, Christopher Seaton

TL;DR

This work proves that for partitions with $\lambda\subsetneq\rho$, the ratio $s_{\lambda}(\boldsymbol{x})/s_{\rho}(\boldsymbol{x})$ has strictly decreasing behavior in each variable $x_j>0$, with the partial derivatives expressible as ratios where the numerator is nonpositive and the denominator nonnegative. The authors achieve this via a combinatorial injection on pairs of skew semistandard Young tableaux, providing a self-contained, elementary proof. They then apply this result to the Hilbert series of linear circle quotients, giving upper bounds and monotonicity results for the first Laurent coefficient $\gamma_0$, and deducing finiteness statements: for any fixed dimension, only finitely many circle quotients can be graded regularly diffeomorphic to $\mathrm{SU}_2$ quotients. These findings offer a practical criterion to distinguish between circle and $\mathrm{SU}_2$ quotients and to identify non-equivalences at the level of graded regular structures.

Abstract

We show that a ratio of Schur polynomials $s_λ/s_ρ$ associated to partitions $λ$ and $ρ$ such that $λ\subsetneqρ$ 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 $\operatorname{SU}_2$.

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

TL;DR

This work proves that for partitions with , the ratio has strictly decreasing behavior in each variable , with the partial derivatives expressible as ratios where the numerator is nonpositive and the denominator nonnegative. The authors achieve this via a combinatorial injection on pairs of skew semistandard Young tableaux, providing a self-contained, elementary proof. They then apply this result to the Hilbert series of linear circle quotients, giving upper bounds and monotonicity results for the first Laurent coefficient , and deducing finiteness statements: for any fixed dimension, only finitely many circle quotients can be graded regularly diffeomorphic to quotients. These findings offer a practical criterion to distinguish between circle and quotients and to identify non-equivalences at the level of graded regular structures.

Abstract

We show that a ratio of Schur polynomials associated to partitions and such that 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 .
Paper Structure (8 sections, 12 theorems, 28 equations, 3 figures)

This paper contains 8 sections, 12 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainthm:PartDerDec']}
  3. The partial derivative in terms of skew Schur polynomials
  4. Reduction to the existence of an injective map
  5. A simple case
  6. An injective map in the case that $\lambda$ and $\rho$ differ by one box
  7. The general case
  8. An application to the Hilbert series of linear symplectic quotients

Key Result

Theorem 1

Let $N \geq n \geq 1$ be fixed integers. Let $\boldsymbol{x} = (x_1,x_2,\ldots,x_N)$ be a set of variables, and let $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_n)$ and $\rho = (\rho_1,\rho_2,\ldots,\rho_n)$ be partitions such that $\lambda_i \leq \rho_i$ for each $i$ and $\lambda_i < \rho_i$ for can be expressed as the ratio of polynomials where the coefficients of the denominator are nonnegat

Figures (3)

  • Figure 1: Illustration of the bijection $\boldsymbol{S}_{(0,1)}\to\boldsymbol{S}_{(1,0)}$.
  • Figure 2: An example of forming $(S_U^\prime, T_U^\prime)$ from $(S, T)$ where $U$ is the set of shaded positions in $S$. Each box in $U$ is replaced by the box in the same position in $T$, or removed if $T$ contains no box in this position, to form $T_U^\prime$. Similarly, each box in $U$ is placed in the same position in $T$ to form $S_U^\prime$. The boxes that have been moved between $S$ and $T_U^\prime$, and $T$ and $S_U^\prime$, respectively, are shaded in $S_U^\prime$ and $T_U^\prime$.
  • Figure 3: The skew SSYT pairs $(S_1, T_1)$ and $(S_2, T_2)$ as well as the pair $((S_1)_{U^\ast(S_1,T_1)}^\prime,(T_1)_{U^\ast(S_1,T_1)}^\prime) = ((S_2)_{U^\ast(S_2,T_2)}^\prime,(T_2)_{U^\ast(S_2,T_2)}^\prime)$ discussed in Example \ref{['ex:U*NotInject']}.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.6: Spanning tree
  • Lemma 2.7
  • proof
  • ...and 15 more