On the generalized multiplicities of maximal minors and sub-maximal pfaffians

TL;DR

This paper investigates the asymptotic multiplicities of maximal minors and sub-maximal pfaffians in polynomial rings, revealing their connection to the degrees of Grassmannian and Orthogonal Grassmannian varieties.

Contribution

It introduces a representation-theoretic approach to analyze the asymptotic behavior of local cohomology multiplicities for determinantal and pfaffian ideals, linking them to classical geometric degrees.

Findings

Multiplicity matches degrees of Grassmannian varieties.

Asymptotic behavior of local cohomology is characterized.

Provides a new perspective on multiplicities via representation theory.

Abstract

Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ generic variables (resp. a polynomial ring of $(2n+1) \times (2n+1)$ skew-symmetric variables) over $\mathbb{C}$ and let $I$ (resp. Pf) be the determinantal ideal of maximal minors (resp. sub-maximal pfaffians) of $S$. Using the representation theoretic techniques introduced in the work of Raicu et al, we study the asymptotic behavior of the length of the local cohomology module of determinantal and pfaffian thickenings for suitable choices of cohomological degrees. This asymptotic behavior is also defined as a notion of multiplicty. We show that the multiplicity in our setting coincides with the degrees of Grassmannian and Orthogonal Grassmannian.