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Hermite Reciprocity and Self-Duality of Generalized Eagon-Northcott Complexes

Ethan Reed

TL;DR

This work establishes a new self-duality instance for generalized Eagon-Northcott complexes by exploiting a special Hermite reciprocity map in the setting of binary forms. It constructs an embedding of a $\Sym^{b-1}(\varphi|_{V_1 \otimes S})$-based complex into a minimal free resolution, translates the problem via the Bernstein-Gel'fand-Gel'fand correspondence to $E$-modules, and derives a module-theoretic isomorphism $P(V_1) \cong \hat{P}(V_1)(-(b-1))$ that embodies Hermite reciprocity. A key contribution is the introduction of a special Hermite isomorphism $\psi_{b,0}(V_1)$ (not equivalent to previously known formulations) that can be realized explicitly in low dimensions and is used to construct an $E$-module morphism reflecting self-duality. The results connect to Weyman modules and Green's conjecture, extending known self-duality phenomena and providing explicit maps between module structures that illuminate the role of Hermite reciprocity in this setting.

Abstract

Previous examples of self-duality for generalized Eagon-Northcott complexes were given by computing the divisor class group for Hankel determinantal rings. We prove a new case of self-duality of generalized Eagon-Northcott complexes with input being a map defining a Koszul module with nice properties. This choice of Koszul module can be specialized to the Weyman module, which was used in a proof of the generic version of Green's conjecture. In this case, the proof uses a version of Hermite Reciprocity not previously defined in the literature.

Hermite Reciprocity and Self-Duality of Generalized Eagon-Northcott Complexes

TL;DR

This work establishes a new self-duality instance for generalized Eagon-Northcott complexes by exploiting a special Hermite reciprocity map in the setting of binary forms. It constructs an embedding of a -based complex into a minimal free resolution, translates the problem via the Bernstein-Gel'fand-Gel'fand correspondence to -modules, and derives a module-theoretic isomorphism that embodies Hermite reciprocity. A key contribution is the introduction of a special Hermite isomorphism (not equivalent to previously known formulations) that can be realized explicitly in low dimensions and is used to construct an -module morphism reflecting self-duality. The results connect to Weyman modules and Green's conjecture, extending known self-duality phenomena and providing explicit maps between module structures that illuminate the role of Hermite reciprocity in this setting.

Abstract

Previous examples of self-duality for generalized Eagon-Northcott complexes were given by computing the divisor class group for Hankel determinantal rings. We prove a new case of self-duality of generalized Eagon-Northcott complexes with input being a map defining a Koszul module with nice properties. This choice of Koszul module can be specialized to the Weyman module, which was used in a proof of the generic version of Green's conjecture. In this case, the proof uses a version of Hermite Reciprocity not previously defined in the literature.

Paper Structure

This paper contains 6 sections, 13 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Embedding $\hbox{Sym}^{b-1} \varphi$ into an exact complex
  4. Bernstein-Gel'fand-Gel'fand Correspondence and Modules Generated in Lowest Degree
  5. Morphisms of $E$-modules via Hermite Reciprocity
  6. Special Hermite Isomorphism

Key Result

Theorem 1.1

Let $U$ be the standard representation of $\hbox{SL}_{2}(\mathbb{C})$, $b \geq 2$, and $S = \hbox{Sym}^{\bullet}(\hbox{Sym}^bU)$. Let $\varphi$ be the linear map defined by the inclusion of representations of second highest weight Then the complex $\hbox{Sym}^{b-1}(\varphi)$ is self-dual.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 19 more