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Powers of binary forms and derived Hermite reciprocity

Claudiu Raicu, Steven V Sam, Jerzy Weyman, Fuxiang Yang

TL;DR

This paper determines the ideal-theoretic description of the coincident-root locus $X_{(a^b)}$ of $a$-th powers of binary forms of degree $b$, proving that $I(X)$ is generated in degree $b{+}1$ by $(b{+}1)$-minors of a specific linear map and has a linear resolution with ${\rm pd}(I(X))=d-1$, where $d=ab$. The authors develop a derived Hermite reciprocity to handle complexes of SL$_2$-representations, proving a self-duality phenomenon for symmetric powers that underpins the maximal-rank property of the Foulkes–Howe maps. They establish the determinantal structure of $I(X)$, compute the full algebraic Betti numbers, and give Hilbert-function formulas for $I(X)$, along with explicit resolutions in small $b$ (notably via instanton techniques for $b=3$). Additional results include a complete regularity analysis of $I_{a,b}$ and a detailed description of $I_{a,b}^{b-1}$, illustrating deep connections between invariant theory, elimination theory, and representation theory of SL$_2$ in the context of binary forms.

Abstract

For $a,b \ge 1$, Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X, showing that it is generated in degree b+1 and that it has a linear minimal free resolution. We do this by generalizing results of Abdesselam and Chipalkatti on an analogue of the Foulkes--Howe map and by establishing a derived analogue of the classical Hermite reciprocity theorem for complexes of ${\rm SL}_2$-representations. In our investigation, we are led to the ideal generated by the subrepresentation ${\rm Sym}^{ab}({\Bbb C}^2) \subset {\rm Sym}^a({\rm Sym}^b {\Bbb C}^2)$. We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.

Powers of binary forms and derived Hermite reciprocity

TL;DR

This paper determines the ideal-theoretic description of the coincident-root locus of -th powers of binary forms of degree , proving that is generated in degree by -minors of a specific linear map and has a linear resolution with , where . The authors develop a derived Hermite reciprocity to handle complexes of SL-representations, proving a self-duality phenomenon for symmetric powers that underpins the maximal-rank property of the Foulkes–Howe maps. They establish the determinantal structure of , compute the full algebraic Betti numbers, and give Hilbert-function formulas for , along with explicit resolutions in small (notably via instanton techniques for ). Additional results include a complete regularity analysis of and a detailed description of , illustrating deep connections between invariant theory, elimination theory, and representation theory of SL in the context of binary forms.

Abstract

For , Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X, showing that it is generated in degree b+1 and that it has a linear minimal free resolution. We do this by generalizing results of Abdesselam and Chipalkatti on an analogue of the Foulkes--Howe map and by establishing a derived analogue of the classical Hermite reciprocity theorem for complexes of -representations. In our investigation, we are led to the ideal generated by the subrepresentation . We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.
Paper Structure (18 sections, 13 theorems, 122 equations)

This paper contains 18 sections, 13 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. A presentation for $\mathcal{O}_{{\mathbf P}^b}(a)$
  5. Maximal minors and hypercohomology
  6. Graded modules associated to coherent sheaves
  7. A self-duality theorem
  8. Character computation
  9. Equations and syzygies for $X_{(a^b)}$
  10. The Foulkes--Howe map has maximal rank
  11. The ideal of $X$ has a linear resolution
  12. The ideal of $X$ is determinantal
  13. The algebraic Betti numbers of $X$
  14. Examples
  15. Regularity and syzygies of some primary ideals
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $a,b\geq 2$, set $d=ab$, and $X=X_{(a^b)}$. The homogeneous ideal $I(X)$ is generated by polynomials of degree $b+1$ (the maximal minors of a matrix of linear forms), and its minimal free resolution is linear. The projective dimension of $I(X)$ is $(d-1)$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.4
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • ...and 17 more