Powers of binary forms and derived Hermite reciprocity

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 $${\rm Sym}^{ab}({\Bbb C}²⁾ \subset {\rm Sym}^a({\rm Sym}^b {\Bbb C}<sup>2)$$. We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.

• The paper proves that the ideal of the binary-form power variety X_(a^b) is generated by (b+1)-minors, has a linear minimal free resolution, and has projective dimension ab−1.
• The authors establish maximal rank for every Foulkes–Howe map, with injectivity through degree b and surjectivity from degree b, using cohomological vanishing and a derived self-duality of complexes.
• The paper computes the exact regularity of the primary ideals I_(a,b), gives resolutions in key cases, and identifies open problems for general resolutions and intermediate powers.

The variety $X_{(a^b)} \subset \mathbb{P}^{ab}$ parametrizing $a$-th powers of binary forms of degree $ab$ was known, since Hilbert's 1886 paper on covariant criteria for complete powers, only set-theoretically: Hilbert's covariant equations vanish precisely on this locus but their ideal-theoretic content was unresolved. The paper under review, by Raicu, Sam, Weyman, and Yang (2602.15175), settles the corresponding ideal-theoretic problem completely and, in doing so, proves a derived analogue of Hermite reciprocity that is of independent representation-theoretic interest.

The main results

Let $U = \mathbb{C}^2$, $\mathbb{P}^d = \mathbb{P}(Sym^d U)$, and let $X = X_{(a^b)}$ be the image of the morphism $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$ sending $[G] \mapsto [G^a]$ (a closed immersion). The central theorem states that for $a,b \geq 2$, setting $d = ab$:

• the homogeneous ideal $a$0 is generated in degree $a$1, namely by the $a$2-minors of a matrix of linear forms;
• its minimal free resolution is linear;
• the projective dimension of $a$3 is $a$4.

This confirms Conjecture 5.1 of Abdesselam–Chipalkatti's work on Hilbert covariants. The number of minimal generators is computed explicitly as

$a$5

and all Betti numbers are determined by a closed alternating-sum formula. In particular $a$6 for $a$7, so the coordinate ring has positive depth but fails to be Cohen–Macaulay; the linear resolution therefore does not arise from an Eagon–Northcott complex. The authors emphasize that even in the simplest case $a$8, where $a$9 is generated by the $ab$0-minors of a $ab$1 matrix of linear forms, the Eagon–Northcott prediction (10 cubics with 9 linear syzygies) differs sharply from the actual Betti table, which begins with $ab$2 cubic generators — the determinantal structure here is highly non-generic.

Derived Hermite reciprocity and maximal rank of the Foulkes–Howe maps

The ideal-theoretic statement reduces to a maximal rank property of the Foulkes–Howe maps

$ab$3

the restriction maps induced by the parametrization $ab$4. The second main theorem asserts that $ab$5 is injective for $ab$6 and surjective for $ab$7. Injectivity below degree $ab$8 shows $ab$9 has no equations of degree $U = \mathbb{C}^2$0; surjectivity from degree $U = \mathbb{C}^2$1 bounds the regularity of the coordinate ring via Cox–Materov regularity of the Veronese ring $U = \mathbb{C}^2$2, giving generation in degree $U = \mathbb{C}^2$3.

At the critical level $U = \mathbb{C}^2$4, the map $U = \mathbb{C}^2$5 is an isomorphism between two spaces abstractly isomorphic by classical Hermite reciprocity. Notably, $U = \mathbb{C}^2$6 is not the classical Hermite isomorphism: in the case $U = \mathbb{C}^2$7, where both sides decompose as $U = \mathbb{C}^2$8, the eigenvalue triples are $U = \mathbb{C}^2$9 versus $\mathbb{P}^d = \mathbb{P}(Sym^d U)$0, so the two isomorphisms are non-proportional. This distinction matters: the authors point out that prior arguments in the literature (e.g., Faenzi's Proposition 3.11) appeal to Hermite reciprocity without specifying which isomorphism is used, and their remark makes precise which instance is required.

The proof of maximality of $\mathbb{P}^d = \mathbb{P}(Sym^d U)$1 is cohomological. One considers the $\mathbb{P}^d = \mathbb{P}(Sym^d U)$2-equivariant presentation

$\mathbb{P}^d = \mathbb{P}(Sym^d U)$3

whose cokernel is $\mathbb{P}^d = \mathbb{P}(Sym^d U)$4, and forms symmetric powers $\mathbb{P}^d = \mathbb{P}(Sym^d U)$5. The map $\mathbb{P}^d = \mathbb{P}(Sym^d U)$6 is the map on global sections induced by the degree-zero differential of the extended complex $\mathbb{P}^d = \mathbb{P}(Sym^d U)$7, so it suffices to prove the hypercohomology of $\mathbb{P}^d = \mathbb{P}(Sym^d U)$8 vanishes. The key technical input is a self-duality theorem: there is a $\mathbb{P}^d = \mathbb{P}(Sym^d U)$9-equivariant isomorphism of complexes

$X = X_{(a^b)}$0

which term-by-term reads as a family of Hermite-type isomorphisms

$X = X_{(a^b)}$1

and whose compatibility with differentials is the difficult part. The proof passes through the graph $X = X_{(a^b)}$2 of the embedding, identifies $X = X_{(a^b)}$3, pushes forward Koszul complexes along the projection, and concludes using Beĭlinson's equivalence between minimal free complexes over the coordinate ring of $X = X_{(a^b)}$4 and bounded derived categories of coherent sheaves. Combined with quasi-isomorphisms $X = X_{(a^b)}$5 for $X = X_{(a^b)}$6 and $X = X_{(a^b)}$7, this yields the vanishing of hypercohomology and hence maximality of all Foulkes–Howe maps.

The character computation behind the self-duality suggests the phenomenon extends beyond $X = X_{(a^b)}$8: the termwise characters agree as products of three $X = X_{(a^b)}$9-binomial coefficients, indicating a larger family of complex-level isomorphisms. The authors state this as a suggestion rather than a theorem.

Regularity of the primary ideals $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$0

Viewing $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$1 as cut out by the subrepresentation $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$2 leads to the primary ideal $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$3. Its Castelnuovo–Mumford regularity is determined exactly:

$\nu_a : \mathbb{P}^b \to \mathbb{P}^d$4

The proof combines Weyman's Gröbner basis description of $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$5 (generated by monomials $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$6 and $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$7) with an explicit extremal monomial of degree $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$8 surviving in the quotient, showing the Hilbert function drops off exactly at the predicted bound. As a corollary for $\nu_a : \mathbb{P}^b \to \mathbb{P}^d$9, one recovers Faenzi's surjectivity statement concerning multiplication maps involving $[G] \mapsto [G^a]$0.

Two partial resolutions are established. For powers, the resolution of $[G] \mapsto [G^a]$1 is given explicitly, with Betti numbers concentrated on two linear strands at degrees $[G] \mapsto [G^a]$2 and $[G] \mapsto [G^a]$3:

$[G] \mapsto [G^a]$4

In particular $[G] \mapsto [G^a]$5, confirming the conjectural formula for $[G] \mapsto [G^a]$6. Note also that $[G] \mapsto [G^a]$7 being an isomorphism implies $[G] \mapsto [G^a]$8 where $[G] \mapsto [G^a]$9 is the maximal homogeneous ideal — the power $a,b \geq 2$0 case of the conjecture holds automatically.

For $a,b \geq 2$1, the resolution of $a,b \geq 2$2 connects to instanton bundles: dualizing the presentation sequence yields the unique $a,b \geq 2$3-equivariant instanton bundle $a,b \geq 2$4 on $a,b \geq 2$5 with second Chern class $a,b \geq 2$6, which admits a linear monad by Faenzi's classification. A mapping-cone construction produces the minimal resolution

$a,b \geq 2$7

with $a,b \geq 2$8, consistent with the general formula. The second linear strand arises from the monad structure, and the syzygy modules carry explicit $a,b \geq 2$9-module structures.

Limitations and open questions

Several points remain open within the paper's own scope. The minimal free resolution of $d = ab$0 is described only for $d = ab$1; the general case is unknown. Conjecturally,

$d = ab$2

with the additional assertion that minimal generators of $d = ab$3 form a Gröbner basis; this is proved for $d = ab$4 and $d = ab$5 and follows for $d = ab$6, but intermediate $d = ab$7 are open. The self-duality is proved only when $d = ab$8 divides $d = ab$9; the character computation indicates an extension to arbitrary $a$00 and a larger family of multi-$a$01-binomial isomorphisms, neither of which is constructed. Finally, the connection between resolutions of $a$02 and coincident root loci $a$03 for hook partitions $a$04 is indicated but not developed.

Conclusion

The paper resolves the ideal-theoretic version of Hilbert's problem on complete powers of binary forms, proving that $a$05 is generated by $a$06-minors of a natural matrix of linear forms and has a linear resolution of projective dimension $a$07, while determining all Foulkes–Howe maps have maximal rank. The mechanism is a genuinely new derived refinement of Hermite reciprocity — an isomorphism between symmetric and exterior power complexes compatible with differentials — together with a precise regularity computation for the associated primary ideals. The remaining gaps, chiefly the full resolution of $a$08 and the conjectured regularity of its intermediate powers, are stated precisely enough to serve as well-defined targets for further work.