Powers of binary forms and derived Hermite reciprocity
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908837332647936 |
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| author | Raicu, Claudiu Sam, Steven V Weyman, Jerzy Yang, Fuxiang |
| author_facet | Raicu, Claudiu Sam, Steven V Weyman, Jerzy Yang, Fuxiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Powers of binary forms and derived Hermite reciprocity Raicu, Claudiu Sam, Steven V Weyman, Jerzy Yang, Fuxiang Commutative Algebra Algebraic Geometry 13D02 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. |
| title | Powers of binary forms and derived Hermite reciprocity |
| topic | Commutative Algebra Algebraic Geometry 13D02 |
| url | https://arxiv.org/abs/2602.15175 |