Powers of binary forms and derived Hermite reciprocity

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 ${\mathrm{S}\mathrm{L}}_2$-representations. In our investigation, we are led to the ideal generated by the subrepresentation ${\mathrm{S}\mathrm{y}\mathrm{m}}^{ab}({\mathbb{C}}^2)\subset {\mathrm{S}\mathrm{y}\mathrm{m}}^a({\mathrm{S}\mathrm{y}\mathrm{m}}^b{\mathbb{C}}^2)$. We determine its Castelnuovo–Mumford regularity in general and the minimal free resolution for small values of b.