The syzygies of some thickenings of determinantal varieties (Q2832806)

Let \(S=\text{Sym}(\mathbb{C}^m\otimes\mathbb{C}^n)(=\mathbb{C}[z_{ij}])\) be the ring of polynomial functions on the vector space of \(m\times n\) matrices with entries in the complex numbers and \(m \geq n\). \(S\) admits an action of the group \(\mathrm{GL}=\mathrm{GL}_m(\mathbb{C})\times \mathrm{GL}_n(\mathbb{C})\), and it decomposes into irreducible \(\mathrm{GL}\)--representations according to Cauchy's formula: NEWLINE\[NEWLINE S=\bigoplus_{\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\geq 0)} S_{\lambda}\mathbb{C}^m\otimes S_{\lambda}\mathbb{C}^n, NEWLINE\]NEWLINE where \(S_{\lambda}\) denotes the Schur functor associated to a partition \(\lambda\). For each \(\lambda\), let \(I_{\lambda}\) denote the ideal in \(S\) generated by the irreducible representation \(S_{\lambda}\mathbb{C}^m\otimes S_{\lambda}\mathbb{C}^n\). The paper deals with the problem to describe the syzygies of the ideals \(I_{\lambda}\), and their \(\mathrm{GL}\)-equivariant structure. The main result of the paper is a short solution to this problem in the case \(\lambda\) is a rectangular partition which means that the Young diagram associated to \(\lambda \) is the \(a\times b\) rectangle. In this situation let \(I_{\lambda} = I_{a\times b}\) and let \(Z = (z_{i,j})\) the generic matrix of indeterminates. Examples of ideals \(I_{a\times b}\) are among others: (1) \(I_{a\times 1}=I_a\), the ideal generated by the \(a\times a\) minors of \(Z\), (2) \(I_{n\times b}=I_n^b\), the \(b\)-th power of the ideal \(I_n\) of maximal minors of \(Z\) and (3) \(I_{1\times b}\), the ideal of \(b\times b\) permanents of \(Z\). -- The authors' methods are a connection between commutative algebra and the representation theory of the superalgebra \(\mathfrak{gl}(m|n)\) and their calculations of \(\text{Ext}\) modules done in the context of describing local cohomology with determinantal support (see [\textit{C. Raicu} and \textit{J. Weyman}, Algebra Number Theory 8, No. 5, 1231--1257 (2014; Zbl 1303.13018)]).