On some modules supported in the Chow variety (Q2135090)

The Chow variety \(Y_{d,n}\) is the set of homogeneous polynomials of degree \(d\) in \(n+1\) variables which decompose into a product of linear forms. The standard parameterization of \(Y_{d,n}\) realizes the coordinate ring \(\pmb k[Y_{d,n}]\) as a subalgebra of \(A_{d,n}\), which is the homogeneous coordinate ring of the Segre product \((\mathbb P^n)^{\times d}\). There is a natural action of the symmetric group \(\mathfrak S_n\) on \(A_{d,n}\) and the invariant subring \(B_{d,n}=A_{d,n}^{\mathfrak S_n}\) is the normalization of \(\pmb k[Y_{d,n}]\). The normalization map has connections to Foulkes' conjecture [\textit{H. O. Foulkes}, J. Lond. Math. Soc. 25, 205--209 (1950; Zbl 0037.14902)] about plethysm coefficients in algebraic combinatorics. Let \(\lambda\) be a partition of \(d\) and \[ M_\lambda=\operatorname{Hom}_{\mathfrak S_d}(V_\lambda,A_{d,n}), \] where \(V_\lambda\) is the irreducible \(\mathfrak S_d\)-representation which corresponds to \(\lambda\). In characteristic zero, \(M_\lambda\) is a maximal Cohen-Macaulay module supported on \(Y_{d,n}\). The space \(Y_{d,1}\) is affine, so the modules \(M_\lambda\) are free in this case, but the description of their generators involves interesting combinatorics related to the statistics of descents and major indices, and to Kostka-Foulkes polynomials. One way to interpret the formulas in the case \(n = 1\) is as generalizations of Hermite Reciprocity, which states that if the dimension of the vector space \(U\) is two, then, for all non-negative \(a\) and \(b\), the \(\operatorname{GL}(U)\)-representations \[ \operatorname{Sym}_a(\operatorname{Sym}_bU)\text{ and }\operatorname{Sym}_b(\operatorname{Sym}_aU) \] are isomorphic. For general parameters \(d\) and \(n\), the paper takes a first step in the study of the syzygies of \(M_\lambda\) by establishing a bound on their Castelnuovo-Mumford regularity. The paper gives a flavor of the questions and results surrounding Chow varieties, with the intention that future research will be sparked in this area.