Linear spaces, transversal polymatroids and ASL domains (Q857751)

Let \(V_1,\dots,V_m\) be vector spaces of linear forms in \(R=K[x_1,\dots,x_n]\) of dimensions \(d=d_1,\dots,d_m\), \(K\) an infinite field, and \(A(V)\) the \(k\)-subalgebra of \(R\) generated by the elements of the product \(V_1\dots V_m\). The author proves several results about the presentation of \(A(V)\) as algebra with straightening law (ASL). In the monomial case, i.e., if the \(V_i\) are generated by variables in subsets \(C_i\), \(A(V)\) is the quotient of a Segre product by an ideal generated by a set of binomials that is in fact a Gröbner basis. Hence \(A(V)\) is Koszul (thm.\ 3.5). \(A(V)\) is explicitly described as a factor of the Hibi ring associated with the lattice \(C=C_1\times\dots\times C_m\) (prop.\ 3.7). Let \(t_{ij}, 1\leq i\leq m, 1\leq j\leq n\) be distinct variables, \(a_{ijk}\in K\) and \(L=(L_{ij})\) the \((m\times n)\) matrix with entries \(L_{ij}=\sum_k{a_{ijk}t_{ik}}\). A conjecture concerns the initial ideal of \(I_2(L)\), the ideal of the 2-minors of \(L\), to be square-free. \(L\) is close related to vector bases of \(V_1,\dots,V_m\). For generic \(V_1,\dots,V_m\) the matrix \(L\) is generic, the conjecture holds (thm.\ 5.1) and a basis of \(I_2(L)\) can be given that is a Gröbner basis with respect to any term order with \(t_{ij+1}>t_{ij}\) for all \(j\) (cor.\ 5.3). Moreover (thm.\ 5.7), in this case \(A(V)\) is a homogeneous ASL over the sublattice \(H_n(d)=\left\{(a_1,\dots,a_m):1\leq a_i\leq d_i, \sum_i{a_i}-m<n\right\}\) of \(\mathbb N^m\) and hence normal, Cohen-Macaulay and Koszul, since \(H_n(d)\) is a wonderful poset.