Initial terms, products of determinants, and the Calculus of Shapes.

Brian D. Taylor
Wayne State University

A generalized { shape} such as

1 2 3 4 5 6
1     x x   x
2 x   x x
3     x
4 x

is a collection of cells in a matrix. Just as { partition} shapes such as

1 2 3 4 5 6
1 x x x x x x
2 x x x x
3 x x x
4 x

are associated to representations of $S_n$ and $GL_n$, so is each generalized shape. If $D$ is a shape, the associated representation ${\cal S}^D$ may be constructed as the linear span inside the polynomial ring ${\bf C}[x_{i,j}]$ of certain products of determinants of minors in the matrix $(x_{i,j})$. Having made this construction, one can order the monomials of ${\bf C}[x_{i,j}]$ and ask to \begin{enumerate} \item %(1) Describe which monomials can show up as the smallest (or {\em initial}) monomial of some $p\in {\cal S}^D$? \item %(2) Produce a list of polynomials $p_i\in {\cal S}^D$ whose initial monomials include all initial monomials of ${\cal S}^D$. \item %(3) Select from the above list of polynomials an explicit basis for ${\cal S}^D$. \end{enumerate}

In this talk, I describe how (under suitable monomial orders) to fully answer the above questions when $D$ is a row-convex'' shape, or even when $D$ is replaced with a union of row-convex'' shapes. I conjecture two structural results concerning the answer to problem~(1) and I describe how to verify these conjectures on a different class of shapes using some new techniques of Bruns and Conca.

Applications include {\sc sagbi}-basis'' algorithms for the coordinate rings of some configuration varieties and (via a result of Sturmfels) some easy proofs that some of these rings are Cohen-Macaulay.