Compressed straight tableaux and a distributive lattice of representations
The paper under review studies the algebraic and combinatorial properties of tableaux which are row-convex and are more general than the Ferrers diagrams of skew partitions. The first main result of the paper is that the set of polynomials associated with compressed straight tableaux forms a basis for the ring of polynomials \({\mathbb{Z}}[t_{ij}\mid 1\leq i\leq m, j\geq 1]\). This basis involves Gröbner bases ideas and is used in the study of the representations of \(\text{GL}_m({\mathbb{Z}})\) corresponding to row-convex shapes. In particular, the basis enjoys the remarkable property that every ``row-convex'' GL\(_m({\mathbb{Z}})\)-submodule of \({\mathbb{Z}}[t_{ij}]\) has a basis of ``compressed straight'' polynomials. This implies that the lattice generated by row-convex representations under intersection and linear span is distributive. The author provides some ring-theoretic applications and studies the Cohen-Macaulay property for rings generated by elements of a representation of the above lattice.
scientific article; zbMATH DE number 1525324
scientific article