Algebraic invariants of the special fiber ring of ladder determinantal modules

TL;DR

This paper derives explicit algebraic invariants for the special fiber rings of ladder determinantal modules, connecting combinatorial data with algebraic properties and generalizing classical formulas for Grassmannian degrees.

Contribution

It provides explicit formulas for invariants like dimension, regularity, and multiplicity of these rings, linking algebraic and combinatorial aspects.

Findings

Computed the dimension, regularity, and a-invariant of the rings.

Derived a formula for multiplicity via counting standard skew Young tableaux.

Connected the invariants to Hibi rings and Sagbi degenerations.

Abstract

We provide explicit formulas for key invariants of special fiber rings of ladder determinantal modules, that is, modules that are direct sums of ideals of maximal minors of a ladder matrix. Our results are given in terms of the combinatorial data of the associated ladder matrix. In particular, we compute its dimension, regularity, $a$-invariant, and multiplicity, which via \textsc{Sagbi} degeneration coincide with those of Hibi rings associated to a distributive lattice. Then, via Gr\"{o}bner degeneration these calculations are reduced to quotients of polynomial rings by monomial ideals. Our formula for the multiplicity of the special fiber ring of these ladder determinantal modules is obtained by counting the number of standard skew Young tableaux associated to a certain skew partition, and so provides a natural generalization of the classical formula for the degree of the Grassmannian.