Algebraic invariants of the special fiber ring of ladder determinantal modules

Alessandra Costantini; Louiza Fouli; Kriti Goel; Kuei-Nuan Lin; Haydee Lindo; Whitney Liske; Maral Mostafazadehfard

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arXiv · 2507.22167

Algebraic invariants of the special fiber ring of ladder determinantal modules

Alessandra Costantini·Louiza Fouli·Kriti Goel·Kuei-Nuan Lin·Haydee Lindo·Whitney Liske·Maral Mostafazadehfard

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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 Gröbner 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.

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Alessandra Costantini, Louiza Fouli, Kriti Goel, Kuei-Nuan Lin, Haydee Lindo, Whitney Liske, Maral Mostafazadehfard. 2025-07-29. Algebraic invariants of the special fiber ring of ladder determinantal modules. https://arxiv.org/abs/2507.22167

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