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Gröbner crystal structures

Abigail Price, Ada Stelzer, Alexander Yong

TL;DR

This work develops the Gröbner crystal structure (GCS) framework for coordinate rings ${\mathbb C}[{\sf Mat}_{m,n}]$ under Levi group actions, unifying Gröbner theory with Kashiwara crystal combinatorics to yield a generalized Littlewood–Richardson rule for multiplicities. It provides a finite, algorithmic criterion (test sets) to decide when an ideal is bicrystalline and introduces a ballot-based multiplicity rule that reads irreducible components from crystal data via RSK, covering broad families such as Gröbner-determinantal, Knutson determinantal, and GL-stable in-KRS ideals. The theory extends classical determinant- and Schubert-theoretic multiplicity rules into a common, Levi-equivariant setting, enabling uniform computation of representation-theoretic data for quotients and ideals. Beyond the commutative setting, the authors sketch extensions to non-commutative algebras using Gröbner–Shirshov bases, highlighting the GCS philosophy as a flexible, far-reaching paradigm for connecting combinatorics, algebraic geometry, and representation theory with computational tools.

Abstract

We develop a theory of bicrystalline ideals, synthesizing Gröbner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and new. The theory concerns ideals in the coordinate ring of matrices, stable under the action of some Levi group, whose quotients admit standard bases equipped with a crystal structure. We construct an effective algorithm to decide if an ideal is bicrystalline. When the answer is affirmative, we provide a uniform, generalized Littlewood-Richardson rule for computing the multiplicity of irreducible representations either for the quotient or the ideal itself.

Gröbner crystal structures

TL;DR

This work develops the Gröbner crystal structure (GCS) framework for coordinate rings under Levi group actions, unifying Gröbner theory with Kashiwara crystal combinatorics to yield a generalized Littlewood–Richardson rule for multiplicities. It provides a finite, algorithmic criterion (test sets) to decide when an ideal is bicrystalline and introduces a ballot-based multiplicity rule that reads irreducible components from crystal data via RSK, covering broad families such as Gröbner-determinantal, Knutson determinantal, and GL-stable in-KRS ideals. The theory extends classical determinant- and Schubert-theoretic multiplicity rules into a common, Levi-equivariant setting, enabling uniform computation of representation-theoretic data for quotients and ideals. Beyond the commutative setting, the authors sketch extensions to non-commutative algebras using Gröbner–Shirshov bases, highlighting the GCS philosophy as a flexible, far-reaching paradigm for connecting combinatorics, algebraic geometry, and representation theory with computational tools.

Abstract

We develop a theory of bicrystalline ideals, synthesizing Gröbner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and new. The theory concerns ideals in the coordinate ring of matrices, stable under the action of some Levi group, whose quotients admit standard bases equipped with a crystal structure. We construct an effective algorithm to decide if an ideal is bicrystalline. When the answer is affirmative, we provide a uniform, generalized Littlewood-Richardson rule for computing the multiplicity of irreducible representations either for the quotient or the ideal itself.

Paper Structure

This paper contains 25 sections, 35 theorems, 273 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivating examples
  3. The GCS thesis
  4. Summary of results; organization
  5. Bicrystalline ideals
  6. Notation and preliminaries
  7. Main definitions
  8. The bicrystalline property is decidable
  9. The algorithms
  10. Proof of Theorem \ref{['thm:testsetalg']}
  11. Combinatorial Representation Theory Preliminaries
  12. Combinatorial preliminaries
  13. Representation theory
  14. Highest weight matrices and tableaux
  15. The ballot rule for the irreducible multiplicities
  16. ...and 10 more sections

Key Result

Lemma 2.6

${\mathcal{S}}\subseteq {\sf Mat}_{m,n}({\mathbb Z}_{\geq 0})$ is $(\mathbf{I}, \mathbf{J})$-bicrystal closed if and only if its complement ${\mathcal{S}}^c$ is $(\mathbf{I}, \mathbf{J})$-bicrystal closed.

Figures (8)

  • Figure 1: The crystal graph with highest weight word $211$.
  • Figure 2: The crystal graph for $\operatorname{SSYT}(\vcenter{\hbox{\ydiagram{2,1}}},3)$.
  • Figure 3: Case 2 in the proof of Theorem \ref{['thm:generalizeddetideals']}, with $(i, j) = (3, 4)$.
  • Figure 4: Part of the crystal for $T(\mathbb{C}^3)$, with elements of $\mathrm{Std}_\prec I$ in blue.
  • Figure 5: Part of the crystal for $T({\operatorname{Sym}}^3(\mathbb{C}^2))$.
  • ...and 3 more figures

Theorems & Definitions (170)

  • Example 1.1: Symmetric algebra
  • Example 1.2: Determinantal varieties
  • Example 1.4: Matrix matroid varieties
  • Example 1.5: Double Bruhat ideals
  • Example 1.6: Buchsbaum--Eisenbud variety of complexes Buchsbaum
  • Example 1.7: Thick determinantal ideals
  • Example 1.8: tensor power of the standard representation
  • Example 1.9: Standard graded case
  • Example 2.1
  • Remark 2.2
  • ...and 160 more