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A minimal Gröbner basis for simple $\mathfrak{sl}_n$- or $\mathfrak{sp}_n$-modules

Ghislain Fourier, León van Eß

Abstract

We explicitly provide minimal Gröbner bases for simple, finite-dimensional modules of complex Lie algebras of types A and C, using a homogeneous ordering that is compatible with the PBW filtration on the universal enveloping algebras.

A minimal Gröbner basis for simple $\mathfrak{sl}_n$- or $\mathfrak{sp}_n$-modules

Abstract

We explicitly provide minimal Gröbner bases for simple, finite-dimensional modules of complex Lie algebras of types A and C, using a homogeneous ordering that is compatible with the PBW filtration on the universal enveloping algebras.
Paper Structure (5 sections, 5 theorems, 14 equations)

This paper contains 5 sections, 5 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PBW filtration
  4. Gröbner bases
  5. Proof of the main result

Key Result

Theorem 1

Let $\mathfrak{g}$ be of type $A$ or $C$, and $\lambda$ a dominant integral weight. $M_\lambda$ is a minimal (left) Gröbner basis of $I_\lambda$ and the induced monomial basis of $V(\lambda)$ is the FFLV basis.

Theorems & Definitions (10)

  • Theorem
  • Theorem 2.2
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Proposition 3.1
  • proof : Step 1
  • proof : Step 2
  • proof : Step 3