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Computing monomial bases in Lie theory using OSCAR

Xin Fang, Ghislain Fourier, Lars Göttgens, Ben Wilop

TL;DR

This guide is presented on using the computer algebra system OSCAR to compute monomial bases for simple, finite-dimensional modules of simple, complex Lie algebras and how to determine monomial bases for the homogeneous coordinate ring of a (partial) flag variety.

Abstract

In this survey, we present a detailed guide on using the computer algebra system OSCAR to compute monomial bases for simple, finite-dimensional modules of simple, complex Lie algebras. We will also demonstrate how to determine monomial bases for the homogeneous coordinate ring of a (partial) flag variety, depending on a chosen birational sequence and a monomial order. This survey will be updated to reflect any advancements in OSCAR's capabilities in these areas.

Computing monomial bases in Lie theory using OSCAR

TL;DR

This guide is presented on using the computer algebra system OSCAR to compute monomial bases for simple, finite-dimensional modules of simple, complex Lie algebras and how to determine monomial bases for the homogeneous coordinate ring of a (partial) flag variety.

Abstract

In this survey, we present a detailed guide on using the computer algebra system OSCAR to compute monomial bases for simple, finite-dimensional modules of simple, complex Lie algebras. We will also demonstrate how to determine monomial bases for the homogeneous coordinate ring of a (partial) flag variety, depending on a chosen birational sequence and a monomial order. This survey will be updated to reflect any advancements in OSCAR's capabilities in these areas.
Paper Structure (12 sections, 2 theorems, 16 equations, 2 tables, 1 algorithm)

This paper contains 12 sections, 2 theorems, 16 equations, 2 tables, 1 algorithm.

Table of Contents

  1. OSCAR
  2. Theoretical background
  3. Representations as functions
  4. Birational sequences
  5. Newton-Okounkov bodies
  6. Identifications of Newton-Okounkov bodies
  7. Essential monomials
  8. Algorithm and first example
  9. More examples
  10. Generators of the monoid for fixed Kodaira embedding
  11. Experiments on generators of the monoid
  12. Comparison of the runtime against GAP

Key Result

Theorem 2.8

Let $\underline{w}_0=s_{i_1}s_{i_2}\ldots s_{i_N}$ be a reduced decomposition of $w_0$.

Theorems & Definitions (14)

  • Conjecture 1
  • Definition 2.3
  • Example 2.4
  • Definition 2.7
  • Theorem 2.8: FaFoLFFL3FN
  • Definition 2.10
  • Theorem 2.11: FaFoL
  • Example 2.12
  • Example 3.1
  • Example 4.1
  • ...and 4 more