Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noncommutative Poisson structure and invariants of matrices

Farkhod Eshmatov, Xabier García-Martínez, Rustam Turdibaev

Abstract

We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two $n\times n$ matrices. We entirely solve the open problem of computing the algebra of invariants of two $4 \times 4$ matrices. As an application, we derive the complete description of the invariant commuting variety of $4 \times 4$ matrices and the fourth Calogero-Moser space.

Noncommutative Poisson structure and invariants of matrices

Abstract

We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two matrices. We entirely solve the open problem of computing the algebra of invariants of two matrices. As an application, we derive the complete description of the invariant commuting variety of matrices and the fourth Calogero-Moser space.
Paper Structure (20 sections, 4 theorems, 37 equations)

This paper contains 20 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generators and defining relations
  4. Hilbert Series and Hironaka Decomposition
  5. Invariant Commuting Variety and Calogero-Moser spaces
  6. Poisson algebra structure on Cn2
  7. Necklaces
  8. Description of the algorithmic approach
  9. Overview
  10. Initial considerations
  11. First non-trivial degree
  12. Degrees less than 12
  13. Higher degrees
  14. Refined execution
  15. Outcome
  16. ...and 5 more sections

Key Result

Theorem 2.1

The algebra $C_{nd}$ is generated over $\mathbb{C}$ by all traces of products of generic matrices $\mathop{\mathrm{Tr}}\nolimits(A_{i_1}A_{i_2}\cdots A_{i_j})$, with $j\leq 2^n-1$.

Theorems & Definitions (7)

  • Theorem 2.1: Pr
  • Theorem 2.2: Ra
  • Definition 2.8
  • Definition 2.9
  • Theorem 3.8
  • proof
  • Corollary 3.11