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Auslander regular algebras and Coxeter matrices

Viktória Klász, Rene Marczinzik, Hugh Thomas

TL;DR

The paper investigates Auslander-Gorenstein and Auslander regular algebras, establishing that the grade bijection on simple modules coincides with the Auslander-Reiten bijection via a new characterization, and showing that the grade bijection also matches the Coxeter permutation appearing in a Bruhat decomposition of the Coxeter matrix. This yields a purely linear-algebraic interpretation of the grade bijection, enabling faster computation, and leads to several applications including that the permanent of the Coxeter matrix is $1$ or $-1$, a new combinatorial criterion for distributive lattices via the Coxeter matrix, and connections to rowmotion in incidence algebras and blocks of category $ ext{O}$. The results unify homological and combinatorial perspectives, offering practical criteria and computations and linking representation theory with lattice theory and dynamical combinatorics. The paper also poses open questions regarding the sufficiency of $U_1= ext{id}$ for Auslander regularity and characterizations of incidence algebras in this framework.

Abstract

We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category $\mathcal{O}$ of semisimple Lie algebras.

Auslander regular algebras and Coxeter matrices

TL;DR

The paper investigates Auslander-Gorenstein and Auslander regular algebras, establishing that the grade bijection on simple modules coincides with the Auslander-Reiten bijection via a new characterization, and showing that the grade bijection also matches the Coxeter permutation appearing in a Bruhat decomposition of the Coxeter matrix. This yields a purely linear-algebraic interpretation of the grade bijection, enabling faster computation, and leads to several applications including that the permanent of the Coxeter matrix is or , a new combinatorial criterion for distributive lattices via the Coxeter matrix, and connections to rowmotion in incidence algebras and blocks of category . The results unify homological and combinatorial perspectives, offering practical criteria and computations and linking representation theory with lattice theory and dynamical combinatorics. The paper also poses open questions regarding the sufficiency of for Auslander regularity and characterizations of incidence algebras in this framework.

Abstract

We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category of semisimple Lie algebras.
Paper Structure (14 sections, 27 theorems, 21 equations)

This paper contains 14 sections, 27 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on Auslander-Gorenstein algebras
  4. The Bruhat decomposition of an invertible matrix
  5. A survey of bijections
  6. The Coxeter permutation
  7. The grade bijection by Iyama
  8. The Auslander-Reiten bijection for Auslander-Gorenstein algebras
  9. Higher Auslander-Reiten translates
  10. The rowmotion bijection
  11. The Nakayama permutation
  12. Grade bijection
  13. Characterisation of distributivity for lattices using the Coxeter matrix
  14. Two open questions

Key Result

Theorem 1.1

(Theorem AGcondition and gradeBijCharacterisation2) A finite dimensional algebra is Auslander-Gorenstein if and only if $\operatorname{grade} S= \operatorname{pdim} I(S)<\infty$ for every simple module $S$ with injective envelope $I(S)$. Furthermore, the grade permutation coincides with the Auslande

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • ...and 44 more