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The Green correspondence for SL(2,p)

Denver-James Logan Marchment

TL;DR

This work analyzes the Green correspondence for the classical group $G = SL_2(\mathbb{F}_p)$ with $p$ odd and its Borel-like subgroup $B$ of upper triangular matrices. By parameterizing non-projective indecomposable modules via Brauer-tree walks and locating them on stable Auslander–Reiten quivers, the authors produce an explicit bijection between $\mathbb{F}[G]$-modules and $\mathbb{F}[B]$-modules, described in detail through the Green correspondents $V_{a,b}$ of $U_{a,b}$. They derive concrete results for induction and restriction, including a lifting decomposition theorem and explicit formulas for $\Ind_B^G$ and $\Res_B^G$ on simple and projective modules, with dimension congruences modulo $p$ guiding the correspondences. The approach yields a complete, computable description of the Green correspondence in this setting and provides tools for equivariant decompositions in related geometric contexts, such as Drinfeld curves, and potentially generalizes to other blocks with cyclic defect via Brauer-tree techniques.

Abstract

Let ${p > 2}$ be an odd prime and ${G = SL_2(\mathbb{F}_p)}$. Denote the subgroup of upper triangular matrices as $B$. Finally, let ${\mathbb{F}}$ be an algebraically closed field of characteristic ${p}$. The Green correspondence gives a bijection between the non-projective indecomposable ${\mathbb{F}[G]}$ modules and non-projective indecomposable ${\mathbb{F}[B]}$ modules, realised by restriction and induction. In this paper, we start by recalling a suitable description of the non-projective indecomposable modules for these group algebras. Next, we explicitly describe the Green correspondence bijection by pinpointing the modules' position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of these descriptions: formulae for lifting the ${\mathbb{F}[B]}$ module decomposition of an ${\mathbb{F}[G]}$ module, and a complete description of ${\text{Ind}_B^G}$ and ${\text{Res}^G_B}$.

The Green correspondence for SL(2,p)

TL;DR

This work analyzes the Green correspondence for the classical group with odd and its Borel-like subgroup of upper triangular matrices. By parameterizing non-projective indecomposable modules via Brauer-tree walks and locating them on stable Auslander–Reiten quivers, the authors produce an explicit bijection between -modules and -modules, described in detail through the Green correspondents of . They derive concrete results for induction and restriction, including a lifting decomposition theorem and explicit formulas for and on simple and projective modules, with dimension congruences modulo guiding the correspondences. The approach yields a complete, computable description of the Green correspondence in this setting and provides tools for equivariant decompositions in related geometric contexts, such as Drinfeld curves, and potentially generalizes to other blocks with cyclic defect via Brauer-tree techniques.

Abstract

Let be an odd prime and . Denote the subgroup of upper triangular matrices as . Finally, let be an algebraically closed field of characteristic . The Green correspondence gives a bijection between the non-projective indecomposable modules and non-projective indecomposable modules, realised by restriction and induction. In this paper, we start by recalling a suitable description of the non-projective indecomposable modules for these group algebras. Next, we explicitly describe the Green correspondence bijection by pinpointing the modules' position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of these descriptions: formulae for lifting the module decomposition of an module, and a complete description of and .

Paper Structure

This paper contains 6 sections, 34 theorems, 95 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. The indecomposables and block structure
  4. Describing the Green correspondence
  5. Induction and restriction
  6. Bibliography

Key Result

Proposition 2.1

For ${a \in \mathbb{Z}}$, let ${S_a}$ denote the one dimensional vector space over ${\mathbb{F}}$ on which $g$ acts trivially and ${\lambda}$ acts as multiplication by ${\zeta^a}$. Then ${S_0, S_1, ..., S_{p-2}}$ gives all the simple modules for ${\mathbb{F}[B]}$. Next, let ${1\leq b \leq p}$. Then

Figures (10)

  • Figure 1: The Brauer trees ${\textrm{\ss}_0}$ (left), ${\textrm{\ss}_1}$ (right) of $B$. Their exceptional vertices are labeled $e$, and they have multiplicity $2$.
  • Figure 2: The Brauer trees ${\mathcal{B}_0}$ (top), ${\mathcal{B}_1}$ (bottom) of $G$.
  • Figure 3: Walk type 1, ${s\geq 2}$
  • Figure 4: Walk type 2, ${s\geq 2}$
  • Figure 5: Stable A.R. quiver of the block ${\textrm{\ss}_i}$ of ${\mathbb{F}[B]}$
  • ...and 5 more figures

Theorems & Definitions (70)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 60 more