{
  "id": "2503.07581",
  "version": "v2",
  "published": "2025-03-10T17:45:49.000Z",
  "updated": "2025-05-14T15:40:35.000Z",
  "title": "The Green correspondence for SL(2,p)",
  "authors": [
    "Denver-James Logan Marchment"
  ],
  "categories": [
    "math.RT"
  ],
  "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 ${\\text{Res}^G_B}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-14T15:40:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper triangular matrices",
      "green correspondence bijection",
      "non-projective indecomposable",
      "module decomposition",
      "odd prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}