The p-Modular Green Correspondence for $${\text {SL}_2(\mathbb {F}_p)}$$

Abstract

<jats:p> Let <jats:italic>p</jats:italic> be an odd prime. Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${G = SL_2(\mathbb {F}_p)}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and let <jats:italic>B</jats:italic> denote the subgroup of upper triangular matrices of <jats:italic>G</jats:italic> . Finally, let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathbb {F}}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be an algebraically closed field of characteristic <jats:italic>p</jats:italic> . The Green correspondence gives a bijection between the non-projective indecomposable <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathbb {F}[G]}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> modules and non-projective indecomposable <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathbb {F}[B]}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>B</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathbb {F}[B]}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>B</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> module decomposition of an <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\mathbb {F}[G]}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> module, and a complete description of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\text { Ind}_B^G}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:msubsup> <mml:mtext>Ind</mml:mtext> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${\text { Res}^G_B}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:msubsup> <mml:mtext>Res</mml:mtext> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>