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The representations of the Lie superalgebra p(3) in prime characteristic

Ye Ren

TL;DR

The paper addresses the modular representation theory of the Lie superalgebra $p(3)$ over an algebraically closed field of characteristic $p>3$, aiming to classify all finite-dimensional irreducible modules and compute their characters. The authors develop restricted Lie superalgebra theory, introduce $p$-characters $\chi$, reduced enveloping algebras $U_\chi(g)$, and Kac modules $K_\chi(\lambda)$, with a PBW-type basis and dimension formula $\dim U_\chi(g)= p^{\dim g_0} 2^{\dim g_1}$. They classify simple modules for $p(3)$ by six orbit types of $\chi$ under GL(3), presenting detailed structures of $L_\chi^0(\lambda)$ for each case and computing maximal vectors in $K_\chi(\lambda)$ to obtain exact composition factors and character formulas. The results provide a complete, case-by-case description of irreducible modules and their characters for $p(3)$ in characteristic $p>3$, advancing the modular representation theory of restricted Lie superalgebras.

Abstract

Let g be the Lie superalgebra p(3) of rank 2 over an algebraically closed field K of characteristic p > 3. We classify all irreducible modules of g, and give the character formulae for irreducible modules.

The representations of the Lie superalgebra p(3) in prime characteristic

TL;DR

The paper addresses the modular representation theory of the Lie superalgebra over an algebraically closed field of characteristic , aiming to classify all finite-dimensional irreducible modules and compute their characters. The authors develop restricted Lie superalgebra theory, introduce -characters , reduced enveloping algebras , and Kac modules , with a PBW-type basis and dimension formula . They classify simple modules for by six orbit types of under GL(3), presenting detailed structures of for each case and computing maximal vectors in to obtain exact composition factors and character formulas. The results provide a complete, case-by-case description of irreducible modules and their characters for in characteristic , advancing the modular representation theory of restricted Lie superalgebras.

Abstract

Let g be the Lie superalgebra p(3) of rank 2 over an algebraically closed field K of characteristic p > 3. We classify all irreducible modules of g, and give the character formulae for irreducible modules.
Paper Structure (2 sections, 15 theorems, 65 equations)

This paper contains 2 sections, 15 theorems, 65 equations.

Table of Contents

  1. Preliminaries
  2. The representations of $\mathfrak{p}(3)$

Key Result

Proposition 1.2

Suppose $x_1,...,x_s$ is a basis of $\mathfrak{g}_{\bar{0}}$, $y_1,...,y_t$ is a basis of $\mathfrak{g}_{-1}$, $z_1,...,z_m$ is a basis of $\mathfrak{g}_{+1}$. Then is a basis of $U_{\chi}(\mathfrak{g})$. In particular, dim$U_{\chi}(\mathfrak{g})= p^{\text{dim}\mathfrak{g}_{\bar{0}}}2^{\text{dim}\mathfrak{g}_{\bar{1}}}$.

Theorems & Definitions (27)

  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • proof
  • ...and 17 more