Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity-free tensor products of irreducible modules over simple algebraic groups in positive characteristic

Gaëtan Mancini

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. In this master thesis, we classify multiplicity-free tensor products of simple modules for the groups $SL_2(k)$ and $SL_3(k)$. We also provide a classification for $SL_n(k)$ when $p=2$ and give partial results in the case of $Sp_4(k)$.

Multiplicity-free tensor products of irreducible modules over simple algebraic groups in positive characteristic

Abstract

Let be an algebraically closed field of characteristic . In this master thesis, we classify multiplicity-free tensor products of simple modules for the groups and . We also provide a classification for when and give partial results in the case of .

Paper Structure

This paper contains 37 sections, 98 theorems, 293 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Weights and alcoves
  3. Chevalley groups and algebraic groups
  4. Linear algebraic groups
  5. Chevalley groups
  6. Modules
  7. First definitions and irreducible modules
  8. Duality
  9. Weyl modules
  10. Filtrations and tilting modules
  11. Characters
  12. Jantzen $p$-sum formula
  13. Linkage principle
  14. An argument to count multiplicities
  15. Properties of tensor products
  16. ...and 22 more sections

Key Result

Lemma 1.7

Let $\lambda \in X^+\cap \widehat{C_1}$ and $\mu\in X^+$ be such that $\mu\leq \lambda$. We have $\mu \in \widehat{C_1}$.

Figures (2)

  • Figure 1: Alcoves for $A_2$ and $p=7$.
  • Figure 2: Alcoves for $B_2$ and $p=7$.

Theorems & Definitions (195)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Lemma 1.7
  • proof
  • Definition 1.8
  • Theorem 1.9: Humphreys_Coxeter
  • ...and 185 more