Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representations with the same degree

Frank Lübeck

TL;DR

The paper addresses the problem of finding infinitely many pairs of finite-dimensional irreducible representations of a connected reductive simply-connected complex group $G$ of rank $>1$ that share the same degree but are not related by any automorphism of $G$, answering a question posed by Serre. It leverage$s$ the Weyl dimension formula and root data to reduce the search to identifying dominant weights with identical dimensions but distinct coordinate patterns in the fundamental weight basis, proceeding through rank- and type-based casework. It provides explicit infinite families of equal-degree pairs for the classical families $A_l$, $B_l$, $D_l$, and describes a Pell-type Diophantine condition governing the $C_l$ case that guarantees infinitely many solutions; exceptional types are handled with direct computations and a table of concrete pairs. The results demonstrate a rich interplay between representation theory and number-theoretic methods, showing that degree coincidences arise abundantly beyond automorphism-related cases and answering Serre’s question in the affirmative.

Abstract

In this short note we show that every connected reductive simply-connected algebraic group of rank $>1$ over the complex numbers has infinitely many pairs of irreducible representations which are not related by an automorphism of the algebraic group and which have the same degree. This answers a question I was asked by J.~P.~Serre.

Representations with the same degree

TL;DR

The paper addresses the problem of finding infinitely many pairs of finite-dimensional irreducible representations of a connected reductive simply-connected complex group of rank that share the same degree but are not related by any automorphism of , answering a question posed by Serre. It leverage the Weyl dimension formula and root data to reduce the search to identifying dominant weights with identical dimensions but distinct coordinate patterns in the fundamental weight basis, proceeding through rank- and type-based casework. It provides explicit infinite families of equal-degree pairs for the classical families , , , and describes a Pell-type Diophantine condition governing the case that guarantees infinitely many solutions; exceptional types are handled with direct computations and a table of concrete pairs. The results demonstrate a rich interplay between representation theory and number-theoretic methods, showing that degree coincidences arise abundantly beyond automorphism-related cases and answering Serre’s question in the affirmative.

Abstract

In this short note we show that every connected reductive simply-connected algebraic group of rank over the complex numbers has infinitely many pairs of irreducible representations which are not related by an automorphism of the algebraic group and which have the same degree. This answers a question I was asked by J.~P.~Serre.
Paper Structure (6 sections, 3 theorems, 3 equations)

This paper contains 6 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction and notation
  2. Proof of Theorem 1
  3. The case of rank $l=1$
  4. The case that all simple direct factors have rank $1$
  5. The case of exceptional types, $A_2$ and $B_2$
  6. Cases $A_l$, $B_l$, $C_l$, $D_l$ for large rank $l$

Key Result

Theorem 1

When the rank of $G$ is $>1$ then $G$ has infinitely many pairs of finite dimensional irreducible rational representations $\rho_1$ and $\rho_2$ such that there is no automorphism $\alpha$ of $G$ with $\rho_2 = \rho_1 \circ \alpha$ and such that $\rho_1$ and $\rho_2$ have the same degree.

Theorems & Definitions (3)

  • Theorem 1
  • Proposition 2
  • Theorem 3