Representations with the same degree

TL;DR

The paper demonstrates that for certain algebraic groups, there are infinitely many pairs of irreducible representations with the same degree that are not related by automorphisms, answering a specific mathematical question.

Contribution

It provides a proof that connected reductive simply-connected algebraic groups of rank greater than one have infinitely many such representation pairs, a previously open question.

Findings

Existence of infinitely many representation pairs with same degree

Representation pairs are not related by automorphisms

Addresses a question posed by J. P. Serre

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.