Flatness of the commutator map over SL_n

TL;DR

This paper investigates the geometric structure of fibers of the commutator map on SL_n, revealing uniform dimensions over non-central elements and detailed behavior over central elements, using character theory and algebraic geometry.

Contribution

It provides a detailed analysis of fiber dimensions of the commutator map on SL_n, including explicit computations for central elements, combining character theory and algebraic geometry techniques.

Findings

Fibers over non-central elements have uniform dimension.

Fibers over central elements can be larger, with computed maximum size.

Uses character tables of finite general linear groups to count solutions.

Abstract

This paper contributes to the study of the fibers of the commutator map on special linear groups in characteristic zero. Specifically, we show that the fibers over non-central elements all have the same dimension. Also we explain that the fibers over central elements can be of larger dimension and compute how large. We use the character tables of finite general linear groups constructed by J.A. Green to count solutions to the commutator equation $[x,y]=g$ over finite fields and use algebraic geometry to go from characteristic $p$ to characteristic $0$. To deal with fibers over central elements, we compute the orbits of the conjugation action of $\mathrm{GL}_n$ on these fibers.