Low-dimensional representations of \(\text{Aut}(F_ 2)\)
type: Item
instance of: scholarly article
title: Low-dimensional representations of \(\text{Aut}(F_ 2)\)
zbMATH Open document ID: 0862.20006
DOI: 10.1007/BF02567531
published in: Manuscripta Mathematica
publication date: 1996-08-21T00:00:00Z
review text: Let \(F_2\) be the free group of rank two and \(\Phi_2\) its automorphism group. The authors consider indecomposable representations \(\rho\) of \(\Phi_2\) of degree \(n\leq 4\), over an algebraically closed field \(K\), such that \(\rho(F_2)\neq 1\). They show that there are no such representations if \(n\leq 2\). If \(\rho(F)\) is infinite then, up to weak equivalence, there exist for \(n=3\) only one such representation, and for \(n=4\) two if \(\text{char }K\neq2,3\), one if \(\text{char }K=3\), and none if \(\text{char }K=2\). All the representations mentioned above are reducible and listed explicitly.
reviewed by: Steven C. Althoen
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 885530
zbMATH Keywords: braid groups
zbMATH Keywords: free groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: indecomposable representations
author: Dragomir Ž. Đoković
author: V. P. Platonov
MaRDI profile type: Publication
cites work: A complete survey of the representations of degree 3 of the automorphism group of the free group of rank 2
cites work: On the linearity of automorphism groups of free groups
cites work: Braid Group Representations of Low Degree
cites work: The automorphism group of a free group is not linear
cites work: Q5513066
cites work: Matrix representation for the braid group \(B_4\)
cites work: Q5668937
full work available at URL: https://doi.org/10.1007/bf02567531
OpenAlex ID: W1973485068
description: scientific article; zbMATH DE number 885530