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Matrix Representations of Automorphism Groups of Free Groups Brigham Young University BYU ScholarsArchive Theses and Dissertations 2005-06-20 Matrix Representations of Automorphism Groups of Free Groups Ivan B. Andrus Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Andrus, Ivan B., "Matrix Representations of Automorphism Groups of Free Groups" (2005). Theses and Dissertations. 530. https://scholarsarchive.byu.edu/etd/530 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. MATRIX REPRESENTATIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS by Ivan Andrus A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Brigham Young University August 2005 Copyright © 2005 Ivan Andrus All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Ivan Andrus This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Stephen Humphries, Chair Date Darrin Doud Date Gregory Conner BRIGHAM YOUNG UNIVERSITY As chair of the candidate’s graduate committee, I have read the thesis of Ivan Andrus in its final form and have found that (1) its format, citations, and biblio- graphical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Stephen Humphries Chair, Graduate Committee Accepted for the Department Tyler J. Jarvis Graduate Coordinator Accepted for the College G. Rex Bryce, Associate Dean College of Physical and Mathematical Sciences ABSTRACT MATRIX REPRESENTATIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS Ivan Andrus Department of Mathematics Master of Science In this thesis, we study the representation theory of the automorphism group Aut (Fn) of a free group by studying the representation theory of three finite sub- groups: two symmetric groups, Sn and Sn+1, and a Coxeter group of type Bn, also known as a hyperoctahedral group. The representation theory of these subgroups is well understood in the language of Young Diagrams, and we apply this knowl- edge to better understand the representation theory of Aut (Fn). We also calculate irreducible representations of Aut (Fn) in low dimensions and for small n. Table of Contents 1 Introduction 1 2 General Results of Representation Theory 5 2.1 Young Diagrams ............................ 11 2.2 Irreducible Representations of Cox (Bn) ................ 15 2.3 The Standard Representation of Cox (Bn) .............. 21 2.4 Frobenius’ Formula ........................... 23 3 Elementary Results for Aut (Fn) 34 3.1 1-Dimensional Representations of Aut (Fn) .............. 34 3.2 General Results for Aut (Fn) ...................... 35 4 Commutators 44 5 Results for specific n 49 5.1 Results for Aut (F2) ........................... 49 5.2 Results for Aut (F3) ........................... 54 5.3 Results for Aut (F4) ........................... 58 5.4 Results for Aut (F5) ........................... 65 A Polynomial Results 75 B GAP Code 76 References 98 vi 1 Introduction We will denote by F = x , x ,...,x n h 1 2 n | i the free group of rank n. In this paper we are interested in studying Aut (Fn), the group of automorphisms of F . That is the group of α : F F such that α is n n → n a bijective homomorphism of groups. Our goal is to understand the representation theory of Aut (Fn) for small values of n. In other words we wish to understand the irreducible homomorphisms ρ : Aut (F ) GL (C). As is customary when n → k studying Aut (Fn), automorphisms shall act on the right. However when studying Sn, and Cox (Bn), we shall act on the left as is customary for these groups. Representation theory for finite groups is well developed, but there are few applicable results for infinite groups such as Aut (Fn). To perform our analysis we consider some finite subgroups of Aut (Fn). The most important of these is the Coxeter group of type Bn, denoted Cox (Bn). Other subgroups of interest include two symmetric groups Sn and Σn ∼= Sn+1. The Coxeter group is the subgroup of Aut (Fn) which permutes elements of the set −1 −1 −1 x1, x1 , x2, x2 ,...,xn, xn while respecting inverses i.e., α(x−1)= α(x )−1 for all α Cox (B ). The hyperoc- i i ∈ n tahedral group, as Cox (Bn) is often called, is generated by the symmetric group Sn, which permutes the indices, together with elements ε : x x−1 which take the i i 7→ i i-th generator to its inverse and leave the others unchanged. This Coxeter group is also the group of symmetries of the n-cube [ 1, 1]n [2], and has order 2nn!. It − is sometimes called the signed permutation group for obvious reasons, and is the 1 wreath product (Z ) S of the cyclic group of order 2 with the symmetric group 2 ≀ n on n letters. It is also known as the Weyl group of type C. A result of Nielsen [15] tells us that Aut (Fn) is generated by four elements, denoted P , Q, σ, U, and defined as follows: P : x x Q : x x σ : x x−1 U : x x x 1 7→ 2 1 7→ 2 1 7→ 1 1 7→ 1 2 x x x x x x x x 2 7→ 1 2 7→ 3 2 7→ 2 2 7→ 2 x x x x x x x x (1) 3 7→ 3 3 7→ 4 3 7→ 3 3 7→ 3 . x x x x x x x x n 7→ n n 7→ 1 n 7→ n n 7→ n In fact, Nielsen gives a presentation of Aut (Fn) with generators P , Q, σ, and U, and the defining relationships given in tables 1.1 and 1.2 (with ⇄ signifying ‘commutes with’). We shall use these relations extensively throughout this paper. Table 1.1: Relations not involving U 1= P 2; (P.1) 1= Qn; (P.2) 1=(QP )n−1 ; (P.3) P ⇄Q−iP Qi i =2, 3,..., n/2 (P.4) ⌊ ⌋ 1= σ2; (P.5) σ ⇄ Q−1P Q; n =2 (P.6) 6 σ ⇄ QP ; (P.7) σ ⇄ Q−1σQ; (P.8) 2 Table 1.2: Relations involving U U ⇄ Q−2P Q2; n =2, 3 (U.1) 6 U ⇄ QP Q−1P Q; n =2 (U.2) 6 U ⇄ Q−2σQ2; n =2 (U.3) 6 U ⇄ Q−2UQ2; n =3 (U.4) 6 U ⇄ σUσ; (U.5) U ⇄ P Q−1σUσQP ; (U.6) U ⇄ P Q−1P QP UP Q−1P QP ; n =2 (U.7) 6 UQ−1U = P Q−1UQ 2 UQ−1 n =2 (U.8) 6 1=(PσPU)2 ; (U.9) U = PUPσUσPσ. n =1 (U.10) 6 Later Neumann [14] gave presentations of Aut (Fn) with fewer generators, but we shall work with Nielsen’s presentation for simplicity. We note that P , and Q, together with the relations (P.1) through (P.4) are a presentation of Sn, while P , Q and σ and the relations (P.1) through (P.8) define Cox (Bn) (though the action is considered to be on the right.) We will often denote by P , Q, σ, and U their images under a representation ρ. Any representation of Aut (Fn) gives, by restriction, a representation of Cox (Bn), so we begin by studying the representation theory of Cox (Bn). From elemen- tary representation theory there exists a finite set R = ρ1, ρ2,...,ρN(n) of irre- ducible representations of Cox (Bn) which have the property that any representation ρ : Cox (B ) GL (C) can be written, up to a change of basis, as the direct sum of n → k (possibly multiple) copies of elements of R. Here N(n) is the number of conjugacy 3 classes of Cox (Bn). Any representation ρ of Aut (Fn) is uniquely determined by ρ Cox (Bn) and ρ (U). Thus we need only find all possible matrices M = ρ(U) which satisfy the equations of table 1.2 for any given representation of Cox (Bn), or determine that no such matrix exists. In an effort to understand the representations of Aut (Fn), we shall begin by studying the representation theory of Sn and Cox (Bn). From there we shall place restrictions on the representations which can be extended to representations of Aut (Fn). We enumerate the irreducible representations of Aut (Fn) in low dimen- sions, and give some general results concerning these representations. One of the motivations for this paper is to understand the relationship between representations of the subgroups Cox (Bn), Sn, and Σn. This is determined in part by theorems 2.59 and 2.65. 4 2 General Results of Representation Theory In this chapter we summarize some general results of representation theory that will be useful in our analysis of representations of Aut (Fn) and Cox (Bn). Definition 2.1. Let F be a field and V a vector space over F. For a group G, an F-representation (often called simply a representation when F is understood) of G is a homomorphism ρ : G GL(V ) from G to the general linear group of V . In → this paper we will only consider the case F = C the complex numbers. Often we shall refer to the vector space V itself as a representation.
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