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Torsion Units of Integral Group Rings and Scheme Rings

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Torsion Units of Integral Group Rings and Scheme Rings TORSION UNITS OF INTEGRAL GROUP RINGS AND SCHEME RINGS A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In Mathematics University of Regina By Gurmail Singh Regina, Saskatchewan August, 2015 c Copyright 2015: Gurmail Singh UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Gurmail Singh, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Torsion Units of Integral Group Rings and Scheme Rings, in an oral examination held on August 26, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: *Dr. Yuanlin Li, Brock University Co-Supervisor: Dr. Allen Herman, Department of Mathematics & Statistics Co-Supervisor: Dr. Shaun Fallat, Department of Mathematics & Statistics Committee Member: Dr. Fernando Szechtman, Department of Mathematics & Statistics Committee Member: Dr. Karen Meagher, Department of Mathematics & Statistics Committee Member: **Dr. Robert Hilderman, Department of Computer Science Chair of Defense: Dr. Christopher Yost, Department of Biology *via SKYPE **Not present at defense Abstract We study torsion units of algebras over the ring of integers Z with nice bases. These include integral group rings, integral adjacency algebras of association schemes and integral C-algebras. Torsion units of group rings have been studied extensively since the 1960’s. Much of the attention has been devoted to the Zassenhaus conjecture for normal- ized torsion units of ZG, which says that they should be rationally conjugate (i.e. in QG) to elements of the group G. In recent years several new restrictions on inte- gral partial augmentations for torsion units of ZG have been introduced that have improved the effectiveness of the Luthar-Passi method for checking the Zassen- haus conjecture for specific finite groups G. We have implemented a computer program that constructs units of QG that have integral partial augmentations that are relevant to the Zassenhaus conjecture. Indeed, any unit of ZG with these par- tial augmentations would be a counterexample to the conjecture. In all but three i exceptions among groups of order less than 160, we have constructed units of QG with these partial augmentations that satisfy a condition which implies they cannot be rationally conjugate to an element of ZG. Currently our package has computational difficulties with the Luthar-Passi method for some of the groups of order 160. As C-algebras are generalization of groups, it is natural to ask about torsion units of C-algebras. We establish some basic results about torsion units of C- algebras analogous to what happens for torsion units of group rings. These results can be immediately applied to give new results for Schur rings, Hecke algebras, adjacency algebras of association schemes and fusion rings. We also investigate the possibility for a conjecture analogous to the Zassenhaus conjecture in the C- algebra setting. ii Acknowledgment I am grateful to my supervisors Professor Allen W. Herman and Professor Shaun M. Fallat for all of the their support, understanding, patience, and knowledge they have provided me over the course of my study. Without their supervision and mentorship this would not have been possible. I am thankful to Dr. Yuanlin Li, my external examiner. I also wish to thank Dr. Karen Meagher and Dr. Fernando Szechtman for their advice and careful reading of the manuscript. Their suggestions have been valuable and helpful for my thesis. Finally, the financial support of my supervisors’s NSERC grants, Department of Mathematics and Statistics, and the Faculty of Graduate Studies and Research during my PhD program allowed me to focus solely on my PhD program during these past years. iii To my family iv Contents Acknowledgment iii Dedication iv 1 Introduction 1 2 Background 5 2.1 Group rings . .5 2.2 Generalized C-algebras and table algebras . .9 2.3 Association schemes and scheme rings . 12 2.4 Representations and characters of semisimple algebras . 18 2.4.1 Semisimple algebras . 18 2.4.2 Representation theory of semisimple algebras . 21 2.4.3 Representation theory of groups . 24 v 3 Basic Tools 28 3.1 Torsion units of ZG and partial augmentations . 29 3.2 Luthar-Passi method . 30 3.3 The standard feasible trace of a C-algebra . 36 4 Normalized Torsion Units of Integral Group Rings 45 4.1 Partial augmentations and rational conjugacy . 46 4.2 Computer implementation of the Luthar-Passi method . 49 4.3 Computer construction of torsion units with prescribed partial aug- mentations . 55 4.4 Partially central torsion units of QG ................ 65 5 Torsion units of C-algebras 69 5.1 Torsion units of RB ......................... 70 5.2 Torsion units for integral C-algebras with a standard character . 75 5.3 Lagrange’s theorem for normalized torsion units of Z¯ B ............................. 82 5.4 Applications to Schur rings and Hecke algebras . 90 5.4.1 Schur rings . 90 vi 5.4.2 Integral Hecke algebras . 92 6 Future Work 93 6.1 Categorical aspects . 94 6.2 When all units of ZB are trivial . 96 6.3 Normalized automorphisms of QB ................. 98 vii Chapter 1 Introduction Let R be a commutative ring with identity. A group ring RG is a ring as well as a free R-module whose basis is a multiplicative group G. When the ring R is replaced with a field K then KG is called a group algebra. The trivial units of a group ring RG are scalar multiples of a single group element in G by a unit of the ring R. We will investigate a conjecture that would characterize the nontrivial torsion units of the integral group ring ZG. To get a broader perspective of what is really going on, we formulate and study the analogous conjecture in the general settings of the integral C-algebras and integral adjacency algebras of association schemes (a.k.a scheme rings). Along the way we establish several basic results to set up the necessary machinery in these new settings. 1 In this dissertation we present new results on torsion units of group rings, C-algebras, and scheme rings. Using the Luthar-Passi method and our new con- struction for partially central units we show how we have verified the Zassenhaus conjecture for all but three groups of order up to 159. In the case of C-algebras we establish some fundamental results for torsion units of integral C-algebras with s- tandard character, and we prove a Lagrange-type theorem for integral C-algebras that applies directly to integral scheme rings. Our initial motivation came from the fact that the Zassenhaus conjecture for integral group rings was open for fairly small groups, and so there was an opportu- nity to raise the lower bound or to find a counterexample . The algebraic study of association schemes, table algebras, and C-algebras has seen a number of signifi- cant advances in the last five years, due to the realization of close connections with finite group theory. As the bases for C-algebras and finite association schemes are close to finite groups, we felt we could formulate and investigate versions of the Zassenhaus conjecture in these settings. The purpose of this dissertation is to shed light upon various properties of torsion units for both group rings, C-algebras and scheme rings, to verify Zassen- haus’s conjecture for group rings in some new cases and to move a step toward establishing the Zassenhaus conjecture for a broader class of rings. 2 The dissertation is organized in the following manner. In Chapter 2, we de- fine the algebraic structures such as group rings, adjacency algebras and torsion units of integral C-algebras. We give a brief review of particular known results for torsion units of integral group rings. In Section 2.1, we define integral group rings, their normalized torsion units, and we give the statement of the Zassenhaus conjecture for the normalized torsion units of integral group rings. In Section 2.2, we define generalized C-algebras and table algebras. In Section 2.3, we define as- sociation schemes and scheme rings and demonstrate how an association scheme is a generalization of a group. In Section 2.4, we define semisimple algebras and state some background results for semisimple algebras that will be used. Then we define representations and characters for algebras, and give a similar treatment for groups. In Chapter 3, we state all the preliminary results that will be used in Chapters 4 and 5 to prove the results. In Section 3.1, we state the results used in the Luthar-Passi method. In Section 3.2, we demonstrate the Luthar-Passi method by applying it to the group A4. In Section 3.3, we introduce standard feasible of a C-algebra that will be needed in Chapters 5. In Chapter 4, we examine the extent to which rational conjugacy of units in QG is determined by partial aug- mentations. Also we show that partially central units are not conjugate in QG to elements of ZG and we construct partially central units of QG for a group of order 3 48. This demonstrates the method we used to verify the Zassenhaus conjecture for all but three groups of orders up to 159. In Chapter 5, we prove a generalization of the Berman-Higman Lemma for integral C-algebras and using this we prove a Lagrange-type theorem for integral C-algebras. We apply these results directly to several familiar settings, including that of integral scheme rings. We try to move a step toward generalizing the Zassenhaus conjecture to integral C-algebras by proving conjugacy of finite subgroups of units of KB in LB implies their conju- gacy in KB, where K and L are infinite subfields of C with K ⊆ L and B is the distinguished basis for the C-algebra.
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