Modular representation theory book

For finite groups the theory comes in two distinct flavours. The modular representation theory of finite groups has its origins in the work of richard brauer. This expository chapter is based on the lectures given at the program modular representation theory of finite and. Modular representation theory of finite and padic groups lecture notes series, institute for mathematical sciences, national university of singapore book 30 kindle edition by gan wee teck, tan kai meng, wee teck gan, kai meng tan. Recall that a nakayama algebra also called serial algebras in the literature is an algebra such that every indecomposable module has a unique composition series. I am working in the field of coding theory and i do not know of any applications of modular representation theory in coding theory. This reprint of a 1983 yale graduate course makes results in modular representation theory accessible to an audience ranging from secondyear graduate students to established mathematicians. This volume contains a concise exposition of the theory of finite groups, including the theory of modular representations. Modular representation theory of finite groups peter schneider.

The book is organized this is a great book on modular representation theory, focusing on the basics of the theory and how the projective, indecomposable, simple, etc. For questions about modular representation theory, the study of representations over a field of positive characteristic. We then describe work done in the eighties and nineties on modular representations in nondefining characteristic of finite groups of lie type. Modular representation theory clay mathematics institute. In fact for any group there will be at least one counterexample. On group rings over a modular field which possess radicals expressible as principal ideals, sci. Whilst the theory over characteristic zero is well understood. Download it once and read it on your kindle device, pc, phones or tablets. Representations of hecke algebras at roots of unity. The chapter provides basic general definitions concerning algebraic groups and their representations. Modular representation theory is a branch of mathematics, and that part of representation. If you want to go into modular representation theory and projective representation theory, look for the books of bertram huppert or b. The modulor is an anthropometric scale of proportions devised by the swissborn french architect le corbusier 18871965 it was developed as a visual bridge between two incompatible scales, the imperial and the metric system. The representation theory of nite groups has a long history, going back to the 19th century and earlier.

This is an extended comment, to put julians answer and comments in perspective. Search for library items search for lists search for contacts search for a library. His work helped the development and utilization of sectors like cryptography, chemistry, and physics. We are concerned with recent results on representation theory and cohomology of klr algebras, with emphasis on standard module theory.

The last of these has become a major theme in representation theory into the 21st century. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. Modular representation theory of finite groups springerlink. Exercises are provided at the end of most sections. This book is a comprehensive presentation of the progress in the modular representation theory achieved especially by the work of a. It gives an excellent overview of the state of the art in this fascinating subject and also of the many challenging and fundamental. Modular representation theory of finite groups comprises this second situation. It was used as a system to set out a number of le corbusiers buildings and was later codified into two books. This book aims to familiarize students at the masters level basic results, tools, and techniques of a beautiful and important algebraic theory. Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field k of positive characteristic p, necessarily a prime number. This book is an outgrowth of a research symposium on the modular representation theory of finite groups, held at the university of virginia in may 1998. Representation theory of finite groups dover books on. After this, much of the basic local representation theory.

Adic groups at the national university of singapore. Here is an overview of the course quoted from the course page. Algebraic and analytic methods in representation theory. Good books in modular representation theory mathoverflow. It begins by establishing the basics of the classical theory of siegel modular forms, and then details more advanced topics. To mention some, there is the systematic use of grothendieck groups leading to the cartan matrix and the decomposition matrix of the group as well as greens direct analysis of indecomposable representations. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Solomon, the representation of finite groups in algebraic number fields, j. A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that f z be holomorphic in the upper halfplane. A brief summary of modular representation theory 3 this is a particuarly useful result, and one that we would hope in general. Modular representation theory of finite and padic groups. Modular representation theory of finite groups by michael. The block theory of finite group algebras by markus. This advanced text develops a new approach to the modular representation theory of finite groups. The action of g on x induces a natural action of g on any block system for x the set of orbits of the gset x is an example of a. It was initially developed by brauer, with a view towards the structure of finite groups.

Galgebras and modular representation theory infoscience. In the semisimple case for exampleover the field of complex numbers one can use character theory to completely. The representation theory of symmetric groups is a special case of the representation theory of nite groups. Let abe a ring and let f abe the category of all left amodules. His contributions provided elementary infrastructure used in algebra, geometry, topology, number theory, and logic.

As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory citation. It emphasizes the similarities between the modular representation theory and the representation theory for quantum groups at roots of unity. Modular representations of finite groups of lie type. It has the lightness of a gentlepaced lecture course and could be used with profit for an introduction to the methods of representation theory of finite groups, either in a formal course or for selfstudy. Walter feit october 26, 1930 july 29, 2004 was an american mathematician who worked in finite group theory and representation theory. In addition to being a leader in the field of modular representation theory, humphreys clarity of exposition is almost universally known. The theory is developed in such a way that almost everything carries over to quantum groups. It is based on the height of a man with his arm raised. Modular representations as an introduction to the local representation theory of finite groups cambridge studies in advanced mathematics book. Good book on representation theory after reading rotman.

Modular representations of symmetric groups charlotte chan sunday, august 7, 2011 1 introduction over the last few weeks, we have seen lots of things in representation theory we began by looking at an onslaught of examples to see explicitly what representations look like, we. Modular theory allows for a wide variety of approaches, including the narrow but fruitful one in alperins book. Galgebras and modular representation theory book, 1995. This 2volume book is a very welcome addition to the existing literature in modular representation theory. The modular representation theory of iwahorihecke algebras and this theory s connection to groups of lie type is an area of rapidly expanding interest. Questions on group and nakayama algebras from a book. It contains a wealth of material much of which is here presented in textbook form for the. Learn more modular representation theory of finite and padic groups lecture notes series, institute for mathematical sciencese, national unversity of singapore.

Modular representation theory of finite groups peter. If i understand you correctly, youre saying that youre not familar with ordinary ie char 0 representation theory. Buy modular representation theory of finite groups paperback at. To mention some, there is the systematic use of grothendieck groups leading to the cartan matrix and the decomposition matrix of the group as well as greens direct analysis. Perhaps the most significant are to number theory, via the theory of automorphic forms, lfunctions and modularity. D j benson the aim of this 1983 yale graduate course was to make some recent results in modular representation theory accessible to an audience ranging from secondyear graduate students to established.

However, in the modular case that is, if p divides the order of g then this result isnt true. The aim of this 1983 yale graduate course was to make some recent results in modular representation theory accessible to an audience. Representation theory studies maps from groups into the general linear group of a finitedimensional vector space. In terms of the associated equivalence relation on x, ginvariance means that. The book is expertly written, though the reader should be warned that it is not meant to be selfcontained. Modular representation theory university of warwick. Anyone wishing to learn the fundamental facts of brauers theory of blocks cannot do better. Modular representation theory is the study of representations of groups and related algebras over fields of positive characteristic. Local representation theory modular representations introduction.

Jantzens representations of algebraic groups is the standard reference for this material. Modular representation theory new trends and methods d. The main themes of this symposium were representations of groups of lie type in nondefining or cross characteristic, and recent developments in block theory. Modular representation theory of finite groups by peter. What alperin does in his book is a direct but somewhat ad hoc treatment of one suggestive small case, which goes back to the pioneering work by brauer and his student nesbitt. Geometric representation theory has close and profound connections to many fields of mathematics, which we expect to play a significant role in the program. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. Hottest modularrepresentationtheory answers mathoverflow. This book presents a new approach to the modular representation theory of a finite group g. Representation theory is applied in number theory, combinatorics and in many areas of algebra. The work concludes with a treatment, new in part, of the brauerdade theory. It provides accessible notes on some complex recent results and introduces the reader to an active area of research in pure mathematics.