Modular representation theory book

Anyone wishing to learn the fundamental facts of brauers theory of blocks cannot do better. The theory is developed in such a way that almost everything carries over to quantum groups. In terms of the associated equivalence relation on x, ginvariance means that. Search for library items search for lists search for contacts search for a library. For finite groups the theory comes in two distinct flavours. This advanced text develops a new approach to the modular representation theory of finite groups. It contains research works in the areas of modular representation theory of padic groups and finite groups and their related algebras.

Modular representation theory of finite groups peter schneider. Modular representation theory clay mathematics institute. Learn more modular representation theory of finite and padic groups lecture notes series, institute for mathematical sciencese, national unversity of singapore. For questions about modular representation theory, the study of representations over a field of positive characteristic. Good book on representation theory after reading rotman. If i understand you correctly, youre saying that youre not familar with ordinary ie char 0 representation theory. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. His contributions provided elementary infrastructure used in algebra, geometry, topology, number theory, and logic. This book presents a new approach to the modular representation theory of a finite group g. This volume is an outgrowth of the program modular representation theory of finite and padic groups held at the institute for mathematical sciences at national university of singapore during the period of 126 april 20. Modular representation theory of finite groups peter. After this, much of the basic local representation theory. In this survey article, we first discuss the work being done on some outstanding conjectures in the theory.

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. Modular representation theory is a branch of mathematics, and that part of representation. It is based on the height of a man with his arm raised. The modular representation theory of finite groups has its origins in the work of richard brauer.

This 2volume book is a very welcome addition to the existing literature in modular representation theory. 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. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. Questions on group and nakayama algebras from a book. 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. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. Here is an overview of the course quoted from the course page. In fact for any group there will be at least one counterexample. As a text, this book contains ample material for a one semester course.

It was used as a system to set out a number of le corbusiers buildings and was later codified into two books. Modular representations of finite groups of lie type. This book aims to familiarize students at the masters level basic results, tools, and techniques of a beautiful and important algebraic theory. In addition to being a leader in the field of modular representation theory, humphreys clarity of exposition is almost universally known.

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. A brief summary of modular representation theory 3 this is a particuarly useful result, and one that we would hope in general. It was initially developed by brauer, with a view towards the structure of finite groups. The block theory of finite group algebras by markus. The chapter provides basic general definitions concerning algebraic groups and their representations. Modular representation theory of finite groups springerlink. 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. 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.

Representation theory studies maps from groups into the general linear group of a finitedimensional vector space. The representation theory of symmetric groups is a special case of the representation theory of nite groups. In mathematics and group theory, a block system for the action of a group g on a set x is a partition of x that is ginvariant. 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 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. His work helped the development and utilization of sectors like cryptography, chemistry, and physics. However, in the modular case that is, if p divides the order of g then this result isnt true. This book is a comprehensive presentation of the progress in the modular representation theory achieved especially by the work of a.

Serres concise book starts out in a fairly elementary style for an audience of chemists, but by the third part on modular representations the sophistication has escalated a lot. The aim of this 1983 yale graduate course was to make some recent results in modular representation theory accessible to an audience. Modular representation theory of finite groups by peter. 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. 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. Modular theory allows for a wide variety of approaches, including the narrow but fruitful one in alperins book. Modular representation theory university of warwick.

Representations of hecke algebras at roots of unity. Exercises are provided at the end of most sections. Modular representations as an introduction to the local representation theory of finite groups cambridge studies in advanced mathematics book. Solomon, the representation of finite groups in algebraic number fields, j. It provides accessible notes on some complex recent results and introduces the reader to an active area of research in pure mathematics. Geometric representation theory has close and profound connections to many fields of mathematics, which we expect to play a significant role in the program.

Let abe a ring and let f abe the category of all left amodules. We are concerned with recent results on representation theory and cohomology of klr algebras, with emphasis on standard module theory. The work concludes with a treatment, new in part, of the brauerdade theory. It gives an excellent overview of the state of the art in this fascinating subject and also of the many challenging and fundamental. 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. Modular representation theory of finite groups comprises this second situation. 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 begins by establishing the basics of the classical theory of siegel modular forms, and then details more advanced topics. Its aim is to provide a comprehensive treatment of the theory of.

Buy modular representation theory of finite groups paperback at. The representation theory of nite groups has a long history, going back to the 19th century and earlier. 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. Adic groups at the national university of singapore. The book is expertly written, though the reader should be warned that it is not meant to be selfcontained. We then describe work done in the eighties and nineties on modular representations in nondefining characteristic of finite groups of lie type. 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. Modular representation theory of finite groups by michael. Local representation theory modular representations introduction. Algebraic and analytic methods in representation theory. I am working in the field of coding theory and i do not know of any applications of modular representation theory in coding theory.

Good books in modular representation theory mathoverflow. Hottest modularrepresentationtheory answers mathoverflow. On group rings over a modular field which possess radicals expressible as principal ideals, sci. 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. Modular representation theory is the study of representations of groups and related algebras over fields of positive characteristic. Galgebras and modular representation theory book, 1995. Perhaps the most significant are to number theory, via the theory of automorphic forms, lfunctions and modularity. Whilst the theory over characteristic zero is well understood. Download it once and read it on your kindle device, pc, phones or tablets. The modular representation theory of iwahorihecke algebras and this theory s connection to groups of lie type is an area of rapidly expanding interest. Galgebras and modular representation theory infoscience. 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. Jantzens representations of algebraic groups is the standard reference for this material. Modular representation theory new trends and methods d.

It emphasizes the similarities between the modular representation theory and the representation theory for quantum groups at roots of unity. Modular representation theory of finite and padic groups and millions of other books are available for amazon kindle. This is an extended comment, to put julians answer and comments in perspective. In the semisimple case for exampleover the field of complex numbers one can use character theory to completely.

It contains a wealth of material much of which is here presented in textbook form for the. The main themes of this symposium were representations of groups of lie type in nondefining or cross characteristic, and recent developments in block theory. Representation theory is applied in number theory, combinatorics and in many areas of algebra. 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. Modular representation theory of finite and padic groups.