Mikhail Muzychuk - Böcker

The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irreducible characters of finite groups. Today, table algebra theory is a well-established branch of modern algebra with various applications, including the representation theory of finite groups, algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of degree 3. Divided into 4 parts, the first gives an outline of the classification approach, while remaining parts separately treat special cases that appear during classification. A particularly unique contribution to the field, can be found in part four, whereby a number of the algebras are linked to the polynomial irreducible representations of the group SL3(C).This book will be of interest to research mathematicians and PhD students working in table algebras, group representation theory, algebraic combinatorics and integral fusion rule algebras.

The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irreducible characters of finite groups. Today, table algebra theory is a well-established branch of modern algebra with various applications, including the representation theory of finite groups, algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of degree 3. Divided into 4 parts, the first gives an outline of the classification approach, while remaining parts separately treat special cases that appear during classification. A particularly unique contribution to the field, can be found in part four, whereby a number of the algebras are linked to the polynomial irreducible representations of the group SL3(C).This book will be of interest to research mathematicians and PhD students working in table algebras, group representation theory, algebraic combinatorics and integral fusion rule algebras.

of of classes of finite Properties products conjugacy groups 4r6radoldbrancf, This of finite was studied in the 1980's. The group theory. topic intensively book "Products of Classes in edited Z. Arad and [22] Conjugacy Groups," by M. a of the results obtained this Herzog, gives comprehensive picture during period. It realized several authors that this research could be extended to was by of irreducible characters. Werefer the reader to the products papers [1, 2, 11, 13-16,21,23,35,40,51,52,651. In several of these the authors found an between analogy pr- papers of classes and of irreducible characters which led to ucts conjugacy products of table introduced H.L.Blau and Z. Arad in in the notion algebra, by [7], in uniform the of of order to a decomposition products study way conjugacy and irreducible characters of finite Since the classes groups. then, theory in of Z. - of table was H. algebras extensively developed papers Arad, F. D. M.R. E. H. J. isha, Blau, B-dnger, Chillag, Darafsheh, Erez, Fisman, V. M. A. C. and B. Xu Miloslavsky, Muzychuk, Rahnamai, Scopolla [3-5,7- 10,12,17-20,25,29-33,35,41]. Table as be onsidered a class of defined, c special algbras, may C-algbras Y.Kawada and G. Hoheisel table introduced a by [49] [48].:More precisely, where the is a structure constants are Each algebra C-algebra nonnegative. finite two natural table the table of yields algebras: algebra conjugacy group classes and the table of characters.

In 2006 a special semester on Gr. obner bases and related methods was or- nized by RICAM and RISC, directed by Bruno Buchberger and Heinz Engl. The main focus of the semester were the development of the formal theory of Gr. obner bases (brie?y GB), the e?cient implementation of all algorithms related to this theory, and the promotion of recent and new applications of GB. The workshop D1 "Gr. obner bases in cryptography, coding theory and - gebraic combinatorics", Linz, May 1-6, 2006 (chairmen M. Klin, L. Perret, M. Sala) was one of the main ingredients of the semester. The last two days of this workshop, devoted to combinatorics, made it possible to bring together experts in algorithmic problems related to coherent con?gurations and as- ciation schemes with a community of people working in the area of GB. Each side was interested in understanding the computational problems and current algorithmicpossibilitiesoftheother,withaparticularobjectiveofintroducing the practical use of GB in algebraic combinatorics. Materials (mainly slides of lectures and posters) available from the site http://www.ricam.oeaw.ac.at/specsem/srs/groeb/schedule D1.htmlprovidea helpful and vivid picture of the successful exchange of scienti?c information during the workshop D1. Asafollow-uptothespecialsemester,10volumesofproceedingsarebeing published by di?erent publishers. The current collection of papers re?ects diverse investigations in the area of algebraic combinatorics (with or without explicit use of GB), but with a de?nite emphasis on algorithmic approaches.