# Guide Ordinary representation theory

As such, their study, which started in the early 19th century, is of fundamental importance, and they have applications in many different domains, such as biology, physics or chemistry.

## Modular representation theory - Wikipedia

They also appear in various areas across mathematics. While groups can be studied abstractly, it is often easier or more interesting to consider them through the way they actually act, especially on particular objects such as vector spaces. Vector spaces are very familiar frameworks, like the 3-dimensional space around us, Einstein's 4-dimensional space-time, or the dimensional space needed for M-theory in string theory.

The study of group actions on vector spaces is known as representation theory, and it has been an active research area in mathematics since the beginning of the 20th century, with ramifications throughout mathematics and other sciences. Of particular interest is the symmetric group, which contains all the possible permutations of a given set of objects. It has long been known that the representation theory of the symmetric group can be described using very elegant combinatorial tools, like integer partitions the different ways to write a given positive integer as a sum of smaller integers.

The study of representations becomes, however, much harder when we let the groups act on vector spaces over fields of prime characteristic, as opposed to the more natural characteristic 0 of the fields of rational, real or complex numbers.

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In this context, we talk about modular representations, while ordinary representations relate to fields of characteristic 0. One long-standing problem in representation theory is the determination of the decomposition matrix, which enlightens the relationship between ordinary and modular representations.

Much work has been done in recent years to try and understand the modular representations of the symmetric groups, and of other related groups such as the alternating groups, or Coxeter groups and complex reflection groups. Part III. Modular representation theory.

The modular theory is more subtle and homological algebra comes into play. We shall consider projective modules, extensions, and filtrations. The ordinary and modular theories are tied together by the study of modules over a local principal ideal domain whose quotient field is of characteristic zero and whose residue field is of characteristic p.

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Block theory leads to deeper arithmetical results on characters through the study of modular representations, and also has many fascinating problems of its own. We hope to end by discussing a famous conjecture of Alperin which is now one of the main focal points in this subject. In the second semester we shall explore the theory of blocks further.

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## ordinary representation theory

To each block there is associated a conjugacy class of p-subgroups called the defect groups. These results yield useful information about the values of characters on p-singular elements. My main area of interest and expertise is the representation theory of Coxeter groups, Iwahori-Hecke algebras and Schur algebras.

Recently my work has focused on the modular representation theory of the Ariki-Koike algebras and the associated cyclotomic q-Schur algebras.

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Contact Research Expert. The representation theory of the symmetric groups is a very dynamic and active field with many challenging open problems. Professor Andrew Mathas. School of Mathematics and Statistics.

The representation theory of the symmetric group over fields of characteristic zero is very well understood. The irreducible modules and their characters have been completely determined for a long time and few problems remain in this area. On contrast, relatively little is known about the representations of the symmetric groups over fields of positive characteristic.