# Decomposition numbers for symmetric groups

• Applications accepted all year round
• Awaiting Funding Decision/Possible External Funding

## Project Description

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2016 (funded students) or at any point in the academic year (self-funded students).

This project concerns the representation theory of the symmetric groups in prime characteristic, and lies at the interface of representation theory and combinatorics. The representations of the symmetric group are naturally labelled by integer partitions, and representation theory of symmetric groups is a key example in "combinatorial representation theory", where combinatorial formulae and algorithms describe algebraic phenomena.

The "decomposition number problem" is the main outstanding problem in this area. The idea is as follows: take an irreducible representation of the symmetric group in characteristic 0, and choose a basis such that the representing matrices all have integer entries. Reduce these integers modulo p, and you have a representation in characteristic p. This is not necessarily irreducible, so what are its composition factors? The "decomposition numbers" record how many times each characteristic p irreducible occurs as a composition factor of the p-modular reduction of a given characteristic 0 irreducible.

This problem is wide open in general, but certain special cases are understood. One particular approach is to consider the "p-weight" of a partition; this is a non-negative integer which gives a rough idea of how complicated the corresponding representation is in characteristic p. The idea of the project is to look at partitions of small weight, especially weight 3, where (although we know that the decomposition numbers are all either 0 or 1, it’s hard to describe exactly what they are. A student taking this on should have a good grounding in algebra, and ideally will have seen some representation theory. A good starting point for reading about this area is Gordon James’s book "The representation theory of the symmetric groups".

This project will be supervised by Doctor Matthew Fayers.

Further details can be found in the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/phd%20projects%202015/Algebra/fayers_project_2016-1.pdf

The application procedure is described on the School website. For further enquiries please contact Dr. Matthew Fayers ().

This project is eligible for several sources of full funding for the 2016/17 academic year, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is January 31st 2016.

There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2016.

This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.

## Funding Notes

If you wish to apply for the above funding slots, please visit the application website and mention that you wish to work on the “Decomposition numbers for symmetric groups” project.

School website: View Website

## How good is research at Queen Mary University of London in Mathematical Sciences?

FTE Category A staff submitted: 34.80

Research output data provided by the Research Excellence Framework (REF)

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