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Representations of the Symmetric Groups and their connections to other algebras (LYLESU19SCIC2)

  • Full or part time
  • Application Deadline
    Thursday, April 04, 2019
  • Competition Funded PhD Project (Students Worldwide)
    Competition Funded PhD Project (Students Worldwide)

Project Description

The representation theory of the symmetric group is a fascinating study in its own right, as well as being intrinsically linked to that of other fundamental objects, including a wealth of diagram algebras and semigroups. Representations of the symmetric group on n letters over the complex field are well understood since the algebra is semisimple. However, the algebra of the full transformation monoid on n letters over the complex field is not semisimple (for n at least 4) or quite so well understood. But ts representations are ultimately controlled by the representations of the symmetric group.
Representations of the symmetric group over fields of positive characteristic are more difficult. Even though it is possible to construct the irreducible modules explicitly as quotients of the Specht modules, their dimensions are not generally known. Even less is understood about the representation of the transformation monoid in positive characteristic. A constructive approach to the first problem was given by James [i] who developed the use of combinatorial tools, such as diagrams, tableaux and abacuses. This approach generalises in a straightforward way to give techniques for studying representations of related algebras including the Hecke algebras of type A and the Ariki-Koike algebras. See the book [ii] and the survey article [iii] for more details.

Recent work has given us a new line of attack. The cyclotomic quiver Hecke algebras of type A, defined independently by Khovanov and Lauda and by Rouquier have been shown to be isomorphic to Ariki-Koike algebras, which include the symmetric group algebra as special cases. A review can be found in the article [iv].

This PhD project will focus on trying to develop new techniques for studying the representations of the symmetric group, in particular the Specht modules, and seeing how these can be used to give information about connected algebras.

For more information on the supervisor for this project, please go here: https://people.uea.ac.uk/en/persons/s-lyle

Type of programme: PhD

Project start date: October 2019

Mode of study: Full time

Entry requirements: Acceptable first degree - Mathematics. The Standard minimum entry requirement is 2:1.


Funding Notes

This PhD project is in a Faculty of Science competition for funded studentships. These studentships are funded for 3 years and comprise UK/EU fees, an annual stipend of £15,009 and £1,000 per annum to support research training. Overseas applicants may apply but they are required to fund the difference between home/EU and overseas tuition fees (which are detailed on the University’s fees pages at View Website . Please note tuition fees are subject to an annual increase).

References

i) G. D. James (1978). The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer.

ii) A. Mathas (1999). Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series 15, American Mathematical Society.

iii) A. Mathas (2004). The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras, Representation theory of algebraic groups and quantum groups, 261-320, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo.

iv) A. Kleshchev (2010) Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. (N.S.) 47 (3), 419-481.

v) M.S. Putcha, (1996) Complex representations of finite monoids. Proc. London Math. Soc. 73 (3), 623–641.

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