Congratulations and thank you to Pavel Turek. Pavel successfully defended his PhD thesis and will soon be leaving for Birmingham and then Japan. The department would like to thank Pavel for his many contributions from helping at outreach events to organising a seminar. 

He writes about his thesis and his time at Royal Holloway: Representation theory studies how to represent symmetries, that is, elements of groups, as matrices. As such, it allows one to transform abstract groups into easy-to-manipulate and concrete objects. When the group in question is a symmetric group, there is a remarkable connection between its representations and symmetric functions. My thesis, written under the supervision of Professor Mark Wildon, builds upon this connection to establish new results about representations of symmetric groups. The examiners were Professor Matthew Fayers (QMUL) and Professor Radha Kessar (Manchester). 

 

The main task of the thesis is to classify induced representations which are multiplicity-free. As the main result, the full classification of subgroups from which an irreducible representation can be induced and remain multiplicity-free is established, and for the majority of these subgroups, all choices of these irreducible representations are found. The previously existing classification of multiplicity-free permutation representations is then an immediate consequence. On the side of the symmetric functions, the thesis contains a new proof of the formula for the product of certain symmetric functions known as the plethystic Murnaghan–Nakayama rule, which uses the labelled abacus.

 

During my stay at Royal Holloway, I had a chance to broaden my mathematical knowledge to (geometric) group theory by taking part in two reading groups and to meet brilliant mathematicians from both inside and outside Royal Holloway at the weekly mathematical seminars.

 

In the second half of 2025, I will be working at the University of Birmingham with Dr. Stacey Law. Next year, I will join the Representation Theory and Algebraic Combinatorics Unit at the Okinawa Institute of Science and Technology, Japan, led by Professor Liron Speyer.