Topics in the representation theory of finite groups and related algebras

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Representation theory is a branch of pure mathematics broadly known as abstract algebra. It is a study of linear group actions, and in particular tries to classify the linear actions in terms of certain basic building blocks known as irreducibles.Classical theory (which goes back over a hundred years to Frobenius and Schur) works over the complex numbers; however modern treatments, and in particular the PhD work of my students work over modular fields of (non-zero) prime characteristic where the prime divides the group order. Taking the group to be the symmetric group (comprising all bijections of a finite set) we can ask a fundamental question: what are the dimensions of the irreducibles over fields of prime characteristic? Our work will use modern techniques from algebra together with some computational techniques in GAP to attack this problem, using related algebraic structures known as Schur algebras. We can also use these methods to consider related questions for more general diagram algebras, such as the Brauer algebra and the partition algebra.