Projective Representations of the Symmetric Group

Summer 2016

Eric Chen : Mathematics

Mentor: Vera Serganova

Lately, we have witnessed increased interest in the study of representations of symmetric groups, and in particular, in their projective representations. In a classic paper, I. Schur introduced what are now known as Schur Q-functions in order to calculate these projective characters; combinatorial formulas for these characters are also available in the early works of D. Littlewood and A. Richardson. On the other hand, the Schur Q-functions admit a natural Hopf algebra structure paralleling the classical case of symmetric polynomials, and it is well known that the Hopf algebra approach is indispensable in understanding the classification of linear representations of the symmetric group. This summer, I will be investigating what new information can be extracted from the Hopf algebra structure on the ring of Schur Q-functions regarding the projective representations of symmetric groups. More specifically, by interpreting the multiplication and comultiplication operations, it is possible to study projective representations in a similar vein to the linear case, where comultiplication (resp. multiplication) is interpreted as restriction (resp. induction) functors.

I would like to thank the Rose Hills Foundation for its generous support, without which I would not be able to carry out the project with undivided attention this summer. Zelevinsky's Hopf algebra approach in studying representations of the symmetric group was one of the most exciting ideas presented to me during my time at the Berkeley math department. I am enthused to be able to work on developing this idea further and contribute to the existing theory with the professor who first introduced me to representation theory.