# Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup Q(n)

*Algebra/Geometry/Topology Seminar*

*by Sasha Kleshchev*

*Institution:*Oregon

*Date: Fri 16th March 2012*

*Time: 3:15 PM*

*Location: 213 Richard Berry*

*Abstract*: There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects representation theory of the supergroup Q(n) and projective representation theory of the symmetric group via appropriate Schur algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type \(A_{p-1}^{(2)}\) . We explain how to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups. This is achieved by developing the theory of modular lowering operators for the supergroup Q(n) which is parallel to (although much more intricate than) the similar theory for GL(n) , first developed by the speaker in mid-1990's. The results are joint with Vladimir Shchigolev.