A graphical calculus for shifted symmetric functions

Abstract

The goal of this thesis is twofold. The fi rst goal is to describe three categori cations of the algebra of symmetric functions and establish relationships between them all. The second goal is to establish an isomorphism between the centre of Khovanov's Heisenberg category [Kho14] and the algebra of shifted symmetric functions defined by Okounkov and Olshanski [OO97]. This isomorphism lends us a graphical description of some important bases of the algebra of shifted symmetric functions. Conversely, we are also able to describe some important generators of the centre of the Heisenberg category in the language of shifted symmetric functions. This turns out to be given in the language of free probability, in particular, the transition and co-transition measures of Kerov [Ker93, Ker00].