Speaker: Andrew Hardt (UIUC); Title: Factorial Schur functions and the boson-fermion correspondence
Abstract: The boson fermion correspondence is an isomorphism of Heisenberg algebra representations which has connections to the KP hierarchy, vertex operators, symmetric functions, and representations of Lie algebras and quantum groups. It has a long history, dating back to the 1970s, and has been studied extensively by the Kyoto school, by Kac and collaborators, and by many others.
We will discuss a new deformation of this correspondence, joint with Daniel Bump and Travis Scrimshaw. By using a shifted-power analog of fermion fields, we can introduce deformed current operators, half-vertex operators, and vertex operators that satisfy analogous formulas to the classical case. Our deformed half vertex operators correspond to the row transfer matrices of a solvable six vertex model for factorial Schur functions, and our vertex operators are closely related–on the bosonic side–to operators studied by Miyaura and Mukaihira. As a consequence, we obtain an isomorphism of Heisenberg algebra representations where the natural basis of fermionic Fock space maps to the factorial Schur functions.
One of the main goals of this talk will be expository--to give a concrete introduction to some of the basic constructions of the subject.