The University of Michigan Combinatorics Seminar


Abstract 

I will discuss a pfaffian analogue of immanants, which we call pfaffinants. Roughly speaking, a pfaffinant is a linear combination of the monomials which occur in a pfaffian. Certain immanants are known to possess positivity properties when evaluated on totallypositive matrices, or on JacobiTrudi matrices. Our aim is to produce pfaffinants with similar properties. Our main object is the TemperleyLieb pfaffinants, a pfaffiananalogue of Rhoades and Skandera's TemperleyLieb immanants. We show that TLpfaffinants are positive when evaluated on planar networks and that products of complementary pfaffians can be expressed in terms of TLpfaffinants in a simple manner. We conjecture that TLpfaffinants are Schur Qpositive when evaluated on a Schur QJacobiTrudi matrix. If time permits I will discuss other pfaffinants which may have positivity properties. This is joint work with Pavlo Pylyavskyy. 