Temperley-Lieb pfaffinants

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 totally-positive matrices, or on Jacobi-Trudi matrices. Our aim is to produce pfaffinants with similar properties.

Our main object is the Temperley-Lieb pfaffinants, a pfaffian-analogue of Rhoades and Skandera's Temperley-Lieb immanants. We show that TL-pfaffinants are positive when evaluated on planar networks and that products of complementary pfaffians can be expressed in terms of TL-pfaffinants in a simple manner. We conjecture that TL-pfaffinants are Schur Q-positive when evaluated on a Schur Q-Jacobi-Trudi matrix. If time permits I will discuss other pfaffinants which may have positivity properties.

This is joint work with Pavlo Pylyavskyy.