Temperley-Lieb Immanants

摘要

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form $ Sigma _{{upsigma in S_{n} }} f(upsigma )x_{{1,upsigma (1)}} cdots x_{{n,upsigma (n)}} $ . The cone generated by these polynomials contains all totally nonnegative polynomials of the form $ Delta _{{J,J' }} (x)Delta _{{L,L' }} (x) - Delta _{{I,I' }} (x)Delta _{{K,K' }} (x) $ , where, $ Delta _{{I,I' }} (x), ldots ,Delta _{{K,K' }} (x) $ are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative.