, Volume 9, Issue 4, pp 451-494

Temperley-Lieb Immanants

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Abstract.

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.

Received September 30, 2004