Temperley-Lieb Immanants

By Brendon Rhoades, M. Skandera. 2005

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Total positivity in loop groups I: whirls and curls (2008, methodology): “-crossing matchings. An example for n = 5, S = {3,6,7,8,10} is shown in the ﬁgure below. For a subset I ⊂ [n] let I¯:= [n] \I and let I∧ := {2n +1 − i | i ∈ I}. The following results were obtained in [RS]. Theorem 10.4. [RS, Proposition 2.3, Proposition 4.4] If Y is a totally nonnegative matrix, the”
Kazhdan–Lusztig immanants and products of matrix minors (2006, background): “More TNN, SNN and MNN immanants related to the Temperley-Lieb algebra and Bruhat order were studied in [6, 7, 26].”
Schur positivity and Schur log-concavity (2005, background): “In this section we give an overview of some results of Rhoades-Skandera [ RS1 ], [RS2], mostly paraphrasing [RS2, Sections 2–3].,For any permutation v ∈ Sn and a 321-avoiding permutation w ∈ Sn, let fw(v) be the coefficient of the basis element tw ∈ TLn (2) in the basis expansion of θ(Tv )=( ti1 −1)”
On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations (2008, background): “By [16], [30], [29], these are TNN and SNN.”