Temperley-Lieb Immanants, Key Positivity, and Demazure Crystals

The main goal of this paper is to extend three important Schur positivity results to key positivity, replacing all Schur polynomials in relevant expressions with flagged Schur polynomials. Namely, we first show that the Temperley-Lieb immanants of (many) flagged Jacobi-Trudi matrices are key positive. Using this result, we give a combinatorial rule for the key expansion of (most) products of flagged skew Schur polynomials, and also give a log concavity result inspired by that of Lam-Postnikov-Pylyavskyy. The main tools in our proofs are Demazure crystals, and the recently defined shuffle tableaux of Nguyen and Pylyavskyy. In order to prove our main results, we must develop a new characterization of Demazure crystals, which builds off of prior work of Assaf and Gonzalez. This characterization may be useful in other contexts.

💡 Research Summary

The paper “Temperley‑Lieb Immanants, Key Positivity, and Demazure Crystals” extends three classic Schur‑positivity results to the setting of key polynomials, replacing Schur functions with flagged Schur functions throughout. The authors’ primary achievement is Theorem 1.1, which states that for a strict partition λ, a non‑decreasing flag β, and any partition µ, all Temperley‑Lieb immanants of the flagged Jacobi‑Trudi matrix A^{β}_{λ,µ} expand as a non‑negative linear combination of key polynomials. The strictness of λ is essential; the authors provide explicit counter‑examples when λ is not strict, showing that the Demazure crystal structure breaks down in those cases.

The proof of Theorem 1.1 proceeds in two main steps. First, the combinatorial definition of Temperley‑Lieb immanants, due to Nguyen and Pylyavskyy, is interpreted in terms of shuffle tableaux—objects that simultaneously encode a pair of skew tableaux interlaced in a prescribed way. Second, the authors develop a new, highly local set of axioms characterizing Demazure crystals, building on earlier work of Assaf and González. These axioms involve (i) the behavior of i‑strings, (ii) the existence of extremal elements, (iii) closure under the flagged shuffle operation, (iv) compatibility of Bruhat order with dominance order, and (v) preservation of braid relations. By verifying that flagged shuffle tableaux satisfy these axioms (under the strict‑λ hypothesis), they conclude that the set of tableaux contributing to a given Temperley‑Lieb immanant forms a Demazure subcrystal, whose character is precisely a key polynomial. Hence each immanant is key‑positive.

With Theorem 1.1 in hand, the authors obtain two significant applications. The first is a key‑positivity result for products of flagged skew Schur polynomials. If two skew shapes λ/µ and ν/ρ satisfy (i) identical row lengths (λ_i = ν_i for all i) and (ii) an interlacing condition (λ_i ≥ ν_{i+1} and ν_i ≥ λ_{i+1}), then the product s^{β}{λ/µ}·s^{β}{ν/ρ} expands positively in the key basis. The authors give an explicit combinatorial algorithm for this expansion, extending a theorem of Reiner‑Shimozono. They also note that without these hypotheses the product can fail to be key‑positive, and they provide several counter‑examples.

The second application is a log‑concavity type inequality for flagged Schur functions in the key setting. Inspired by the conjecture of Lam, Postnikov, and Pylyavskyy on Schur log‑concavity, the authors prove that for skew shapes satisfying the same two conditions as above, the expression \