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1. Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

   https://doi.org/10.1093/imrn/rnz261  

   Billey, Sara C.; Rhoades, Brendon; Tewari, Vasu (November 2019, International Mathematics Research Notices)

   Abstract Let $$k \leq n$$ be positive integers, and let $$X_n = (x_1, \dots , x_n)$$ be a list of $$n$$ variables. The Boolean product polynomial$$B_{n,k}(X_n)$$ is the product of the linear forms $$\sum _{i \in S} x_i$$, where $$S$$ ranges over all $$k$$-element subsets of $$\{1, 2, \dots , n\}$$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $$B_{n,k}(X_n)$$ for certain $$k$$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $$B_{n,n-1}(X_n)$$ to a bigraded action of the symmetric group $${\mathfrak{S}}_n$$ on a divergence free quotient of superspace. 

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2. On Hodge-Riemann Cohomology Classes

   Julius Ross and Matei Toma (May 2023, Birational Geometry, Kähler–Einstein Metrics and Degenerations)

   Ivan Cheltsov, Xiuxiong Chen (Ed.)

   We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian. 

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3. Principal Specialization of Dual Characters of Flagged Weyl Modules

   https://doi.org/10.37236/10460  

   Mészáros, Karola; St. Dizier, Avery; Tanjaya, Arthur (October 2021, The Electronic Journal of Combinatorics)

   Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We prove a lower bound on the principal specialization of dual characters of flagged Weyl modules. Our result yields an alternative proof of a conjecture of Stanley about  the principal specialization of Schubert polynomials, originally proved by Weigandt. 

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4. Polynomials from Combinatorial -theory

   https://doi.org/10.4153/S0008414X19000464  

   Monical, Cara; Pechenik, Oliver; Searles, Dominic (February 2021, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $$K$$ -theoretic deformation of the quasi-key basis and also a lift of the $$K$$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $$K$$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $$K$$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $$K$$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations. 

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5. Logarithmic concavity of Schur and related polynomials

   https://doi.org/10.1090/tran/8606  

   Huh, June; Matherne, Jacob; Mészáros, Karola; St. Dizier, Avery (January 2022, Transactions of the American Mathematical Society)

   We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients in the special case of Kostka numbers. 

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