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   https://doi.org/10.1112/plms.12592  

   Eur, Christopher; Fink, Alex; Larson, Matt; Spink, Hunter (March 2024, Proceedings of the London Mathematical Society)

   Abstract We establish a connection between the algebraic geometry of the type permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases. 

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2. Logarithmic concavity of Schur and related polynomials

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   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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3. Stellahedral geometry of matroids

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   Eur, Christopher; Huh, June; Larson, Matt (January 2023, Forum of Mathematics, Pi)

   Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

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4. 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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5. K-theoretic Catalan functions

   https://doi.org/10.1016/j.aim.2022.108421  

   Blasiak, Jonah; Morse, Jennifer; Seelinger, George H (August 2022, Advances in Mathematics)

   We prove that the K-k-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for K-k-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism. 

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