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1. Dens, nests and the Loehr-Warrington conjecture

   https://doi.org/10.1090/jams/1057  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G (June 2025, Journal of the American Mathematical Society)

   We prove and extend the longest-standing conjecture in ‘ $q,t$-Catalan combinatorics,’ namely, the combinatorial formula for ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$conjectured by Loehr and Warrington, where $s_{\mathrm{\mu <\#comment/>}}$is a Schur function and $\mathrm{\nabla <\#comment/>}$is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of $G{L}_l$characters involvingSchur Catalanimals; these were recently shown by the authors to represent Schur functions $s_{\mathrm{\mu <\#comment/>}}[-<\#comment/>M{X}^{m,n}]$in subalgebras $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)\subset <\#comment/>\mathcal{E}$isomorphic to the algebra of symmetric functions $\mathrm{\Lambda <\#comment/>}$over $\mathbb{Q}(q,t)$, where $\mathcal{E}$is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths callednests, having endpoints and bounding constraints controlled by data called aden. The special case for $\mathrm{\Lambda <\#comment/>}\left(X^{m,1}\right)$proves the Loehr-Warrington conjecture, giving ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)$our formula implies a new $(m,n)$version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$Loehr-Warrington formula generalize the $(km,kn)$shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations. 

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2. An orthodontia formula for Grothendieck polynomials

   https://doi.org/10.1090/tran/8529  

   Mészáros, Karola; Setiabrata, Linus; StDizier, Avery (January 2021, Transactions of the American Mathematical Society)

   We give a new operator formula for Grothendieck polynomials that generalizes Magyar’s Demazure operator formula for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. 

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3. The Combinatorics Behind the Leading Kazhdan-Lusztig Coefficients of Braid Matroids

   https://doi.org/10.37236/12778  

   Gao, Alice LL; Proudfoot, Nicholas James; Yang, Arthur LB; Zhang, Zhong-Xue (July 2024, The Electronic Journal of Combinatorics)

   Ferroni and Larson gave a combinatorial interpretation of the braid Kazhdan-Lusztig polynomials in terms of series-parallel matroids. As a consequence, they confirmed an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with odd rank, as conjectured by Elias, Proudfoot, and Wakefield. Based on Ferroni and Larson’s work, we further explore the combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids. The main results of this paper include an explicit formula for the leading Kazhdan-Lusztig coefficients of braid matroids with even rank, a simple expression for the number of simple series-parallel matroids of rank $k + 1$ on $2k$ elements, and explicit formulas for the leading coefficients of inverse Kazhdan-Lusztig polynomials of braid matroids. The binomial identity for the Abel polynomials plays an important role in the proofs of these formulas. 

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4. New structure on the quantum alcove model with applications to representation theory and Schubert calculus

   https://doi.org/10.4171/JCA/77  

   Kouno, Takafumi; Lenart, Cristian; Naito, Satoshi (November 2023, Journal of Combinatorial Algebra)

   NA (Ed.)

   The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariantK-group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. 

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5. A Proof of the Extended Delta Conjecture

   https://doi.org/10.1017/fmp.2023.3  

   Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. (January 2023, Forum of Mathematics, Pi)

   Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $$\Delta _{h_l}\Delta ' _{e_k} e_{n}$$ , where $$\Delta ' _{e_k}$$ and $$\Delta _{h_l}$$ are Macdonald eigenoperators and $$e_n$$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $$\operatorname {\mathrm {GL}}_m$$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions. 

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