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1. Doppelgängers: Bijections of Plane Partitions

   https://doi.org/10.1093/imrn/rny018  

   Hamaker, Zachary; Patrias, Rebecca; Pechenik, Oliver; Williams, Nathan (March 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract We say two posets are doppelgängers if they have the same number of P-partitions of each height k. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing K-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman’s rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the 1st bijective proof of a 1983 theorem of R. Proctor—that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a shifted trapezoid. 

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2. Decompositions of Grothendieck Polynomials

   https://doi.org/10.1093/imrn/rnx207  

   Pechenik, Oliver; Searles, Dominic (September 2017, International Mathematics Research Notices)

   null (Ed.)

   Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $$K$$-theory of flag varieties (in type $$A$$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $$K$$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $$n$$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $$\beta$$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $$K$$-theory, and we state our results in this more general context. 

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3. An inverse Grassmannian Littlewood–Richardson rule and extensions

   https://doi.org/10.1017/fms.2024.65  

   Pechenik, Oliver; Weigandt, Anna (December 2024, Forum of Mathematics, Sigma)

   Chow rings of flag varieties have bases of Schubert cycles \sigma_u, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood-Richardson rules solve this problem for special products \sigma_u \cdot \sigma_v where u and v are p-Grassmannian permutations. Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product \sigma_u \cdot \sigma_v when u is p-inverse Grassmannian and v is q-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for \sigma_u \cdot \sigma_v in the case that u is covered in weak Bruhat order by a p-inverse Grassmannian permutation and v is a q-inverse Grassmannian permutation. 

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4. Cluster algebra structures on Poisson nilpotent algebras

   https://doi.org/10.1090/memo/1445  

   Goodearl, K; Yakimov, M (October 2023, Memoirs of the American Mathematical Society)

   Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements. 

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5. Structure constants in equivariant oriented cohomology of flag varieties

   https://doi.org/10.1007/s00209-024-03485-w  

   Goldin, Rebecca; Zhong, Changlong (July 2024, Mathematische Zeitschrift)

   We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results. 

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