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1. 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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2. 𝐾-classes of Brill–Noether Loci and a Determinantal Formula

   https://doi.org/10.1093/imrn/rnab025  

   Anderson, Dave; Chen, Linda; Tarasca, Nicola (April 2021, International Mathematics Research Notices)

   Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $$\rho $$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $$\rho =1$$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $$K$$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux. 

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3. Castelnuovo-Mumford regularity of matrix Schubert varieties

   https://doi.org/10.1007/s00029-024-00959-x  

   Pechenik, Oliver; Speyer, David E; Weigandt, Anna (July 2024, Selecta Mathematica)

   Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order. 

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4. Frozen pipes: lattice models for Grothendieck polynomials

   https://doi.org/10.5802/alco.277  

   Brubaker, Ben; Frechette, Claire; Hardt, Andrew; Tibor, Emily; Weber, Katherine (June 2023, Algebraic combinatorics)

   We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$$ -- {\it biaxial double} $$(\beta,q)$$-{\it Grothendieck polynomials} -- which specialize at $$q=0$ and $v=1$ to double $$\beta$$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $$n$$ pairs of variables is a Drinfeld twist of the $$U_q(\widehat{\mathfrak{sl}}_{n+1})$$ $$R$$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $$\beta$$-Grothendieck polynomials and dual double $$\beta$$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $$\beta$$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $$\beta$$-Grothendieck polynomials, and prove a new branching rule for double $$\beta$$-Grothendieck polynomials. 

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5. Motives of melonic graphs

   https://doi.org/10.4171/AIHPD/156  

   Aluffi, Paolo; Marcolli, Matilde; Qaisar, Waleed (July 2023, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions)

   We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space\mathcal M_{0,4}. We also conjecture that the corresponding polynomials arelog-concave, on the basis of hundreds of explicit computations. 

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