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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. Schubert line defects in 3d GLSMs. Part II. Partial flag manifolds and parabolic quantum polynomials

   https://doi.org/10.1007/JHEP04(2026)075  

   Closset, Cyril; Gu, Wei; Khlaif, Osama; Sharpe, Eric; Zhang, Hao; Zou, Hao (April 2026, Journal of High Energy Physics)

   A<sc>bstract</sc> We construct Schubert line defects in the 3d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifoldX= Fl(k;n), generalizing our construction for complete flag manifolds given in a companion paper (part I) [1]. In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varietiesX_w⊆Xin the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of [𝒪_w] — more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes [X_w] in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials. 

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3. 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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4. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (June 2025, FPSAC Proceedings)

   We study random permutations corresponding to pipe dreams. Our main model is motivated by the Grothendieck polynomials with parameter ß = 1 arising in the K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of its Grothendieck polyno- mial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process, we describe the limiting permuton and fluctuations around it as the order n of the permutation grows to infinity. The fluctuations are of order n$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for B = 1 Grothendieck polynomials, and provide bounds for general B. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond. 

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5. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}$ {\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m$ {A_{m}}with $m\le n$ {m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}$ {\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$ \square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4$ {D_{4}}which do not arise from an embedding of $A_r$ {A_{r}}with $r\le 3$ {r\leq 3}in $D_4$ {D_{4}}. 

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