#Grothendieck polynomials
Grothendieck polynomials are closely related to the Schubert polynomials. They are the polynomial representatives naturally attached to \(K\)-theoretic Schubert calculus. They were introduced by A. Lascoux in 1990 [Las90].
N. J. Y. Fan and P. L. Guo give formulas for Grothendieck polynomials as generating functions of set-valued Rothe tableaux [FG21]. Their tableaux extend the usual Rothe-diagram description of Schubert polynomials and include specializations for vexillary, dominant, and Grassmannian permutations.
In [BFHT+20], the authors introduce biaxial Grothendieck polynomials, which generalize both the double Grothendieck polynomials and the double dual Grothendieck polynomials. They prove several results about Grothendieck polynomials, including branching rules and Cauchy identities. They also provide solvable lattice models whose partition functions give Grothendieck polynomials and their duals.
H. Tamvakis defines skew Grothendieck polynomials for skew elements in Weyl groups of classical type and proves set-valued tableau formulas for the skew double Grothendieck polynomials in types \(A,\) \(B,\) \(C,\) and \(D\) [Tam24]. This also gives tableau formulas for Grassmannian Grothendieck polynomials and \(K\)-theoretic skew Stanley functions in these types. E. Marberg and B. Pawlowski study structural properties of symplectic Grothendieck polynomials [MP21]. This fits with the type \(C\) side of \(K\)-theoretic Schubert calculus and the nearby \(K\)-theoretic Schur \(P\)/\(Q\) families. For a fixed-point-free involution \(z,\) J. C.-A. Chou defines a symplectic Rajchgot index and uses it to compute the degree of the symplectic Grothendieck polynomial and the regularity of the corresponding skew-symmetric matrix Schubert variety. He also identifies its top homogeneous component with that of an ordinary Grothendieck polynomial determined by the match code of \(z\) [Cho26, Thms. 1.1–1.2 and Cor. 1.3]. E. Marberg derives transition equations for classical double Grothendieck polynomials, extending the type \(A\) \(K\)-theoretic transition relations to types \(B,\) \(C,\) and \(D\) [Mar25]. As an application, the corresponding \(K\)-Stanley symmetric functions expand positively in \(K\)-theoretic Schur \(P\)- and \(Q\)-functions.
#Operator definition
Define \[\partial_i(f)=\frac{f-s_i(f)}{x_i-x_{i+1}}, \qquad 1 \leq i \leq n-1,\] as on the Schubert-polynomial page, and let \[\pi_i(f) \coloneqq \partial_i((1-x_{i+1})f).\]
The Grothendieck polynomials are then defined as \[\grothendieck_\omega(x_1,\dotsc,x_n) \coloneqq \pi_{\omega^{-1}\omega_0}(x_1^{n-1}x_2^{n-2}\dotsm x_{n-1}).\]
Note that the lowest-degree homogeneous part of \(\grothendieck_\omega(\xvec)\) is the Schubert polynomial \(\schubert_\omega(\xvec).\)
There is a connection with Newton polytopes just as for Schubert polynomials; see [MD17].
#Pipe dreams
As with Schubert polynomials, there is a way to describe Grothendieck polynomials using pipe dreams; see [FK94, KM04]. The double Grothendieck polynomials are given as \[\grothendieck^{(\beta)}_\omega(\xvec,\yvec) = \sum_{P \in PD(\omega)} \beta^{-\length(\omega)} \prod_{(i,j)\in S(P)} (\beta x_i \oplus y_j)\] where we sum over pipe dreams and \(a \oplus b = a+b+\beta ab.\) Here, \(S(P)\) is the set of crosses of the pipe dream; this particular version of the formula is from [Wei21, Thm. 6.1].
A. Weigandt expresses the double Grothendieck polynomials as a sum over bumpless pipe dreams [Wei21, Thm. 1.1]: \[\grothendieck^{(\beta)}_\omega(\xvec,\yvec) = \beta^{-\length(\omega)} \sum_{P \in BPD(\omega)} \prod_{(i,j)\in D(P)} \beta (x_i \oplus y_j) \prod_{(i,j)\in NW(P)} \left(1+ \beta (x_i \oplus y_j)\right)\] where \(BPD(\omega)\) is the set of bumpless pipe dreams of type \(\omega\) and \(NW(P)\) is the set of West-to-North pipe bends.
However, this formula does not immediately give the same pipe dream expression as the Fomin–Kirillov formula, and it is an open problem to find a bijection between the two models.
V. Buciumas and T. Scrimshaw give a lattice model and relate it to bumpless pipe dreams [BS20].
Karola Mészáros, L. Setiabrata, and Avery St. Dizier give a Grothendieck orthodontia formula for Grothendieck polynomials [MSD22]. It extends Magyar’s orthodontia formula for Schubert polynomials to the \(K\)-theoretic setting.
C.-A. Chou and T. Yu give an explicit algorithm for the unique maximal pipe dream whose row and column weights are determined by the \(\mathsf{rajcode}\) statistic of O. Pechenik, D. Speyer, and A. Weigandt [CY23]. Their construction uses ladder moves and solves the direct-recipe problem for this top-degree term of double Grothendieck polynomials. C.-A. Chou and T. Yu also study Grothendieck polynomials of inverse fireworks permutations from the pipe-dream viewpoint [CY24]. C.-A. Chou and L. Setiabrata prove saturated Newton polytope results for fireworks Grothendieck polynomials [CS25]. They characterize the support by divisibility between the Schubert support and the maximal Grothendieck monomial, prove \(M\)-convexity after homogenization, and compare supports with those of suitable layered permutations.
For the principal specialization \(\Upsilon_w(\beta)=\mathfrak G_w^{(\beta)}(1,1,\dotsc),\) H. Bai, F. Gu, P. L. Guo, and J. Liu give pattern-occurrence expansions with nonnegative coefficients when \(w\) avoids \(1423\) or \(1342\) [BGGL26]. Their pipe-dream reduction algorithm partially resolves conjectures of Gao, Gaetz, Mészáros–Tanjaya, and Dennin.
M. Dreyer, Karola Mészáros, and A. S. Dizier give an alternative proof of the Pechenik–Speyer–Weigandt degree formula for Grothendieck polynomials, using the climbing-chain model [DMD24]. They also determine the leading monomials of the homogeneous components for term orders with \(x_1 \lt{} x_2 \lt{} \dotsb \lt{} x_n.\) J. Rajchgot, Y. Ren, C. Robichaux, Avery St. Dizier, and A. Weigandt relate degrees of symmetric Grothendieck polynomials to Castelnuovo–Mumford regularity [RRRD+21]. E. S. Hafner studies supports of Castelnuovo–Mumford polynomials, the top-degree components of Grothendieck polynomials [Haf26]. She proves the expected \(M\)-convex support property for new families of permutations and studies when the support is the set of integer points in the relevant Newton polytope, a schubitope.
C. Robichaux gives a minimal counterexample in \(\symS_8\) to the Ross–Yong \(K\)-Kohnert conjecture for Grothendieck polynomials [Rob24]. The same paper proposes a revised \(K\)-theoretic Kohnert rule for Grothendieck polynomials and proves both the original and revised rules in the \(321\)-avoiding case.
Avery St. Dizier subsequently gives further counterexamples to Robichaux’s ghost \(K\)-Kohnert rule [Diz26]. Both the Ross–Yong and ghost rules nevertheless hold for \(1432\)-avoiding permutations, and the paper characterizes when a ghost \(K\)-Kohnert diagram contributes to the Grothendieck polynomial.
E. S. Hafner studies vexillary Grothendieck polynomials through bumpless pipe dreams [Haf22]. She characterizes the bumpless pipe dreams contributing maximal-degree terms and compares the Pechenik–Speyer–Weigandt and Rajchgot–Robichaux–Weigandt degree formulas in the vexillary case. A. Gregory, Z. Hamaker, and T. Yu give a tableau formula for vexillary double \(\beta\)-Edelman–Greene coefficients [GHY24]. The formula is manifestly \(\beta\)-Graham positive and refines known positivity for these coefficients.
O. Pechenik and M. Satriano prove a conjectured Möbius-inversion formula for Grothendieck polynomials [PS24]. In the case where the corresponding Schubert polynomial is a multiplicity-free monomial sum, the Grothendieck polynomial is recovered from it by Möbius inversion on an associated poset.
#Alternating sign matrices
There is an expression for Grothendieck polynomials using alternating sign matrices, due to A. Lascoux.
#Stable Grothendieck polynomials
The stable Grothendieck polynomials were introduced by S. Fomin and A. Kirillov in [FK94, FK96]. Similar to Stanley symmetric functions, the stable Grothendieck polynomials are defined as the limit \[\grothendieckStable_\omega(\xvec) \coloneqq \lim_{k\to \infty} \grothendieck_{1^k \times \omega}(\xvec).\]
D. Yeliussizov studies positive specializations of symmetric Grothendieck polynomials, giving a \(K\)-theoretic counterpart to the Edrei–Thoma style parametrization for Schur-positive specializations [Yel20].
J. Morse, J. Pan, W. Poh, and A. Schilling construct a \(K\)-theoretic type \(A\) crystal on decreasing factorizations in the \(0\)-Hecke monoid [MPPS20]. Their insertion map relates this model to Hecke insertion, set-valued tableaux, and uncrowding.
B. Brubaker, A. Suki Dasher, M. Hu, N. Jain, Y. Li, Y. Lin, M. Mihaila, V. Tran, and I. Deniz Unel give a solvable lattice model for twisted Kirillov polynomials obtained from universal divided-difference operators [BDHJ+26]. Their model specializes to Schubert, \(\beta\)-Grothendieck, dual \(\beta\)-Grothendieck, and Di Francesco–Zinn-Justin polynomials, and proves Kirillov’s nonnegativity conjecture for the Hecke–Grothendieck specialization.
#Grassmann Grothendieck polynomials
See Grassmannian permutations for the definition of a Grassmannian permutation, and for how to associate a partition to such a permutation. For a paper with many references and results, see [MPS18].
Grassmann Grothendieck polynomials are the stable Grothendieck polynomials indexed by Grassmannian permutations. They are symmetric polynomials and serve as a \(K\)-theoretic analogue of the Stanley symmetric functions. They can be computed by a combinatorial formula using set-valued tableaux: \[\begin{aligned} \grothendieckStable_\lambda(\xvec) = \sum_{T \in \mathrm{SVT}(\lambda)} (-1)^{|T|-|\lambda|} \xvec^T. \end{aligned}\] Equivalently, let \[\grothendieckStable^{(\beta)}_\lambda(\xvec) \coloneqq \sum_{T \in \mathrm{SVT}(\lambda)} \beta^{|T|-|\lambda|}\xvec^T.\] Thus the unadorned stable Grothendieck polynomial above is the specialization \(\beta=-1.\) The set \(\mathrm{SVT}(\lambda)\) consists of all set-valued tableaux of shape \(\lambda,\) where each box is filled with a nonempty set of natural numbers. Rows must be weakly increasing and columns must be strictly increasing. The sum of the sizes of the sets in \(T\) is denoted \(|T|.\)
Example (Monical’s Lascoux-atom expansion).
Fix three variables and pad partitions with zeros to length \(3.\) Monical’s Lascoux atoms refine the beta Grothendieck polynomial by \[\grothendieckStable^{(\beta)}_\lambda(x_1,x_2,x_3) = \sum_{\sort(\alpha)=\lambda} \lascouxAtom^{(\beta)}_\alpha(x_1,x_2,x_3),\] where the sum is over the weak compositions obtained by permuting the parts of \(\lambda,\) including the padded zeros [Mon17, Thm. 1.2]. For example, \[\begin{aligned} \grothendieckStable^{(\beta)}_{(1)}(x_1,x_2,x_3) &= \lascouxAtom^{(\beta)}_{100} +\lascouxAtom^{(\beta)}_{010} +\lascouxAtom^{(\beta)}_{001},\\ \grothendieckStable^{(\beta)}_{(1,1)}(x_1,x_2,x_3) &= \lascouxAtom^{(\beta)}_{110} +\lascouxAtom^{(\beta)}_{101} +\lascouxAtom^{(\beta)}_{011},\\ \grothendieckStable^{(\beta)}_{(2,1)}(x_1,x_2,x_3) &= \lascouxAtom^{(\beta)}_{210} +\lascouxAtom^{(\beta)}_{201} +\lascouxAtom^{(\beta)}_{120} +\lascouxAtom^{(\beta)}_{102} +\lascouxAtom^{(\beta)}_{021} +\lascouxAtom^{(\beta)}_{012}. \end{aligned}\] The last line is the decomposition attached to the rearrangements of \((2,1,0),\) the small family illustrated in Monical’s examples.
A Gelfand–Tsetlin pattern-type combinatorial formula is described in [MS14], where marked GT-patterns are used to give the monomial expansion of \(\grothendieckStable_\lambda.\) See also [MPS18], where they give an alternative proof of this.
#Homogeneous components and 0-Hecke modules
At \(\beta=1,\) the set-valued tableau formula is unsigned. For \(m\geq |\lambda|,\) let \(G_{\lambda,m}^{(1)}\) be the homogeneous component of degree \(m.\) If \(\operatorname{StdSVT}_m(\lambda)\) denotes the standard set-valued tableaux of shape \(\lambda\) with \(m\) entries, then Y.-H. Kim proves [Kim26, Thm. 3.6] \[G_{\lambda,m}^{(1)} = \sum_{T\in\operatorname{StdSVT}_m(\lambda)} \gessel_{\operatorname{comp}(\DES(T))}.\]
He equips the span of these tableaux with an \(H_m(0)\)-action whose quasisymmetric characteristic is \(G_{\lambda,m}^{(1)}\) [Kim26, Thm. 3.13]. Moreover, this module decomposes as a direct sum of weak Bruhat interval modules [Kim26, Thm. 4.12]. This realizes each homogeneous component by a combinatorial module rather than merely by a sum of one-dimensional simple modules.
#Set-valued tableaux, cells, and crystals
E. Smirnov constructs an explicit weight-preserving bijection between set-valued tableaux of shape \(\lambda\) and the efficient cells in a cellular decomposition of the Gelfand–Tsetlin polytope [Smi26, Thm. 4.3.1]. Under this bijection, the number of excess tableau entries is the dimension of the cell. Consequently, Buch’s tableau formula and the cellular formula for \(G_\lambda^{(\beta)}\) agree term by term. Transported square-root crystal operators alternate between two adjacent cell dimensions along double strings, although consecutive cells need not be incident [Smi26, Thms. 6.1.1 and 6.2.2].
For a skew shape \(\lambda/\mu,\) T. Shimazaki gives weight-preserving bijections among semistandard set-valued tableaux, marked Gelfand–Tsetlin patterns, and decorated states of the five-vertex model [Shi26, Thms. 4.6 and 5.11]. Tableau excess becomes the number of marks or nontrivial bumps, and the ordinary type \(A\) crystal operators correspond to local moves on the lattice-model states [Shi26, Thm. 6.9]. Fibers with fixed minimum or maximum tableau are graded Boolean lattices, and the sign-reversing involution moves along one Boolean edge whenever it is nontrivial.
#Weyl identity
A Weyl-type bialternant formula is given by the following quotient, see [IN13, Kir16, Iwa19].
\[\begin{aligned} \grothendieckStable_\lambda(x_1,\dotsc,x_n) = \frac{ \det \left( x_i^{\lambda_j+n-j}(1+\beta x_i)^{j-1} \right)_{1\leq i,j \leq n} }{ \prod_{1 \leq i \lt j \leq n} (x_i - x_j) } \qquad (\beta = -1). \end{aligned}\]
#Jacobi–Trudi identity
There is a Jacobi–Trudi type identity, given by [Kir16, Thm. 1.10], \[\begin{aligned} \grothendieckStable_\lambda(\xvec) = \det\left( \sum_{m=0}^\infty \binom{i-1}{m} \beta^m \completeH_{\lambda_i-i+j+m}(\xvec) \right)_{1\leq i,j \leq n}. \end{aligned}\]
#Pieri rule
C. Lenart [Len00] proved the following Pieri rule for stable Grassmann Grothendieck polynomials: \[\grothendieckStable_{(a)}\grothendieckStable_{\lambda} = \sum_{\mu / \lambda \text{ horizontal strip}} (-1)^{|\mu/\lambda|-a} \binom{ \textrm{rows}(\mu/\lambda)-1 }{ |\mu/\lambda|-a } \grothendieckStable_{\mu}\] where \(\textrm{rows}(\mu/\lambda)\) is the number of rows in the horizontal strip.
#Murnaghan–Nakayama rule
In [NHST23], the following Murnaghan–Nakayama type rule is the main result. \[\begin{aligned} \grothendieckStable_\lambda(\xvec) \powerSum_k(\xvec) = \sum_{\mu} (\beta)^{|\mu|-|\lambda|-k} (-1)^{k-c(\mu/\lambda)} \binom{r(\mu/\lambda)-1}{k-c(\mu/\lambda)} \grothendieckStable_\mu(\xvec) \end{aligned}\] where the sum runs over all connected skew shapes \(\mu/\lambda,\) containing at least one \(k\)-ribbon along its northwestern border. Here, \(c(\mu/\lambda)\) is the number of columns spanned by the shape. Likewise, \(r(\mu/\lambda)\) is the number of rows spanned by the shape.
Note that at \(\beta=0,\) we recover the classical Murnaghan–Nakayama rule for Schur polynomials.
#Schur expansion
C. Lenart proves [Len00, Thm. 2.8] that whenever \(k\geq \length(\lambda),\) \[\grothendieckStable_{\lambda}(x_1,\dotsc,x_k) = \sum_{\lambda \subseteq \mu \subseteq \hat{\lambda}} (-1)^{|\mu|-|\lambda|} a_{\lambda\mu} \schurS_{\mu}(\xvec)\] where the sum is taken over partitions \(\mu,\) and \(\hat{\lambda}\) is the unique maximal partition of length \(k\) obtained from \(\lambda\) by adding at most \(j-1\) boxes to the \(j\)-th row of the Young diagram \(\lambda.\)
The coefficients \(a_{\lambda\mu}\) are nonnegative, and count the number of increasing tableaux of shape \(\mu/\lambda.\) These tableaux are strictly increasing along rows and columns, and for each \(i\geq 1,\) row \(i\) only contains entries \(\leq i-1.\)
Alternatively, in [MPS18, Cor. 3.11], it is proved that \(a_{\lambda\mu}\) count the number of set-valued tableaux of shape \(\lambda\) and weight \(\mu,\) which are also Yamanouchi. The authors prove this by defining a type \(A\) crystal graph on set-valued fillings.
#Littlewood–Richardson rule
There is a Littlewood–Richardson rule for the product of stable Grothendieck polynomials: \[\grothendieckStable_{\lambda}(\xvec;\yvec) \grothendieckStable_{\mu}(\xvec;\yvec) = \sum_{\nu} (-1)^{|\lambda|+|\mu|-|\nu|} c_{\lambda\mu}^{\nu} \grothendieckStable_{\nu}(\xvec;\yvec)\] where \(c_{\lambda\mu}^{\nu}\) is the number of set-valued tableaux of shape \(\lambda * \mu\) with content \(\nu.\)
S. Kundu gives another model for these \(K\)-theoretic Littlewood–Richardson coefficients using set-valued contratableaux [Kun25]. This gives a tableau rule in the same signed structure-constant setting as the stable Grothendieck product.
#Skew stable Grothendieck
In [Buc02, LMS16], the skew version of the stable Grothendieck polynomials are defined as \[\begin{aligned} \grothendieckStable_{\nu/\lambda}(\xvec) = \sum_{\mu} \sum_{T \in \mathrm{SVT}(\nu/\lambda,\mu)} (-1)^{|\lambda|-|\nu|-|\mu|} \monomial_\mu(\xvec). \end{aligned}\]
From a result in [BJS93], it follows that the skew stable Grothendieck polynomials are precisely the stable limit of Grothendieck polynomials indexed by 321-avoiding permutations.
A recent result in [CP19] gives an expansion of the skew stable Grothendieck polynomials in terms of skew Schur polynomials. Here, \(\sigma\) and \(\mu\) are skew shapes. \[\begin{aligned} \grothendieckStable_{\sigma}(\xvec) = \sum_{\mu \supseteq \sigma} (-1)^{|B(\mu/\sigma)|} a_{\sigma,\mu} \schurS_\mu(\xvec). \end{aligned}\] The motivation behind this formula is the connection with Euler characteristics of certain varieties; see [CP17].
The coefficients \(a_{\sigma,\mu}\) are nonnegative integers with an explicit combinatorial interpretation. M. Chan and N. Pflueger also prove a similar inverse formula, for connected skew shapes \(\sigma\): \[\begin{aligned} \schurS_{\sigma}(\xvec) = \sum_{\mu \supseteq \sigma} (-1)^{|A(\mu/\sigma)|} b_{\sigma,\mu} \grothendieckStable_\mu(\xvec). \end{aligned}\]
S. Kundu [Kun24] studies a flagged refined skew stable Grothendieck polynomial \(G_{\lambda/\mu}(X_\Phi;t).\) Using a Demazure crystal structure on flagged semistandard set-valued tableaux, he obtains an expansion of this polynomial in key polynomials. The coefficients carry the usual K-theoretic signs, and at \(t=0\) the formula recovers the Reiner–Shimozono key expansion of flagged skew Schur functions.
P. L. Guo, M. Kang, and J. Liu introduce hybrid Grothendieck polynomials \(H_{\lambda/\mu}(\xvec;\tvec;\wvec)\) as generating functions over set-valued reverse plane partitions [GKL25]. Such a filling assigns a nonempty finite set to each box, with the maximum in one box at most the minimum in the box directly to its right or below; singleton sets recover ordinary reverse plane partitions. For a filling \(T,\) the statistic \(\operatorname{wt}_i(T)\) counts the columns containing \(i,\) \(\operatorname{ceq}_i(T)\) counts vertical equalities in row \(i,\) and \(\operatorname{ex}_i(T)\) counts the entries beyond the first in row \(i.\) Thus \[H_{\lambda/\mu}(\xvec;\tvec;\wvec) =\sum_T \xvec^{\operatorname{wt}(T)} \tvec^{\operatorname{ceq}(T)} \wvec^{\operatorname{ex}(T)}.\] Setting \(\tvec=\mathbf{0}\) and replacing \(\wvec\) by \(-\wvec\) gives the refined stable Grothendieck polynomial, while \(\wvec=\mathbf{0}\) gives the refined dual stable Grothendieck polynomial.
Example (A hybrid weight).
The following set-valued reverse plane partition of shape \((2,1)\) uses entries at most \(3.\)
The two occurrences of \(2\) lie in different columns, the equality between the two left boxes contributes \(t_1,\) and each row has one excess entry. Hence this filling contributes \[t_1w_1w_2x_1x_2^2x_3\] to \(H_{(2,1)}(x_1,x_2,x_3;\tvec;\wvec).\)
These polynomials admit a Schur expansion from a crystal structure and have saturated Newton polytopes in straight shape.
S. Kundu equips the flagged set-valued reverse plane partitions indexing \(H_{\lambda/\mu}(\xvec_\Phi;\tvec;\wvec)\) with a Demazure crystal structure [Kun26, Thm. 4]. This simultaneously extends the known crystals on flagged set-valued tableaux and flagged reverse plane partitions, and gives an explicit key-polynomial expansion of the flagged hybrid polynomial [Kun26, Cor. 2]. Removing the flag yields a Schur expansion and, by rectification, expansions in both stable and dual stable Grothendieck polynomials [Kun26, Cor. 3 and Sec. 4]. The Schur and stable Grothendieck coefficients are polynomials in \(\tvec\) and \(\wvec\) with nonnegative integer coefficients. The dual stable Grothendieck expansion is instead signed, with sign \((-1)^{|\operatorname{ex}(S)|}\) for a set-valued tableau \(S.\)
S. Iwao gives a fermionic description of flagged skew Grothendieck polynomials [Iwa23]. These are \(K\)-theoretic analogues of flagged skew Schur polynomials, and the proof uses Matsumura’s Jacobi–Trudi type formula.
#More information
A. S. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong describe the expansion of stable Grothendieck polynomials into Grassmann Grothendieck polynomials [BKST+07]. T. Matsumura expresses flagged Grothendieck polynomials, defined by flagged set-valued tableaux, by a Jacobi–Trudi type determinant [Mat18]. The same paper introduces flagged skew Grothendieck polynomials and shows that the tableau and determinant definitions coincide. D. Yeliussizov proves skew Cauchy identities for symmetric Grothendieck polynomials and studies the associated dual filtered Young graphs [Yel19].
M. Wheeler and P. Zinn-Justin compute Littlewood–Richardson coefficients for double Grothendieck polynomials of Grassmannian permutations using an integrable vertex model [WZ19]. The Yang–Baxter equation gives product rules for these polynomials and their duals, and the resulting coefficients are expressed by puzzles without gashes.
Problem (See [MPS18]).
Describe a crystal structure, whose connected components give stable Grothendieck polynomials. This would be analogous to the type \(A\) crystals that produce Schur polynomials.
E. Marberg, K. H. Tong, and T. Yu introduce normal square root crystals as a \(K\)-theoretic analogue of normal crystals [MTY25]. Their main theorem shows that the character of such a crystal is a sum of symmetric Grothendieck polynomials, giving a crystal-theoretic route to Grothendieck positivity.
#Double Grothendieck polynomials
The double Grothendieck polynomials are defined via divided difference operators, \(\pi_i,\) defined by \[\pi_i(f) \coloneqq \frac{ (1+\beta x_{i+1})f-(1+\beta x_i)s_i(f)}{x_i - x_{i+1}},\] where \(s_i\) permutes \(x_i\) and \(x_{i+1}.\) We then have the base case \[\grothendieck_{\omega_0}(x|b) = \prod_{i+j \leq n}(x_i + b_j + \beta x_i b_j),\] and whenever \(\length(\omega s_i) = 1 + \length(\omega)\) the recursive definition \[\grothendieck_{\omega}(x|b) \coloneqq \pi_i \left( \grothendieck_{\omega s_i}(x|b) \right).\]
R. Xiong proves Cauchy formulas for generalized double \(\beta\)-Grothendieck polynomials in all types [Xio21]. These formulas imply a \(K\)-theoretic comodule structure map for flag varieties and lead to a generalized Billey localization formula. G. Hawkes proves a generalized Cauchy identity for double Grothendieck polynomials and gives tableau models for their stable double limits [Haw24]. The paper also introduces a half weak stable double Grothendieck polynomial interpolating between stable and weak stable versions. H. Dennin proves Cauchy identities for double Grothendieck polynomials using pipe-dream rectification [Den25]. The rectification algorithm gives a weight-preserving bijection between the pipe-dream families suggested by the identity, and also produces an insertion algorithm recovering a dual RSK-type correspondence. A. Hardt and D. Wallach prove stability results for products of double Grothendieck polynomials [HW25]. The back- and forward-stability numbers agree with the corresponding Schubert product stability numbers, and their work gives another proof of a finiteness conjecture for products of back stable Grothendieck polynomials. For the cohomological stable-limit theory, see back stable Schubert polynomials. A. Knutson, E. Miller, and A. Yong obtain a formula for double Grothendieck polynomials of vexillary permutations using flagged set-valued tableaux [KMY09]. The proof gives a geometric interpretation through Gröbner degenerations of vexillary matrix Schubert varieties and geometric vertex decompositions. T. Matsumura gives a tableau formula for double Grothendieck polynomials attached to \(321\)-avoiding permutations [Mat20]. The proof checks compatibility with the \(K\)-theoretic divided difference operators. D. Yeliussizov proves skew Cauchy identities for symmetric Grothendieck polynomials and studies the associated dual filtered Young graphs [Yel19].
#Canonical stable Grothendieck polynomials
D. Yeliussizov introduces a two-parameter deformation of the stable Grothendieck polynomials, \(\grothendieckStable^{(a,b)}_\lambda(\xvec)\) and the dual stable Grothendieck polynomials, \(\grothendieckDual^{(a,b)}_\lambda(\xvec)\) with the following properties.
They specialize to Schur polynomials, \(\schurS_\lambda(\xvec) = \grothendieckStable^{(0,0)}_\lambda(\xvec),\)
They specialize to classical Grothendieck polynomials, \(\grothendieckStable_\lambda(\xvec) = \grothendieckStable^{(0,-1)}_\lambda(\xvec)\) and \(\grothendieckDual_\lambda(\xvec) = \grothendieckDual^{(0,1)}_\lambda(\xvec),\)
They satisfy \(\omega(\grothendieckStable_\lambda(\xvec)) = \grothendieckStable^{(-1,0)}_{\lambda'}(\xvec)\) and \(\omega(\grothendieckDual_\lambda(\xvec)) = \grothendieckDual^{(1,0)}_{\lambda'}(\xvec),\)
The highest, respectively lowest, degree term is a Schur polynomial, \[\grothendieckStable^{(a,b)}_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{higher terms} \quad \grothendieckDual^{(a,b)}_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{lower terms}\]
They are dual with respect to the Hall inner product, \(\langle \grothendieckStable^{(-a,-b)}_\lambda, \grothendieckDual^{(a,b)}_\mu \rangle = \delta_{\lambda\mu}\)
They have nice involution properties, \[\omega( \grothendieckStable^{(a,b)}_\lambda ) = \grothendieckStable^{(b,a)}_{\lambda'} \text{ and } \omega( \grothendieckDual^{(a,b)}_\lambda ) = \grothendieckDual^{(b,a)}_{\lambda'}\]
They have the same structure constants \(c^{\nu}_{\lambda\mu}\) as the usual stable Grothendieck polynomials, \[\grothendieckStable^{(a,b)}_\lambda \grothendieckStable^{(a,b)}_\mu = \sum_{\nu} (a+b)^{|\nu|-|\lambda|-|\mu|} c^{\nu}_{\lambda\mu}\grothendieckStable^{(a,b)}_\nu.\]
There is also a skew version, a Weyl-type alternant formula, tableau descriptions, and operator descriptions.
The \(\grothendieckDual^{(a,b)}_\mu\) are Schur-positive, and a crystal structure on the canonical stable Grothendieck polynomials and their duals is given by G. Hawkes and T. Scrimshaw [HS20].
J. Pan, J. Pappe, W. Poh, and A. Schilling introduce an uncrowding algorithm for hook-valued tableaux [PPPS22]. This gives a useful bridge between hook-valued tableaux and ordinary semistandard-tableau data in the Grothendieck setting.
J. Jang, J. S. Kim, J. Pan, J. Pappe, and A. Schilling relate the hook-valued-tableau model for refined canonical stable Grothendieck polynomials to the Schur-expansion model using uncrowding, Goulden–Greene jeu de taquin, and tableau switching [JKPP+26]. Their bijection identifies the hook-valued-tableau model with pairs of a semistandard tableau and an exquisite tableau, and yields a third model in terms of biflagged tableaux.
B.-H. Hwang, J. Jang, J. S. Kim, M. Song, and U.-K. Song give further combinatorial models for refined canonical stable Grothendieck polynomials and their duals [HJKS+24]. Their formulas use generalized set-valued tableaux and reverse plane partitions, and include flagged and skew versions.
S. Iwao, K. Motegi, and T. Scrimshaw give a free-fermion presentation of refined canonical Grothendieck polynomials and their duals, including skew versions [IMS24]. They derive skew Cauchy identities, branching rules, expansion formulas, and integral formulas.
In [GZ20], solvable vertex models are used to prove Cauchy identities for the canonical Grothendieck polynomials: \[\sum_{\lambda} \grothendieckStable^{(-a,-b)}_\lambda(x_1,\dotsc,x_m) \grothendieckDual^{(a,b)}_\lambda(y_1,\dotsc,y_n) = \prod_{\substack{ 1 \leq i \leq m \\ 1 \leq j \leq n }} \frac{1}{1 - x_i y_j}\]
\[\sum_{\lambda} \grothendieckStable^{(-a,-b)}_{\lambda'}(x_1,\dotsc,x_m) \grothendieckDual^{(a,b)}_\lambda(y_1,\dotsc,y_n) = \prod_{\substack{ 1 \leq i \leq m \\ 1 \leq j \leq n }} (1 + x_i y_j ).\]
S. Kundu gives a new proof of the Murnaghan–Nakayama type rule for stable Grothendieck polynomials and proves a Murnaghan–Nakayama type rule for canonical stable Grothendieck functions [Kun25].
#Quantum double Grothendieck polynomials
In [LM06], the authors introduce the quantum double Grothendieck polynomials.
S. Naito and D. Sagaki prove a Pieri-type multiplication formula for quantum Grothendieck polynomials [NS25]. This proves a conjecture of C. Lenart and S. Maeno, and gives an explicit method for quantum products of Schubert classes in the small quantum \(K\)-theory ring of the full flag manifold.
#Flagged factorial Grothendieck polynomials
In [MS19], the authors consider a generalization of the Grassmann Grothendieck polynomials. They introduce the flagged factorial Grothendieck polynomials, indexed by a partition \(\lambda\) with \(r\) parts and a flag \(f = (0 \lt f_1 \leq f_2 \leq \dotsb \leq f_r).\) We then let \[\grothendieckStable_{\lambda,f}(\xvec|b) \coloneqq \sum_{T \in SVT(\lambda,f)} \beta^{|T|-|\lambda|}[x|b]^T\] where the sum runs over all flagged set-valued tableaux of shape \(\lambda\) and flag \(f.\) This means that the sets in row \(i\) may only contain elements from \(\{f_1,\dotsc,f_i\}.\) Furthermore, \[[x|b]^T \coloneqq \prod_{e\in T} x_{val(e)} \oplus b_{val(e)-r(e)+c(e)}.\] Here, \(a\oplus b \coloneqq a+b+\beta ab.\)
In [MS19], the authors give a Jacobi–Trudi type identity for flagged factorial Grothendieck polynomials, and show that these are equal to double Grothendieck polynomials for vexillary (\(2143\)-avoiding) permutations. P. L. Guo and S. C. C. Sun prove Gustafson–Milne and Fehér–Némethi–Rimányi type identities for factorial Grothendieck polynomials [GS18]. Specializing to Schur functions gives a combinatorial proof of the Fehér–Némethi–Rimányi identity.
#Dual stable Grothendieck polynomials
The dual stable Grothendieck polynomials were first studied in [LP07]. They are defined through the Hall inner product \[\langle \grothendieckDual_\lambda, \grothendieckStable_\mu \rangle = \delta_{\lambda \mu}.\]
Let \(\mathrm{RPP}(\lambda)\) be the set of fillings of \(\lambda\) with weakly increasing rows and columns. Then \[\grothendieckDual_\lambda(\xvec) \coloneqq \sum_{T \in \mathrm{RPP}(\lambda)} \xvec^{ev(T)}\] where \(ev(T)_i\) is the number of columns of \(T\) where \(i\) appears. Note that \[\grothendieckDual_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{lower order terms}.\] The skew version \(\grothendieckDual_{\lambda/\mu}(\xvec)\) is defined in a similar manner. One can prove that \(\grothendieckDual_{\lambda/\mu}(\xvec)\) is symmetric by using a version of the Bender–Knuth involutions; see [GGL16].
J. Bandlow and J. Morse expand modified Hall–Littlewood functions in the dual stable Grothendieck basis [BM10]. They extend the charge statistic to set-valued tableaux, giving a signed combinatorial expansion in this basis.
In [LMS16], it is shown that the constants appearing in the expansion \[\grothendieckDual_{\nu/\mu}(\xvec) = \sum_{\lambda} (-1)^{|\lambda|+|\mu|-|\nu|} c^{\nu}_{\mu\lambda} \grothendieckDual_{\lambda}(\xvec),\] agree with the coefficients \(c_{\lambda\mu}^{\nu}\) in the Littlewood–Richardson rule for stable Grothendieck polynomials. M. Takigiku explains Pieri coefficients for stable and dual stable Grothendieck polynomials [Tak18]. In this description, the lower-degree terms are obtained from the leading terms by alternating sums of meets or joins.
#Bialternant formula
A bialternant formula for the dual stable Grothendieck polynomials is proved in [Yel19, Thm. 17]. It states that \[\begin{aligned} \grothendieckDual_{\lambda}(\xvec) = \frac{1}{ \prod_{i \lt j} (x_i-x_j)} \det\left[ \phi^{i-1}x_j^{\lambda_i+n-i} \right]_{1\leq i,j \leq n} \end{aligned}\] where \(\phi^k x^n = \sum_{j=0}^n \binom{k+j-1}{j}x^{n-j}.\)
#Littlewood-type identities
Analogous to the Schur function Littlewood identities, we have \[\sum_{\lambda : \length(\lambda)\leq n} \grothendieckDual_\lambda(x_1,\dotsc,x_m) = \prod_{i=1}^m \frac{1}{(1-x_i)^n}.\]
#Jacobi–Trudi identity
There is a Jacobi–Trudi type identity, given by [Iwa19, Prop. 4.4], \[\begin{aligned} \grothendieckDual_\lambda(\xvec) = \det\left[ \sum_{m=0}^\infty \binom{1-i}{m} \beta^m \completeH_{\lambda_i-i+j-m}(\xvec) \right]_{1\leq i,j \leq n}. \end{aligned}\]
Alternatively, by [Yel17, AY20], we have \[\begin{aligned} \grothendieckDual_\lambda(\xvec) = \det\left[ \completeH_{\lambda_i-i+j}(1^{\lambda_i-1},\xvec) \right]_{1\leq i,j \leq \length(\lambda)} = \det\left[ \elementaryE_{\lambda'_i-i+j}(1^{\lambda'_i-1},\xvec) \right]_{1\leq i,j \leq \lambda_1}. \end{aligned}\]
D. Yeliussizov applies dual stable Grothendieck polynomials and normalized Schur functions to random lozenge tilings and plane partitions [Yel21]. In particular, the paper determines certain corner distributions and introduces related distributions connected with the corner growth model.
There is also a Jacobi–Trudi identity for the skew dual stable Grothendieck polynomials, originally conjectured by D. Grinberg. A proof was later presented by J. S. Kim [Kim21]. A different proof was obtained independently by A. Amanov and D. Yeliussizov [AY20]. Both approaches use techniques with non-intersecting lattice paths.
The skew Jacobi–Trudi identity states that \[\begin{aligned} \grothendieckDual_{\lambda/\mu}(\xvec) = \det\left[ \elementaryE_{\lambda'_i-i-\mu'_j+j}(1^{\lambda'_i-\mu'_j-1},\xvec) \right]_{1\leq i,j \leq \lambda_1}. \end{aligned}\]
The skew Jacobi–Trudi identity extends to the family of refined dual stable Grothendieck polynomials, introduced in [GGL16], see [Kim21, AY20] for proofs. J. S. Kim later proves Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials [Kim22], using plethystic substitution and resolving a conjecture of D. Grinberg.
#Schur expansion
In [LP07, Thm. 9.8], the authors show that \[\grothendieckDual_\lambda(\xvec) = \sum_\mu f^\mu_\lambda \schurS_\mu(\xvec)\] where \(f^\mu_\lambda\) is the number of elegant fillings of shape \(\lambda/\mu.\) A filling is elegant if it is semistandard, and all entries in row \(i\) are in \(1,\dotsc,i-1.\) See also [Gal17] for a crystal proof.
A flagged version of the skew dual Grothendieck polynomials is shown to be key-positive in [Kun23].
#Last passage percolation
D. Yeliussizov shows [Yel20] that the dual Grothendieck polynomials can be realized as a column distribution of a directed last-passage percolation model.
#Generalized and refined dual stable Grothendieck polynomials
The refined dual stable Grothendieck polynomials were introduced in [GGL16]. The authors define them as \[\grothendieckRefinedDualStable_\lambda(\xvec;\tvec) \coloneqq \sum_{T \in RPP(\lambda)} \xvec^{ev(T)} \tvec^{b(T)}\] where \(ev(T)_i\) is the number of columns containing \(i,\) and \(b(T)_j\) is the number of boxes in row \(j\) whose entry is equal to the entry in the box immediately below it.
In [Yel19], the generalized dual stable Grothendieck polynomials are defined: \[\grothendieckDual_\lambda(\xvec;\zvec) \coloneqq \sum_{\pi \in PP(\lambda)} \prod_{(i,j) \in \DES(\pi)} x_i z_{\pi(i,j)}.\] Note that \(\grothendieckDual_\lambda(1^n;\zvec) = \grothendieckDual_\lambda(\zvec).\) Yeliussizov observes that \[\grothendieckDual_\lambda(\xvec;\zvec) = \xvec^{\lambda} \grothendieckRefinedDualStable_\lambda(\xvec^{-1},\zvec),\] where \(\grothendieckRefinedDualStable_\lambda\) is a refined dual stable Grothendieck polynomial.
D. Yeliussizov proves [Yel19, Eq. (11)] that \[\sum_{\lambda : \length(\lambda)\leq n} \grothendieckDual_\lambda(x_1,\dotsc,x_m; z_1,\dotsc,z_n) = \prod_{i=1}^m \prod_{j=1}^n \frac{1}{1-x_iz_j}.\] Yeliussizov proves several new \(q\)-identities involving plane partitions and dual Grothendieck polynomials in [Yel19]. A. Amanov and D. Yeliussizov introduce higher-dimensional analogues of dual Grothendieck polynomials [AY23]. These are quasisymmetric functions whose specializations enumerate higher-dimensional partitions of fixed shape.
K. Motegi and T. Scrimshaw introduce an integrable vertex model whose partition function is the refined dual stable Grothendieck polynomial [MS25]. They prove Cauchy and Jacobi–Trudi identities for skew refined dual stable Grothendieck polynomials, and relate the same model to a Schur measure.
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