Stable Grothendieck Polynomials

• Stable Grothendieck polynomials are K-theoretic analogues of Schur functions defined via set-valued tableaux with corrections that reflect deeper geometric structures.
• They exhibit rich algebraic structures including dual bases, determinantal and Jacobi–Trudi formulas, and variations like canonical and double versions.
• Modern extensions such as refined, flagged, and skew versions enhance their utility in K-theoretic Schubert calculus and algebraic geometry.

Stable Grothendieck polynomials are $K$-theoretic analogues of Schur functions in symmetric function theory, central to $K$-theoretic Schubert calculus, representation theory, and algebraic geometry. They arise as generating functions over set-valued tableaux, interpolate between Schur polynomials and more general symmetric functions, and support rich combinatorial, Hopf-algebraic, and determinantal structures. Modern developments include canonical two-parameter deformations, double and back-stable versions, flagged and skew extensions, and refined dual bases.

1. Definitions and Combinatorial Models

Let $\lambda$ be a partition. The stable Grothendieck polynomial $G_\lambda(x)$ is defined as the generating function over semistandard set-valued tableaux (SVT) of shape $\lambda$: $$G_\lambda(x) = \sum_{T \in \mathrm{SVT}(\lambda)} (-1)^{|T| - |\lambda|} x^T$$ where $$x^T = \prod_{i\geq 1} x_i^{\# \text{ of entries }i}$$, and $|T|$ is the total number of entries in all cells of $T$ (Takigiku, 2018). The lowest-degree part of $G_\lambda(x)$ coincides with the Schur function $s_\lambda(x)$, while higher-degree terms incorporate $K$-theoretic corrections.

For skew shapes $\lambda/\mu$, the definition generalizes: $$G_{\lambda/\mu}(x) = \sum_{T \in \mathrm{SVT}(\lambda/\mu)} (-1)^{|T|-|\lambda/\mu|} x^T$$ These functions are symmetric, and relate geometrically to $K$-theory Schubert classes on the infinite Grassmannian (Alwaise et al., 2016).

Duals and Reverse Plane Partitions

The dual basis $\{g_\lambda(x)\}$ under the Hall inner product is indexed by reverse plane partitions (RPP): $$g_\lambda(x) = \sum_{\pi \in \mathrm{RPP}(\lambda)} x^\pi$$ where $x^\pi = \prod_{c} x_{\pi(c)}$ (Takigiku, 2018). Extensions to refined duals introduce countable sequences of extra parameters weighted by vertical adjacencies (Galashin et al., 2015).

Generalizations: Canonical, Double, and Refined Versions

• Canonical two-parameter functions: $G_\lambda^{(\alpha, \beta)}(x)$ deform $G_\lambda$ with parameters $(\alpha, \beta)$, often presented by determinantal formulas involving $(1+\beta x_i)$ and $(1-\alpha x_i)$ factors (Yeliussizov, 2016).
• Double stable Grothendieck polynomials: $G_\lambda(x;y)$ incorporate two alphabets $x$ and $y$ and exhibit enhanced stability and symmetry properties (Rimanyi et al., 2018).
• Refined canonical stable Grothendieck polynomials: $G_\lambda(x; a, \beta)$ admit infinite sequences of parameters, unifying multiple interpolations and flagged/skew models (Hwang et al., 2021, Hwang et al., 2024).

2. Algebraic and Determinantal Structures

Bi-Alternant and Jacobi–Trudi Formulas

Stable Grothendieck polynomials admit determinantal expressions analogous to cohomological Schur functions, with $K$-theoretic correction: $$G_\lambda(x_1,\dots,x_n; \beta) = \frac{\det[x_i^{\lambda_j + n - j}(1+\beta x_i)^{j-1}]_{1\leq i,j\leq n}}{\prod_{1\leq i<j\leq n} (x_i - x_j)}$$ Letting $n \to \infty$ yields the stable version (Kundu, 2 Nov 2025, Yeliussizov, 2016).

Canonical versions $G_\lambda^{(\alpha, \beta)}(x)$ take

$$G^{(\alpha,\beta)}_\lambda(x_1,\dots,x_n) = \frac{\det\bigl[x_i^{\lambda_j+n-j}(1+\beta x_i)^{j-1}(1-\alpha x_i)^{n-j}\bigr]}{\prod_{1\leq i<j\leq n}(x_i-x_j)(1-\alpha x_i)(1-\alpha x_j)}$$

and stabilize as $n\to\infty$ (Yeliussizov, 2016).

Jacobi–Trudi-Type Identities for Duals and Refined Polynomials

Refined dual versions $g_\lambda(x; t)$ and flagged analogues $g_{\lambda/\mu}(x; t)$ admit plethystic Jacobi–Trudi determinant formulas involving additional parameters, proved via lattice-path bijections and path-wise expansions (Kim, 2020, Kim, 2020): $$g_{\lambda/\mu}(x; t) = \det\Big[ e_{\lambda'_i-\mu'_j-i+j}(x_1, x_2, \ldots; t_{\mu'_j+1}, \ldots, t_{\lambda'_i-1}) \Big]_{1 \leq i,j \leq n}$$

Free-Fermionic Presentations

Stable and skew stable Grothendieck polynomials admit operator-theoretic descriptions via free-fermionic Fock space, with vacuum expectation values and explicit shift operators encoding the $K$-theoretic deformation (Iwao, 2020).

3. Product and Expansion Formulas

Murnaghan–Nakayama Type Rules

A $K$-theoretic analogue of the classical Murnaghan–Nakayama rule is established for both stable Grothendieck and canonical functions (Kundu, 2 Nov 2025): $$p_k(X^n) \, G^\beta_\lambda(X^n) = \sum_{\nu \supseteq \lambda} (-\beta)^{|\nu/\lambda|-k} (-1)^{k-c(\nu/\lambda)} \binom{r(\nu/\lambda)-1}{k-c(\nu/\lambda)} G^\beta_\nu(X^n)$$ where $\nu/\lambda$ runs over connected ribbons, $c(\cdot)$ counts columns, $r(\cdot)$ counts rows.

Pieri Rules

Pieri-type multiplication for $G_\lambda$ and $g_\lambda$ involves alternations and binomial coefficients over horizontal strips: $$G_{(a)} \cdot G_\lambda = \sum_{\mu \supseteq \lambda: \text{a-strip}} (-1)^{|\mu/\lambda|-a} {\binom{r(\mu/\lambda)-1}{|\mu/\lambda|-a}} G_\mu$$ with dual versions for $g_\lambda$ (Takigiku, 2018).

Change of Basis and Schur Expansions

Lenart's and Lascoux's results, recently expressed via bumpless pipe dreams (BPDs), provide explicit expansions of Grothendieck polynomials in the Schur basis and vice versa, with combinatorial sign assignments and Bruhat chain interpretations (Weigandt, 8 Jun 2025).

4. Structural Properties and Symmetry

Duality and Hall Inner Product

$G_\lambda$ and $g_\lambda$ form dual bases under the Hall inner product on symmetric functions: $(G_\lambda, g_\mu) = \delta_{\lambda\mu}$ (Takigiku, 2018, Hwang et al., 2021). This duality persists under refinements and extensions, including canonical and flagged versions (Hwang et al., 2024).

Involutions and Stembridge-Type Equalities

The standard involution $\omega$ on symmetric functions relates $G_\lambda$ and $g_\lambda$ to their conjugate shapes: $\omega(G_\lambda) = (-1)^{|\lambda|} G_{\lambda^T}$ and for canonical functions,

$$\omega(G_\lambda^{(\alpha,\beta)}(x)) = G_{\lambda^T}^{(\beta,\alpha)}(x)$$

Analogues of the Stembridge equality hold for skew Grothendiecks: $$G_{\rho/\mu} = G_{\rho/\mu^T}, \quad g_{\rho/\mu} = g_{\rho/\mu^T}$$ for staircase shapes $\rho$ (Abney-McPeek et al., 2021).

Symmetry and $Q$-Function Connections

In shifted settings, stable Grothendieck polynomials generate symmetric subalgebras of $K$-theoretic peak Hopf algebras and generalize Schur $Q$- and $P$-functions (Lewis et al., 2019). Double stable and "half-weak" stable Grothendieck polynomials evaluated at $x=y$ are $Q$-Schur positive by degree, connecting to type B Stanley symmetric functions (Hawkes, 2020).

5. Refined, Flagged, and Skew Extensions

Refined Canonical and Flagged Models

Refined canonical stable Grothendieck polynomials $G_\lambda(x;a,\beta)$ and their duals $g_\lambda(x;a,\beta)$ admit infinite sequences of parameters. Combinatorial models use marked multiset-valued tableaux and marked reverse plane partitions, encoding both algebraic structure and parameter specializations (Hwang et al., 2021, Hwang et al., 2024).

Flagged extensions restrict entries of tableaux by row or column, with determinantal formulas involving plethystic substitutions and bounds on indices (Kim, 2020, Hwang et al., 2024).

Skew and Double Versions

Skew stable and double Grothendieck polynomials generalize classical cases and admit expansions via non-commutative Schur operators and supersymmetric analogues, possessing Pieri, Jacobi–Trudi, and Cauchy identities (Iwao, 2020, Hawkes, 2020).

Shifted Grothendiecks and Peak Functions

Shifted stable Grothendieck polynomials, built over shifted Young diagrams and indexed by strict partitions, generate symmetric subalgebras in $K$-theoretic peak Hopf algebras, generalizing shifted $Q$-Schur theory (Lewis et al., 2019).

6. Coincidences, Classification, and Open Problems

Classification of when distinct partitions yield identical $G_\lambda$ or $g_\lambda$ involves combinatorial invariants like bottleneck numbers and ribbon equivalence. Ribbon shapes are classified (up to reversal) for dual stable Grothendieck polynomials (Alwaise et al., 2016). Open questions remain regarding full characterization for general skew shapes, survival of nesting-type coincidences, and the interplay between conjugation invariance and $K$-theory.

7. Analytical and Computational Techniques

Iterated Residue Techniques

Iterated residue calculus provides new proofs of straightening laws, multiplication formulas, and alternation of sign in Schur expansions, simplifying computations and furnishing effective tools for positivity and stability analysis in $K$-theory (Allman et al., 2014, Rimanyi et al., 2018).

Pipe Dream and Path Expansion Methods

Pipe dream combinatorics affords explicit algorithms for basis change and indexing of expansions, compatible with Bruhat order recurrences and supported by explicit combinatorial interpretations in both finite and back-stable settings (Weigandt, 8 Jun 2025).

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Summary Table: Families and Their Key Features

Family	Tableau Model(s)	Determinantal Formula(s)
$G_\lambda$ (standard)	Set-valued tableaux (Buch)	Alternating determinant (Lenart)
$g_\lambda$ (dual)	Reverse plane partitions (RPP)	Jacobi–Trudi, plethystic form
Canonical $G_\lambda^{(\alpha,\beta)}$	Hook-valued tableaux	$K$-theoretic Jacobi–Trudi, Cauchy
Refined/flagged polynomials	Multiset/marked tableaux	Flagged plethystic determinant
Double/back-stable versions	Triples of tableaux, pipe dreams	Iterated residue, BPD expansion
Shifted stable Grothendiecks	Shifted set-valued tableaux	Peak algebra generators

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Stable Grothendieck polynomials and their modern generalizations constitute a central theme in $K$-theoretic symmetric function theory, unifying combinatorial, algebraic, and geometric perspectives, with continuing developments in structure, classification, and computational methodology.