Kohnert Polynomials: Combinatorial & Algebraic Insights

• Kohnert polynomials are multivariate polynomials defined through specific moves on diagrams, capturing combinatorial structures in algebraic settings.
• They generalize classical bases like Demazure (key) and Schubert polynomials, providing unified models in algebraic combinatorics and representation theory.
• Extensions via K-Kohnert moves introduce ghost cells and parameters, enabling connections to Grothendieck polynomials and advances in computational and geometric applications.

Kohnert polynomials are a class of multivariate polynomials constructed by applying specific combinatorial moves—known as Kohnert moves—to diagrams of unit cells in the first quadrant of the plane. The family of Kohnert polynomials encompasses and generalizes foundational bases in algebraic combinatorics, including Demazure (key) polynomials and Schubert polynomials, as well as their deformations and $K$-theoretic analogues. The theory provides deep connections among algebraic geometry, representation theory, and combinatorial models such as crystal graphs, posets, and tableaux.

1. Foundational Definitions and Construction

A Kohnert polynomial $\mathfrak{K}_D$ is defined for a diagram $D$—a finite set of unit cells in $\mathbb{N} \times \mathbb{N}$ arranged with prescribed row and column counts. A classical Kohnert move selects the rightmost cell in a row and "drops" it downward to the first available position in its column, possibly jumping past other cells. The set $\mathcal{K}(D)$ of all diagrams reachable from $D$ by sequences (possibly empty) of such moves forms the basis for the polynomial: $$\mathfrak{K}_D(x_1,\ldots,x_n) = \sum_{T \in \mathcal{K}(D)} x_1^{\text{wt}(T)_1} x_2^{\text{wt}(T)_2}\cdots x_n^{\text{wt}(T)_n}$$ where $\text{wt}(T)_i$ records the number of cells in row $i$ of $T$ (Assaf et al., 2017). If $D$ is taken as a left-justified composition diagram, $\mathfrak{K}_D$ specializes to a Demazure (key) polynomial; if $D$ is a Rothe diagram for a permutation, it yields a Schubert polynomial.

Diagrams can be chosen to encode different combinatorial or geometric data. The existence of many admissible choices for $D$ allows Kohnert polynomials to form bases for the polynomial ring $\mathbb{Z}[x_1,x_2,\ldots]$, with different choices yielding distinct families such as "skew polynomials" or "lock polynomials."

2. Diagram Models, Moves, and Parameters

Classical Kohnert moves act on diagrams (skyline, key, Rothe, and their generalizations) to produce the underlying polynomial. An extension—the $K$-Kohnert move—arises in $K$-theoretic settings (Ross et al., 2013, Pan et al., 2022, Robichaux, 21 Mar 2024). Here, a cell may either move as in the classical rule or "freeze" in place, leaving behind a so-called "ghost cell," denoted $g$ or $\circ$. The parameter $\beta$ records the cost of freezing, giving the generating function: $$J_a = \sum_{D} \beta^{\#g\text{ in }D} x^{\text{wt}(D)}$$ For $\beta=0$, only classical moves survive and the sum reduces to the key polynomial; for $\beta=-1$, the rule conjecturally recovers Grothendieck polynomials (Ross et al., 2013, Robichaux, 21 Mar 2024).

Diagrams constructed via sequences of Kohnert or $K$-Kohnert moves may encode additional data, such as excess (number of ghost cells) for Lascoux polynomials, or stabilization properties relevant to quasisymmetric and symmetric function theory (Assaf et al., 2016, Assaf et al., 2017).

3. Algebraic and Combinatorial Bases

The framework of Kohnert polynomials supports various combinatorial bases within $\mathbb{Z}[x_1,x_2,\ldots]$:

• Key (Demazure) polynomials: $\mathfrak{K}_{D_a}$ for left-justified diagrams.
• Schubert polynomials: via Rothe diagrams, matched bijectively to compatible sequences of reduced words (explicitly, a weight-preserving bijection exists between the diagram and BJS models (Assaf, 2020)).
• Skew polynomials: constructed by prescribed shifts; conjecturally Schubert-positive, and upon stabilization, yield skew-Schur functions (Assaf et al., 2017).
• Lock polynomials: arising from right-justified diagrams; coincide with key polynomials for weakly decreasing compositions and stabilize to extended Schur (quasisymmetric) functions (Assaf et al., 2017).
• Fundamental slide and monomial slide bases: Kohnert polynomials expand nonnegatively in these bases, providing positive combinatorial rules (Assaf et al., 2017, Assaf et al., 2016).

For $K$-theoretic analogues, extended diagrams with ghost cells allow for the definition of Lascoux polynomials and Grothendieck polynomials via $K$-Kohnert moves (Ross et al., 2013, Pan et al., 2022, Robichaux, 21 Mar 2024).

4. Crystal Structures and Combinatorial Expansions

Crystal graphs arising from Kohnert diagrams encode type A Demazure modules, with vertices corresponding to Kohnert tableaux (labeled tableaux encoding move histories). Raising and lowering crystal operators can be defined, matching the action of Demazure divided difference operators in Lie theory (Assaf, 2020, Wang, 2020).

For diagrams satisfying "southwest" or "northwest" conditions (closure under certain moves), the associated Kohnert polynomial admits a nonnegative expansion in Demazure characters (key polynomials) (Assaf, 2020, Armon et al., 2020). Explicit labeling and rectification algorithms provide necessary and sufficient conditions for a diagram to belong to a given expansion. Multiplicity-freeness of these expansions is classified via forbidden pattern avoidance in the indexing weak compositions (Hodges et al., 2020, Kerr et al., 26 Apr 2024).

Ta

Diagram Type	Polynomial	Stabilization/Limit
Key/Skyline	Demazure, Key	Schur polynomials
Rothe	Schubert	Stanley symmetric functions
Lock (right-justified)	Lock	Extended Schur/quasisymmetric
Skew	Skew polynomial	Skew-Schur functions

5. Poset Structures and Combinatorial Puzzles

The diagrams generated by Kohnert (and $K$-Kohnert) moves can be organized into posets, $\mathcal{P}(D)$, with order given by move sequences. These posets possess rich structure: boundedness, rankedness, and shellability are characterized combinatorially (Colmenarejo et al., 2023, Bingham et al., 29 Jan 2025, Kerr et al., 26 Apr 2024). Notably, if the initial diagram has weakly decreasing column counts, the poset is bounded with a unique minimal element (Colmenarejo et al., 2023).

For $K$-Kohnert and ghost moves, the associated posets ($\mathcal{P}_G(D)$) form ranked join semi-lattices, and boundedness is determined by increasing free cell sequences in the initial diagram (Hanser et al., 11 Mar 2025, Hanser et al., 4 Oct 2024). Algorithms exist for computing maximal numbers of ghost cells (relevant for total degree computation in Lascoux polynomials), often via reductions and labeling functions in so-called "snow diagrams" (Hanser et al., 4 Oct 2024).

Combinatorial puzzles (e.g., maximizing/minimizing Kohnert moves to reach fixed diagrams) find explicit solutions based on room-counting functions and minimal/maximal path constructions within the poset (Koss et al., 21 Sep 2025).

6. K-Kohnert Rules, Extensions, and Controversies

The extension of Kohnert's rule (the $K$-Kohnert move) introduces ghost cells and powers of $\beta$ in polynomial expansions, capturing $K$-theoretic deformations of key, Schubert, and Grothendieck polynomials (Ross et al., 2013, Pan et al., 2022, Robichaux, 21 Mar 2024).

A conjectural combinatorial rule for Grothendieck polynomials using $K$-Kohnert moves (Ross–Yong conjecture) was formulated, though explicit counterexamples have demonstrated its failure in certain cases (Robichaux, 21 Mar 2024). A revised "ghost" $K$-Kohnert rule now matches all known $K$-theoretic data and agrees with tableaux models for 321-avoiding permutations.

7. Geometric, Representation-Theoretic, and Algorithmic Impact

Kohnert polynomials furnish a unified combinatorial framework for polynomial bases relevant to the geometry of flag varieties and torus equivariant $K$-theory. Key polynomials serve as Demazure characters for type A modules, Schubert polynomials encode structure sheaf classes, and their deformations arise as $K$-theoretic characteristics (Assaf et al., 2017, Assaf, 2020, Armon et al., 2020).

Stable limits of Kohnert polynomials recover symmetric and quasisymmetric functions, aligning combinatorial expansions with algebraic positivity and module decompositions. Classification of multiplicity-freeness and poset shellability ties into spherical Schubert geometry and representation theory (Hodges et al., 2020, Kerr et al., 26 Apr 2024).

Algorithmically, explicit labeling, room-count, and tableau models facilitate computation of monomials, structure constants, and crystal decompositions. These algorithms are foundational for both theoretical analyses and practical enumeration in combinatorial algebraic geometry.

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In summary, Kohnert polynomials and their extensions—through diagrammatic move rules, poset structures, and combinatorial tableaux—provide a powerful, unified machinery bridging algebraic combinatorics, representation theory, and $K$-theory. The ongoing refinement of rules and counterexamples underscores the subtlety of the combinatorial–algebraic correspondence and continues to motivate investigations into deeper geometric and combinatorial phenomena.