Kohnert Moves in Combinatorial Algebra

• Kohnert moves are fundamental combinatorial operations that act on diagrams by dropping the rightmost cell in a row to the first available empty spot below.
• The method generates key polynomial bases—such as Schubert, key, and quasi-key polynomials—via explicit algorithms and poset structures with rankedness and boundedness properties.
• Extensions using ghost moves introduce K-theoretic analogues, linking combinatorial rules to geometric stratifications and representation theory.

Kohnert moves are fundamental combinatorial operations defined on diagrams of unit cells in the first quadrant, in which the rightmost cell in a row is "dropped" to the first available empty position in the same column strictly below. These moves underpin an extensive theory that connects the structure and generation of several bases for the polynomial ring, including Schubert polynomials, key polynomials, Kohnert polynomials, quasi-key polynomials, and their $K$-theoretic analogues. The analysis of diagrams under Kohnert moves yields rich poset structures, efficient rank and boundedness statistics, and explicit combinatorial algorithms for transition, expansion, and enumeration in algebraic combinatorics, representation theory, and geometry.

1. Definition and Combinatorial Framework

A Kohnert move acts on a diagram $D$ (a finite subset of $\mathbb N \times \mathbb N$) by selecting the rightmost cell in a given row $r$ and moving it downward within its column to the first available empty position below. The move may be elementary (by one step if the cell below is empty) or a jump, skipping over occupied positions, but always remains within the same column. This process induces a set $\mathrm{KD}(D)$ of all diagrams reachable by any (possibly empty) sequence of such moves.

For a weak composition $a$, the canonical (key) diagram consists of left-justified rows with $a_i$ cells in row $i$, and Kohnert's algorithm generates all diagrams representing the key polynomial: $$\kappa_a = \sum_{D \in \mathcal{K}(a)} x^{\mathrm{wt}(D)}$$ where $\mathrm{wt}(D)$ is the weak composition of row-counts in $D$. Expanding this to general initial diagrams (e.g., Rothe, skew, lock, or key diagrams), one obtains various Kohnert polynomial bases, each with precise combinatorial content and implications for representation theory (Assaf et al., 2017).

2. Algorithmic Models and Diagram Rules

Kohnert moves underpin explicit diagrammatic rules for generating classical and new polynomial bases:

• Schubert polynomials: Derived from Rothe diagrams by applying Kohnert moves; established with a bijective proof equating the generating function over compatible reduced word sequences with the sum over Kohnert diagrams (Assaf, 2020).
• Key polynomials (Demazure characters): Produced from key diagrams, encoding the Demazure module representations on the polynomial ring.
• Quasi-key and quasi-Schur polynomials: Lifted via Kohnert tableaux, refined labeling (entry content), and combinatorial destandardization. Quasi-Yamanouchi and quasi-Kohnert conditions further partition tableau expansions, yielding positive formulas and stability theorems (Assaf et al., 2016).
• $K$-Kohnert diagrams: Extend Kohnert moves with "ghost moves," producing diagrams for Lascoux polynomials and encoding set-valued tableau structures. Weight-preserving bijections translate these diagrams into tableau models (Pan et al., 2022).

The formalism often connects to crystal, Demazure, and divided difference operators, with diagram recurrence matching algebraic structures (e.g., Magyar's recurrence for flagged Schur modules (Armon et al., 2020)).

3. Poset Structures: Rankedness, Boundedness, and Shellability

The collection of diagrams reachable via Kohnert moves from an initial seed diagram organizes into a Kohnert poset $\mathcal{P}(D)$ by the covering relation of a single move. Theoretical work provides necessary and sufficient conditions for:

• Rankedness: Existence of a rank function ($\operatorname{rowsum}(D) - b(D_0)$) decreasing by one for each move, with forbidden subdiagrams precisely characterized (Colmenarejo et al., 2023).
• Boundedness: Existence of a unique minimal element, often linked to the monotonicity of cell counts across columns.
• Multiplicity-freeness: When each monomial in the polynomial expansion occurs with coefficient $0$ or $1$; combinatorial configuration conditions are given for northeast diagrams (Bingham et al., 29 Jan 2025).
• Shellability: The poset's order complex is (EL-)shellable under diagram restrictions, which is equivalent to multiplicity-freeness of the associated Kohnert polynomial (Kerr et al., 26 Apr 2024).

Special cases (diagrams with at most one cell per column, two-row diagrams, key diagrams) are classified with explicit formulas and combinatorial descriptions for minimal and maximal elements, as well as descent grades.

4. Extensions: Ghost Moves, $K$-Theory, and Lascoux Polynomials

Ghost moves are a generalization whereby the moved cell leaves a "ghost cell" in its place; this produces $K$-Kohnert diagrams associated with Lascoux polynomials, a $K$-theoretic nonhomogeneous analogue of key polynomials. The posets generated solely by ghost moves ($\mathcal{P}_G(D)$) are always ranked join-semilattices, become lattices under specific free cell sequence conditions, and admit canonical labeling inherited from the original diagram (Hanser et al., 11 Mar 2025, Hanser et al., 4 Oct 2024).

The maximum ghost cell count possible from an initial diagram can be computed via local labeling and recursive reduction functions, with explicit combinatorial algorithms provided for arbitrary diagrams and refined bounds for diagram families (Hanser et al., 4 Oct 2024).

5. Enumeration, Algorithmic Puzzles, and Computational Aspects

Recent work investigates enumerative and algorithmic puzzles:

• Maximum/Minimum Moves Puzzles: Given a diagram $D$, the maximal or minimal number of Kohnert moves needed to reach a fixed-point diagram is computed via the "room" statistic for each cell and a bottom-justifying algorithm, respectively. Explicit combinatorial formulas and greedy algorithms are provided (Koss et al., 21 Sep 2025).
• Enumeration of Minimal Elements: Closed binomial formulas and partition bijections enumerate minimal diagrams for checkerboard and other special cases (Colmenarejo et al., 2023).

These puzzles reflect rank and diameter in Kohnert posets, yield efficient computation of expansion terms in polynomial bases, and clarify the transformation classes induced by Kohnert's algorithm.

6. Geometric and Representation-Theoretic Significance

Kohnert moves unify and generalize the combinatorics of Schubert polynomials (flag variety cohomology), Demazure characters (representation theory of algebraic groups), and their extensions to $K$-theory (via Lascoux polynomials). The poset properties (boundedness, shellability, multiplicity-freeness) correspond to geometric features such as spherical variety structure, multiplicity-free module decomposition, and stratifications in flag varieties. Explicit labeling and stability results connect Kohnert models to quasisymmetric and symmetric limits, sliding polynomial bases, and stability indices (e.g., $\eta(a)$ for key polynomial expansion (Assaf et al., 2016)).

In summary, Kohnert moves form the central combinatorial mechanism for generating, expanding, and analyzing polynomial bases in algebraic combinatorics. The diagram rules, poset structures, labeling procedures, and their enumeration not only provide transparent algorithms but also encode deep algebraic, geometric, and representation-theoretic properties across classical and modern polynomial families.