Ghost Kohnert posets
Kelsey Hanser, Nicholas Mayers
TL;DR
The paper investigates ghost Kohnert posets, posets on diagrams generated by ghost moves as an intermediate step in understanding Lascoux polynomials. It defines the ghost-move generated diagram set $ ext{GKD}(D)$ and the combined ghost–Kohnert set $ ext{KKD}(D)$, and proves that when the seed diagram has no ghost cells ($|G(D)|=0$), the ghost Kohnert poset $oldsymbol{ ho}_G(D)$ is always ranked and a join-semilattice. It further derives a necessary condition for when these posets are bounded, hence lattices, in terms of the free-cell sequence $ ext{FR}(D)$, clarifying connections between seed diagrams and poset properties. These results establish a well-behaved, structured class of posets in contrast to general Kohnert posets and contribute to the broader understanding of Lascoux polynomials via diagrammatic representations. The work also suggests directions for fully characterizing boundedness and potential links to associated polynomials $ rak{G}_D$ and their combinatorial interpretations.
Abstract
Recently, Pan and Yu showed that Lascoux polynomials can be defined in terms of certain collections of diagrams consisting of unit cells arranged in the first quadrant. Starting from certain initial diagrams, one forms a finite set of diagrams by applying two types of moves: Kohnert and ghost moves. Both moves cause at most one cell to move to a lower row with ghost moves leaving a new "ghost cell" in its place. Each diagram formed in this way defines a monomial in the associated Lascoux polynomial. Restricting attention to diagrams formed by applying sequences of only Kohnert moves in the definition of Lascoux polynomials, one obtains the family of key polynomials. Recent articles have considered a poset structure on the collections of diagrams formed when one uses only Kohnert moves. In general, these posets are not "well-behaved," not usually having desirable poset properties. Here, as an intermediate step to studying the analogous posets associated with Lascoux polynomials, we consider the posets formed by restricting attention to those diagrams formed by using only ghost moves. Unlike in the case of Kohnert posets, we show that such "ghost Kohnert posets" are always ranked join semi-lattices. In addition, we establish a necessary condition for when ghost Kohnert posets are bounded and, consequently, lattices.