Locks Fit into Keys: A Crystal Analysis of Lock Polynomials
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Annals of Combinatorics
Locks Fit into Keys: A Crystal Analysis of Lock Polynomials George Wang Abstract. Lock polynomials and lock tableaux are natural analogues to key polynomials and Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and give an explicit description of a crystal structure on lock tableaux. Furthermore, we construct an injective, weight-preserving map from lock tableaux to Kohnert tableaux that intertwines with their respective crystal operators. As a result, we see that the crystal structure on lock tableaux has a natural embedding into the Demazure crystal. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric. Mathematics Subject Classification. Primary 05E05. Keywords. Crystals, Lock polynomials, Key polynomials, Kohnert diagrams.
1. Introduction Assaf and Searles [5] in their work on Kohnert diagrams and tableaux defined lock tableaux and lock polynomials as analogues to Kohnert tableaux and to the ubiquitous key polynomials. In this paper, we ask and partially answer a natural question about such an analogue: what properties do locks and keys share, and how are they related to each other? We begin by examining the conditions for lock polynomials and key polynomials to be symmetric or quasisymmetric. These results follow from a careful examination of the combinatorial definitions of lock and Kohnert tableaux. Our main result is that there exists a connected crystal structure on lock tableaux that has a natural embedding into the Demazure crystal. We prove this directly using an injective, weight-preserving algorithm from lock tableaux to Kohnert tableaux that intertwines with their crystal operators. Supported by NSF DGE-1845298. 0123456789().: V,-vol
G. Wang
This algorithm is constructed using what we call unlock operators, which turn out to be a translation of the rectification operators of Assaf and Gonz´ alez [1] from the world of unlabeled diagrams to the world of labeled diagrams. However, this translation is crucial in allowing us to track the movements of labels through our algorithm.
2. Key Polynomials There are many bases for the polynomial ring that have deep geometric and representation theoretic significance. We begin with one such basis by defining it combinatorially using certain diagrams indexed by weak compositions. A diagram is an array of finitely many cells in N×N, and a labeled diagram is a diagram for which each cell contains a natural number, possibly with repetition. We draw all diagrams throughout this paper in French notation, that is, row indices will increase from bottom to top. The location of a cell in a diagram will be denoted using Cartesian coordinates. A weak composition is an ordered sequence of nonnegative integers written a = (a1 , a2 , . . . , an ) for some n ∈ N, and we call ai , i ∈ N, a part of a. The length of a weak composition is the number of parts it has. We write rev(a) to denote (an , an−1 , . . . , a1 ) and sort(a) to denote the weak composi
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