We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.
We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.
We introduce balanced shifted tableaux, as an analogue of balanced tableaux of Edelman and Greene, from the perspective of root systems of type B and C. We show that they are equinumerous to standard Young tableaux of the corresponding shifted shape by presenting an explicit bijection.
In this paper, a natural bijection between multichains of binary paths and shifted tableaux is presented, and it is used for the enumeration of the chains with maximum length from a given path $P$ to the maximum path $\mathbf{1}_{|P|}$. By mapping chains to shifted tableaux, the main formulas given in a recent paper by the authors for the enumeration of the $P - \mathbf{1}_{|P|}$ chains having only small intervals and minimum length are proved, using some new bijections on shifted tableaux.
A bijective proof of the hook-length formula for shifted standard tableaux
本文基于修正的jeu de taquin和Novelli，Pak和Stoyanovskii关于普通标准表的钩长公式的双射证明思想，给出了固定形状移位标准表的钩长公式的双射证明.在Novelli、Pak和Stoyanovskii的证明中，他们定义了费勒斯图的任意填充与整数$1,2,...,n$和标准tableaux和hook tabloids对之间的双射。在我们的移动版本的算法的映射从一组任意填充的移动费勒斯图到一组移动的标准tableaux是类似的建设Novelli，Pak和Stoyanovskii，然而，与他们的算法不同，我们被迫使用'rowwise'总顺序的细胞在移动费勒斯图，而不是'columnwise'总顺序的算法中的基本顺序。不幸的是，在移位的情况下，移位的钩子小报的构造更加复杂。作为一个副作用，我们得到了一个简单的随机算法生成一个给定形状的移位标准tableaux，它产生每一个这样的tableaux同样可能。
We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their shapes, provided that the representation character ratios and their cumulants converge to zero at some prescribed speed. Our class of examples includes uniformly random shifted standard tableaux with prescribed shape as well as shifted tableaux generated by some natural combinatorial algorithms (such as shifted Robinson-Schensted-Knuth correspondence) applied t…
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(…
We prove cases of a conjectural rule of H. Yadav, A. Yong, and the author for structure coefficients of the D. Anderson-W. Fulton ring. In particular, we give a combinatorial description for certain localization coefficients of this ring, which is related to the equivariant cohomology of isotropic Grassmannians.
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…
We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.
Symmetric Dyck tilings and ballot tilings are certain tilings in the region surrounded by two ballot paths. We study the relations of combinatorial objects which are bijective to symmetric Dyck tilings such as labeled trees, Hermite histories, and perfect matchings. We also introduce two operations on labeled trees for symmetric Dyck tilings: symmetric Dyck tiling strip (symDTS) and symmetric Dyck tiling ribbon (symDTR). We give two definitions of Hermite histories for symmetric Dyck tilings, and show that they are equivalent by use of the correspondence between symDTS operation and an Hermite…
A standard barely set-valued tableau of shape $λ$ is a filling of the Young diagram $λ$ with integers $1,2,\dots,|λ|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to the coincidental down-degree expectations (CDE) of lower intervals in Young's lattice. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shape. We show how it can be used to recover two formulas, or…
In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(λ)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $λ$ is a Young diagram with $\ell(λ) < n$ and $\gcd( n, |λ| )=1$, then the triple $\left( \mathsf{SST}_n(λ), \mathsf{C}, q^{- κ(λ)} s_λ(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and sho…
Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply $\mathfrak{q}$-crystal structure. In this paper, we explore the $\mathfrak{q}$-crystal structure of primed tableaux (semistandard marked shifted tableaux) and that of signed unimodal factorizations of reduced words of type $B$. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We also give the explicit algorit…
We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \{ 1, \ldots n\}$. We model this crystal with perforated tableaux (ptableaux), extending this identification isomorphically to biwords, RSK $(P,Q)$ tableaux pairs, and matrix models. In the ptableaux setting, crystal operators are more simply defined and we can identify highest weights visually without computation. We generalize the tenso…
Lock polynomials and lock Kohnert tableaux are natural analogues to key polynomials and key Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and show that the difference of a key polynomial and lock polynomial for the same composition is monomial positive. We also examine the conditions for which key and lock polynomials are symmetric or quasisymmetric. We accomplish these goals combinatorially using key Kohnert tableaux and lock Kohnert tableaux. In particular, for the difference of a key minus a lock, we focus on the behavior of c…
Using Henriques' and Kamnitzer's cactus groups, Schützenberger's promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals. For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram's map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general l…
We continue work begun in \cite{ptab} which introduced \emph{perforated tableaux} as a combinatorial model for crystals of type $A_{n-1}$, emphasizing connections to the classical Robinson-Schensted-Knuth (RSK) correspondence and Lusztig involutions, and, more generally, exploring the role of insertion schemes in the analysis of crystal graphs. An essential feature of our work is the role of \emph{dual} crystals (\cite{GerberLecouvey,vanLeeuwen}) from which we obtain new results within and beyond the classic RSK theory.
We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proo…
We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also g…
We use crystals of tableaux and descent compositions to understand the decomposition of Schur functions $s_λ$ into Gessel's fundamental quasisymmetric functions $F_α$. The connected crystal of tableaux $B(λ)$, associated to $s_λ$, is shown to be partitionned into a disjoint union of connected induced subgraphs $B(T_α)$ corresponding to the $F_α$'s. We show that these subgraphs, which we call quasicrystals, are isomorphic (as graphs) to specific crystals of tableaux. This allows us to give a formula for the number of tableaux of shape $λ$ and maximal entry $n$. We also use this setting to give …
We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in a previous paper by Gillespie, Levinson, and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.
