Axioms for Shifted Tableau Crystals

We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed by the authors 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.

Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of six vertex model as the product of a $t$-deformed Vandermonde and a Schur function. Here we provide an extension of their result by exploiting the language of primed shifted tableaux, with its proof based on the use of non-interesecting lattice paths.

On the Relationship between Pipe Dreams and Permutation Words

Pipe dreams represent permutations pictorially as a series of crossing pipes. Recent applications of pipe dreams include the calculation of Schubert polynomials, fillings of moon polyominoes, and in the combinatorics of antidiagonal simplicial complexes. These applications associate pipe dreams to words of elementary symmetric transpositions via a canonical mapping. However, this canonical mapping is by no means the only way of mapping pipe dreams to permutation words. We define sensical mappings from pipe dreams to words and prove sensical mappings are in bijection with standard shifted tableaux of triangular shape. We characterize the set of pipe dreams associated to a given word (under any sensical map) using step ladder moves. These moves induce a partial order on the set of pipe dreams mapping to a given word, yielding a distributive lattice.

Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux

Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between a pair of distinguished antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of standard Young tableaux of a truncated shifted staircase shape.

The Weighted Hook Length Formula III: Shifted Tableaux

Recently, a simple proof of the hook length formula was given via the branching rule. In this paper, we extend the results to shifted tableaux. We give a bijective proof of the branching rule for the hook lengths for shifted tableaux; present variants of this rule, including weighted versions; and make the first tentative steps toward a bijective proof of the hook length formula for $d$-complete posets.

Bumping algorithm for set-valued shifted tableaux

International audience We present an insertion algorithm of Robinson–Schensted type that applies to set-valued shifted Young tableaux. Our algorithm is a generalization of both set-valued non-shifted tableaux by Buch and non set-valued shifted tableaux by Worley and Sagan. As an application, we obtain a Pieri rule for a K-theoretic analogue of the Schur Q-functions. Nous présentons un algorithme d'insertion de Robinson–Schensted qui s'applique aux tableaux décalés à valeurs sur des ensembles. Notre algorithme est une généralisation de l'algorithme de Buch pour les tableaux à valeurs sur des ensembles et de l'algorithme de Worley et Sagan pour les tableaux décalés. Comme application, nous obtenons une formule de Pieri pour un analogue en K-théorie des Q-functions de Schur.