Promotion of Kreweras words

Kreweras words are words consisting of n$$\mathrm {A}$$ A ’s, n$$\mathrm {B}$$ B ’s, and n$$\mathrm {C}$$ C ’s in which every prefix has at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {B}$$ B ’s and at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {C}$$ C ’s. Equivalently, a Kreweras word is a linear extension of the poset $$\mathsf{V}\times [n]$$ V × [ n ] . Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the $$\mathrm {B}$$ B ’s and $$\mathrm {C}$$ C ’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s $$\mathfrak {sl}_3$$ sl 3 -webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.

Updating the Belief Promotion Operator

In this note, we introduce the local version of the operator for belief promotion proposed by Schwind et al. We propose a set of postulates and provide a representation theorem that characterizes the proposal. This family of operators is related to belief promotion in the same way that updating is related to revision, and we provide several results that allow us to show this relationship formally. Furthermore, we also show the relationship of the proposed operator with features of credibility-limited revision theory.

Promotion of Increasing Tableaux: Frames and Homomesies

A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but observed that this key fact does not generally extend to $K$-promotion of such increasing tableaux. Here, we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectangular increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.

Promotion Operator on Rigged Configurations of Type $A$

In an earlier paper of the first author, the analogue of the promotion operator on crystals of type $A$ under a generalization of the bijection of Kerov, Kirillov and Reshetikhin between crystals (or Littlewood–Richardson tableaux) and rigged configurations was proposed. In this paper, we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type $A_n^{(1)}$ crystals and (unrestricted) rigged configurations is an affine crystal isomorphism.

A promotion operator on rigged configurations

International audience In a recent paper, Schilling proposed an operator $\overline{\mathrm{pr}}$ on unrestricted rigged configurations $RC$, and conjectured it to be the promotion operator of the type $A$ crystal formed by $RC$. In this paper we announce a proof for this conjecture.