# Promotion of increasing tableaux: frames and homomesies

## TL;DR

This paper explores the behavior of a $K$-promotion operator on increasing tableaux, showing that boundary labels are preserved under iteration and deriving homomesy results for statistics over promotion orbits.

## Contribution

It establishes that boundary labels of rectangular increasing tableaux are invariant under $K$-promotion after a certain number of iterations, extending previous homomesy results.

## Key findings

- Boundary labels are preserved under $K$-promotion after q iterations.
- Homomesy results are extended to arbitrary rectangular shapes.
- The key property holds on the boundary, not the entire tableau.

## Abstract

A key fact about M.-P. Sch\"{u}tzenberger'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 showed that the 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 rectanglar 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.