Promotion permutations for tableaux

TL;DR

Promoting a broad class of fluctuating tableaux, the paper develops a comprehensive framework of promotion matrices and promotion permutations, unifying six equivalent characterizations via BK involutions, local rules, and jeu de taquin. It establishes structural results for rectangular fluctuating tableaux, including that promotion has order dividing $n$ and that promotion/evacuation realize a dihedral action, with a diagrammatic dihedral model. The work links fluctuating tableaux to $GL_r(\mathbb{C})$ and $SL_r(\mathbb{C})$ representation theory, Grassmannians, and crystal bases, providing a crystal-theoretic interpretation of promotion, reduced promotion matrices, and invariants. Together, these results furnish a unified combinatorial and representation-theoretic toolkit for promotion, with potential to extend diagrammatic web bases and symmetric-group actions in higher rank settings such as $\mathrm{SL}_4$ webs. The synthesis of local rules, growth diagrams, and crystals broadens the applicability of promotion concepts across families of tableaux and their diagrammatic counterparts.

Abstract

In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux. In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux.

Figures (8)

• Figure 1: A generalized partition with $r=5$ rows. The right border has been drawn in bold.
• Figure 2: The visualization of our running example fluctuating tableau $T$ with $4$ rows, length $7$, shape $\mathbf{1} = (1, 1, 1, 1)$, and type $(2, \overline{1}, 3, 1, \overline{2}, 2, \overline{1})$. Cells in the (final) shape are in light grey. The thick line indicates the outline of the initial shape $\mathbf{0} = (0, 0, 0, 0)$.
• Figure 3: The oscillization of the fluctuating tableau $T$ from \ref{['fig:ft-example']}.
• Figure 4: An example of using a local rule to fill in the lower right corner of a local rule diagram. Here, note that $21\overline{1}0\overline{1} = 32000 + 11\overline{1}\overline{1}\overline{2} - 220\overline{1}\overline{1}$.
• Figure 5: Interactions between the $\mathop{\mathrm{\mathsf{BK}}}\nolimits_i$ involutions and the $\mathop{\mathrm{\mathsf{switch}}}\nolimits_j$ involutions. Free cells are highlighted in light blue $\blacksquare$. Forced cells are pink $\blacksquare$, moving cells are light green $\blacksquare$, and open cells are darker green $\blacksquare$.

Theorems & Definitions (145)

• Example 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Example 2.5
• Corollary 2.6
• Definition 2.7
• Example 2.8
• Proposition 2.9