Combinatorial Theory

We introduce fluctuating tableaux, which subsume many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, standard, and transpose semistandard tableaux. Our main contribution is the introduction of promotion permutations and promotion matrices, which are new even for standard tableaux. We provide characterizations in terms of Bender-Knuth involutions, jeu de taquin, and crystals. We prove key properties in the rectangular case about the behavior of promotion permutations under promotion and evacuation. We also give a full development of the basic combinatorics and representation theory of fluctuating tableaux.

Our motivation comes from our companion paper, where we use these results in the development of a new rotation-invariant \(\operatorname{SL}_4\)-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. These planar graphs are indexed by fluctuating tableaux, tableau promotion corresponds to graph rotation, and promotion permutations correspond to key graphical information.

Mathematics Subject Classifications: 05E10, 05E18

Keywords: Tableaux, promotion, jeu de taquin, Bender-Knuth involutions, growth diagrams, crystals