Tableau stabilization and rectangular tableaux fixed by promotion powers

Abstract

We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization, the shape of stabilized tableaux, and tableau stabilization as a permutation statistic. We apply tableau stabilization to construct the sufficiently large rectangular tableaux fixed by powers of promotion, which were counted by Brendon Rhoades via the cyclic sieving phenomenon [Rho10, Theorem 1.3].

Tableau Stabilization and Lattice Paths

“…Tableau stabilization was introduced in [1] in order to construct sufficiently large rectangular tableaux fixed by given powers of promotion. Rhoades first counted the number of rectangular tableaux fixed by the powers of promotion by exhibiting a remarkable cyclic sieving phenomenon for the action of promotion on rectangular tableaux [3].…”

“…But his representation-theoretic technique said nothing about what the actual fixed points were. In [1], Ahlbach exhibited these fixed points for sufficiently rectangular tableaux by applying the rectification operator to skew tableaux formed by attaching shifted copies of a skew tableaux to itself. This naturally gives rise to the notion of tableaux stabilization.…”

“…Ahlbach proved that once a tableaux stabilizes, it continues to stabilize, [1, Lemma 3.9]. In the special case where the rows of T have the same size, he derived a formula for the shape of Rect(T (k) ) for k ≥ r − 1, [1,Theorem 1.6], and used it to show that any tableaux with r rows of the same size stabilizes at r, [1,Theorem 1.4]. He conjectured that the same bound still holds when the rows have weakly decreasing sizes but was only able to prove a bound of max(1, 2r − 2).…”

“…The proof of the same-size-rows case of Theorem 1.0.4 in [1] relied on a formula for the shape of the stabilized tableaux. Unfortunately, an analogous formula for the general case would have to involve new terms not present in the previous shape formula, and we have not found such a formula.…”

“…Yet, we generalize the lattice path argument in the proof of Lemma 4.2 in [1] to prove Theorem 1.0.4. Our argument relies on Greene's Theorem characterizing the shape of the insertion tableaux of words in terms of their increasing subsequences and a careful analysis of the increasing subsequences of reading words of T (k) for standard skew tableaux T .…”