Tower tableaux

Abstract

We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only for reduced words which makes the algorithm a test for reducibility. Using the algorithm, a bijection between tower diagrams and finite permutations is obtained and it is shown that this bijection specializes to a bijection between certain labellings of a given tower diagram and reduced expressions of the corresponding permutation.

Tower diagrams and Pieri’s rule

“…Recall that the sliding algorithm in [4] determines the rules for sliding an adjacent transposition into a tower diagram so that when applied on a sequence of such transpositions, only the reduced expressions of permutations produce tower diagrams. Moreover this property provides a bijection between the set of all permutations and the set of all tower diagrams.…”

“…In the following Figure we illustrate the possible cases. Note that in the third case the deleted cell is labeled by ∅ and in the other cases the cells labeled by • represent the new positions of c. We remark that the only difference between the generalized sliding and the one in [4] appears in Condition 3. In case sld(c, d) = 0, the algorithm terminates without a result (that is, no tower diagram is produced) in the previous version, whereas in the new one, a tower diagram is produced by removing d. Since the new algorithm is an extension of the previous one, we still call it the sliding algorithm.…”

“…, 2, Remark 3.1. 1) Contrary to the one given in [4], the above definition assigns a flight path to any cell in the first quadrant, not necessarily contained in T . Also we generalized the notion of flight number to any top cell (not necessarily a corner cell) by adding also a hook number.…”

Tower tableaux and Schubert polynomials

“…In this section, we recall the necessary background from [3] without details. To begin with, by a tower T of size k ≥ 0 we mean a vertical strip of k squares of side length 1.…”

“…To achieve this, we first describe how we can remove an initial segment from a given standard tower tableaux. This will lead us to a generalization of the Rothification algorithm described in [3,Section 7].…”