Let SYT n be the set of all standard Young tableaux with n cells. After recalling the definitions of four partial orders, the weak, KL, geometric and chain orders on SYT n and some of their crucial properties, we prove three main results:• Intervals in any of these four orders essentially describe the product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. • The map sending a tableau to its descent set induces a homotopy equivalence of the proper parts of all of these orders on tableaux with that of the Boolean algebra 2 [n−1] . In particular, the Möbius function of these orders on tableaux is (−1) n−3 . • For two of the four orders, one can define a more general order on skew tableaux having fixed inner boundary, and similarly analyze their homotopy type and Möbius function.

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.

The work of C. Bonnafé, M.Geck, L. Iancu and T. Lam [3] shows through two conjectures that r-domino tableaux have an important role in Kazhdan-Lusztig theory of type B with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion r-domino tableaux in Garfinkle's algorithm [4]. Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation [4] on r-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of [3] and provide necessary tool for their proofs. MÜGE TAŞKINwhere s 0 is the transposition (−1, 1) andRecently, the role of r-domino tableaux in this theory is revealed in the work of Bonnafé, Geck, Iancu, and Lam [3] through two main conjectures:• Conjecture A: If ra < b < (r + 1)a for some r ≥ 0 then two signed permutations lie in the same Kazhdan-Lusztig right (left) cell if and only if their insertion (recording) r-domino tableau are the same. • Conjecture B: If b = ra for some r ≥ 1 then two signed permutations lie in the same Kazhdan-Lusztig right (left) cell if and only if their insertion (recording) r − 1-domino tableau are equivalent through the notion of open cycles. In order to establish the proofs of these conjecture one definitely needs the plactic relations between signed permutations which determines when the insertion r-domino tableaux of two signed permutations are the same or equivalent through the notion of open cycles. Our aim here is to fill this gap.This paper is organized as follows: The descriptions of Barbash-Vogan and Garfinkle's algorithms can be found in Section 2 together with some lemmas which are essential in the following section. In Section 3 the definition of plactic relations are given and they are shown to be necessary and sufficient for describing plactic classes of r-domino tableaux.Remark 1.1. Recently T. Pietraho [18] has found another set of generators which can be shown to be equivalent to D r 1 , D r 2 , D r 3 and D r−1 3 given in the Definition 3.1. On the other hand these relations describes a larger set, namely the set of all permutations whose insertion r-domino tableaux are equivalent through the notion of open cycles. Finally, by using his results and an earlier version of the present work, C. Bonnafé provides a partial result towards the previous conjectures [2].

We prove that the well-known condition of being a balanced labeling can be characterized in terms of the sliding algorithm on tower diagrams. The characterization involves a generalization of authors' Rothification algorithm. Using the characterization, we obtain descriptions of Schubert polynomials and Stanley symmetric functions.

This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.