Knuth relations for the hyperoctahedral groups

Abstract

C. Bonnafé, M. Geck, L. Iancu, and T. Lam have conjectured a description of Kazhdan-Lusztig cells in unequal parameter Hecke algebras of type B which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle. We adapt her methods and construct a family of operators which generate the equivalence classes on pairs of arbitrary rank domino tableaux described in the above conjecture.

On Kazhdan-Lusztig cells in type B

“…Recently, M. Taskin [14] and T. Pietraho [12] have independently provided plactic/coplactic relations for the domino insertion algorithm. Our methods rely heavily on their results: we show that, if two elements of W n are directly related by a plactic relation, then they are in the same Kazhdan-Lusztig cell.…”

Erratum to: On Kazhdan–Lusztig cells in type B

“…Also, [2, Theorem 1.5(a)] is still a conjecture. However, [2, Theorem 1.5(b)] is still correct: its proof must only be adapted, using Pietraho's results [5]. …”

“…7.1], we have introduced, following [6], three elementary relations 1 , r 2 and r 3 : for adapting our argument to the setting of [5], we shall need to introduce another relation, which is slightly stronger than r 3 .…”

“…This has no consequence on most of the intermediate results in [2], but changes the scope of validity of [2, Theorem 1.5]. Indeed, for some special cases of the parameters, T. Pietraho [5] …”

Character formulas and descents for the hyperoctahedral group

“…. , w(a k )) be the subsequence of unbarred elements of w. Applying Schensted's correspondence to the two-line array a 1 a 2 · · · a k w(a 1 ) w(a 2 ) · · · w(a k ) results in a pair A Knuth class of (type B and) shape (λ, µ) is a set of the form {w ∈ B n : P B (w) = T } for some fixed T ∈ SYT(λ, µ) (these Knuth classes should not be confused with the ones considered in the study of Kazdhan-Lusztig cells of type B; see [12,31]). The first part of the following corollary, already discussed in Section 4, is a restatement of Theorem 4.1.…”