Erratum to: J Algebr Comb (2010) 31:53–82 DOI 10.1007/s10801-009-0183-2

In [2], we have found, using brute force computations, some (not all) Kazhdan–Lusztig relations (let us call them the elementary relations) between very particular elements of a Weyl group of type B. This shows in particular that the equivalence classes generated by the elementary relations are contained in Kazhdan–Lusztig cells.

It was announced in [6, Theorems 1.2 and 1.3] that the elementary relations generate the equivalence classes defined by the domino insertion algorithm (let us call them the combinatorial cells). As a consequence, we “deduced” that the combinatorial cells are contained in the Kazhdan–Lusztig cells [2, Theorem 1.5], thus confirming conjectures of Geck, Iancu, Lam and the author [3, Conjectures A and B]. However, as was explained in a revised version of [6] (see [7]), the equivalence classes generated by the elementary relations are in general strictly contained in the combinatorial cells. 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] has found that the elementary relations generate the combinatorial cells. So part of [2, Theorem 1.5] can be saved: the aim of this note is to explain precisely what is proved and what remains to be proved.

FormalPara Remark

The fact that [6, Theorems 1.2 and 1.3] is false does not imply that the result stated in [2, Theorem 1.5] is also false: it just means that its proof is not complete and we still expect the statement to be correct (as conjectured in [3, Conjectures A and B]).

1 Proved and unproved results from [2]

Unproved results

We keep the notation of [2]. First of all, the proof of the Theorem stated in the introduction of [2], so its statement remains a conjecture (and similarly for the Corollary stated at the end of this introduction). 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].

Theorem 1

Let r≥0 and assume that b=ra>0. Let ?∈{L,R,LR} and x, y∈W n be such that \(x \approx_{?}^{r} y\). Then x∼? y.

The proof of Theorem 1 will be given in the next section. It must also be noted that [2, Theorem 1.5] is also valid if b>(n−1)a (see [4, Theorem 7.7] and [1, Corollaries 3.6 and 5.2]).

Proved results

Apart from the above mentioned results, all other intermediate results (about computations of Kazhdan–Lusztig polynomials, structure constants, elementary relations) are correct.

2 Proof of Theorem 1

In [2, Sect. 7.1], we have introduced, following [6], three elementary relations ⌣1, \(\smile_{2}^{r}\) and \(\smile_{3}^{r}\): for adapting our argument to the setting of [5], we shall need to introduce another relation, which is slightly stronger than \(\smile_{3}^{r}\).

Definition 2

If w and w′ are two elements of W n , we shall write \(w \smallfrown_{3}^{r} w'\) whenever w′=tw and |w(1)|>|w(2)|>⋯>|w(r+2)|. If r≥n−1, then, by convention, the relation \(\smallfrown_{3}^{r}\) never occurs.

Using this definition, Pietraho’s Theorem [5, Theorem 3.11] can be stated as follows:

Pietraho’s Theorem

The relation \(\approx_{R}^{r}\) is the equivalence relation generated by ⌣1, \(\smile_{2}^{r}\) and \(\smallfrown_{3}^{r-1}\).

It is easy to check that, if \(w \smallfrown_{3}^{r} w'\), then \(w \smile_{3}^{r} w'\). Therefore, Theorem 1 follows from [2, Lemmas 7.1, 7.2 and 7.3] and the argument in [2, Sect. 7.2].