**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.

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]).

## 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.

## 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].

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*B*_{ n }. In: Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics, vol. 284, pp. 33–54. Birkhäuser, New York (2010) - 4.
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## Additional information

The author is partly supported by the ANR (Project No JC07-192339).

The online version of the original article can be found at http://dx.doi.org/10.1007/s10801-009-0183-2.

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### Cite this article

Bonnafé, C. Erratum to: On Kazhdan–Lusztig cells in type *B*
.
*J Algebr Comb* **35, **515–517 (2012) doi:10.1007/s10801-012-0351-7

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