# Erratum to: On Kazhdan–Lusztig cells in type B

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

In , 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  (see ), 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 , but changes the scope of validity of [2, Theorem 1.5]. Indeed, for some special cases of the parameters, T. Pietraho  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]).

## Proved and unproved results from 

### Unproved results

We keep the notation of . First of all, the proof of the Theorem stated in the introduction of , 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 .

### Theorem 1

Let r≥0 and assume that b=ra>0. Let ?∈{L,R,LR} and x, yW 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 , three elementary relations ⌣1, $$\smile_{2}^{r}$$ and $$\smile_{3}^{r}$$: for adapting our argument to the setting of , 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 rn−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 by1, $$\smile_{2}^{r}$$ 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].

## References

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Bonnafé, C.: Two-sided cells in type B (asymptotic case). J. Algebra 304, 216–236 (2006)

2. 2.

Bonnafé, C.: On Kazhdan–Lusztig cells in type B. J. Algebr. Comb. 31, 53–82 (2010)

3. 3.

Bonnafé, C., Geck, M., Iancu, L., Lam, T.: On domino insertion and Kazhdan–Lusztig cells in type B n . In: Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics, vol. 284, pp. 33–54. Birkhäuser, New York (2010)

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Bonnafé, C., Iancu, L.: Left cells in type B n with unequal parameters. Represent. Theory 7, 587–609 (2003)

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Pietraho, T.: Knuth relations for the hyperoctahedral groups. J. Algebr. Comb. 29, 509–535 (2009)

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Taskin, M.: Plactic relations for r-domino tableaux, preprint (2008). Available at arXiv:0803.1148v1

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Taskin, M.: Plactic relations for r-domino tableaux (revised version), preprint (2011). Available at arXiv:0803.1148v3

## Author information

Correspondence to Cédric Bonnafé.

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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