Blocks and Families for Cyclotomic Hecke Algebras

Chlouveraki, Maria

doi:10.1007/978-3-642-03064-2

Cited by 37 publications

(96 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We can now prove the following lemma that helps us to "replace" inside the definition of U the elements ω 5 …”

Section: The Results Follows From Proposition 44(i)mentioning

confidence: 99%

“…The Schur elements belong to R k K (see [9], Proposition 7.3.9) and they depend only on the symmetrizing form t k and the isomorphism class of the representation. Moreover, M. Chlouveraki has shown that these elements are products of cyclotomic polynomials over K k evaluated on monomials of degree 0 (see theorem 4.2.5 in [5]). In the following section we are going to use these elements in order to determine the irreducible representations of CH k (for more details about the definition and the properties of the Schur elements, one may refer to §7.2 in [9]).…”

Section: The Results Follows From Proposition 44(i)mentioning

confidence: 99%

“…θ(s φ ) = 0, where φ is the character that corresponds to M . The Schur elements have been determined in [11] (see also Appendix A.3 in [5]) and one can also find them in CHEVIE package of GAP3; they are where r is a 5th root of u 1 u 2 u 3 u 4 u 5 . However, due to the assumption detA = −λ 5 i , i ∈ {1, 2, 3, 4, 5} (case where i = j), we have that θ(r) + λ i = 0.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Universal deformations of the finite quotients of the braid group on 3 strands

Chavli

2016

Journal of Algebra

View full text Add to dashboard Cite

Abstract. We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequences of this case, we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5, thus recovering a classification of Tuba and Wenzl in a more general framework.Acknowledgments. I would like to thank my supervisor Ivan Marin for his help and support during this research, and Maria Chlouveraki, for fruitful discussions on the last section of this paper. I would also like to thank the University Paris Diderot -Paris 7 for its financial support.

show abstract

“…We can now prove the following lemma that helps us to "replace" inside the definition of U the elements ω 5 …”

Section: The Results Follows From Proposition 44(i)mentioning

confidence: 99%

Section: The Results Follows From Proposition 44(i)mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Universal deformations of the finite quotients of the braid group on 3 strands

Chavli

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…This technique has been extended by Geck and Rouquier [14] to a decent class of algebras over integral domains. It became an important tool for studying algebras involving parameters, so for example Hecke algebras (see [13], [12], and [7]) and, more recently, rational Cherednik algebras (see [2], [15], and [28]). We list several further examples in §2A.…”

Section: Introductionmentioning

confidence: 99%

Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

View full text Add to dashboard Cite

ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

show abstract

“…However, there is one case where this is not possible, that is, when r = 2 and e is even. In that case, we apply the same methods as in [5] in order to determine the Rouquier blocks of the cyclotomic Hecke algebras of G(de, 2, 2), and then we apply Clifford theory in order to obtain the Rouquier blocks for G (de, e, 2).Finally, to every irreducible character of a cyclotomic Hecke algebra of a complex reflection group we can attach integers a and A, as Lusztig has done for Weyl groups. In [15], Lusztig shows that these integers are constant on families.…”

mentioning

confidence: 99%

Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Chlouveraki¹

2010

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

Abstract. The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series G (de, e, r), thus completing their calculation for all complex reflection groups. IntroductionUntil recently, the lack of Kazhdan-Lusztig bases for the non-Coxeter complex reflection groups did not allow the generalization of the notion of families of characters from Weyl groups to all complex reflection groups. However, thanks to the results of Gyoja [12] and Rouquier [21], we have obtained a substitute for the families of characters that can be applied to all complex reflection groups. In particular, Rouquier has proved that the families of characters of a Weyl group W coincide with the Rouquier blocks of the Iwahori-Hecke algebra of W , that is, its blocks over a suitable coefficient ring. This definition generalizes to all complex reflection groups, and we are grateful for this for the following reasons.On the one hand, since the families of characters of a Weyl group play an essential role in the definition of the families of unipotent characters of the corresponding finite reductive group (see [14]), the families of characters of the cyclotomic Hecke algebras could play a key role in the organization of families of unipotent characters in general. On the other hand, for some (non-Coxeter) complex reflection groups W , we have data that seem to indicate that behind the group W , there exists another mysterious objectthe Spets (see [3], [18] G(d, 1, r). Using the generalization of some classic results, known as Clifford theory, they were able to obtain the Rouquier blocks for G (d, d, r) from those of G (d, 1, r). Later, Kim [13] generalized the methods used in [2] in order to obtain the Rouquier blocks of the cyclotomic Hecke algebras of G (de, e, r) from those of G (de, 1, r).As far as the exceptional complex reflection groups are concerned, some special cases were treated by Malle and Rouquier in [19]. Finally, in [5], the author gives the complete classification of the Rouquier blocks of the cyclotomic Hecke algebras for all exceptional complex reflection groups.However, recently it was realized that the algorithm of [2] for G(d, 1, r) does not work, unless d is a power of a prime number. In [7], the author gives the correct algorithm, which is more complicated than the one in [2]. Now, it remains to recalculate the Rouquier blocks of the cyclotomic Hecke algebras of G (de, e, r) in order to complete the determination of the Rouquier blocks for all complex reflection groups.Using the same idea as in [13], we apply Clifford theory in order to obtain the Rouquier blocks for G (de, e, r) from those of G (de, 1, r). However, there is one case where this is not possible, that is, when r = 2 and e is even. In that case, we apply the same methods as in [5] in order to determine ...

show abstract

Blocks and Families for Cyclotomic Hecke Algebras

Cited by 37 publications

References 38 publications

Universal deformations of the finite quotients of the braid group on 3 strands

Universal deformations of the finite quotients of the braid group on 3 strands

Decomposition Matrices are Generically Trivial

Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Contact Info

Product

Resources

About