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

Abstract

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.

The BMR freeness conjecture for the tetrahedral and octahedral families

“…Among the 28 remaining cases, we encounter 6 groups whose associated complex braid group is an Artin group; the groups G 4 , G 8 and G 16 , which are related to the Artin group of Coxeter type A 2 , and the groups G 25 , G 26 and G 32 , which are related to the Artin group of Coxeter type A 3 , B 3 and A 4 , respectively. The next theorem summarizes the results found in [8], [21] and [22]. Theorem 2.3.…”

Decomposition Matrices for the Generic Hecke Algebras on 3 Strands in Characteristic 0

“…Hence, from now on only the cases k = 3, 4, 5 will be considered. The following theorem is Theorem 1.1 in [6] and proves the BMR freeness conjecture for the generic Hecke algebras on 3 strands.…”

“…Using the methodology described in this paper we recover the same decomposition matrices determined in [8] by Chlouveraki and Miyachi. (2) In [6] §5 we have classified the simple representations of the braid group B 3 for dimension k = 2, 3, 4, 5. Let s 1 → A and s 2 → B be such a representation.…”

“…gap> H:=Hecke(ComplexReflectionGroup(16),[[Mvp("x")^10,Mvp("y")^10,Mvp("z")^10,Mvp("t")^10,Mvp("w")^10]]);; gap> T:=CharTable(H).irreducibles;; gap> t:=List(T,i->List(i,j->Value(j,["y", E(12)^7*Mvp("x")])));; gap> t[6]=t[1]+t[2]; true…”

“…gap> t:=List(T,i->List(i,j->Value(j,["w", E(8)*Mvp("x")^(3/2)*Mvp("y")^(3/2)*Mvp("z")^-1*Mvp("t")^-1])));; gap> t[40]=t [6]…”