Blocks and families for cyclotomic Hecke algebras by Maria Chlouveraki

Author:Maria Chlouveraki
Language: eng
Format: epub
Published: 2011-07-17T16:00:00+00:00

• If the nc,j belong to exactly one p-essential hyperplane, corresponding to the p-essential monomial M, then the blocks of OpoTi,^ coincide with the blocks of Aq^,^H, where qM ■= pA+{M - 1)A.

• If the ricj belong to more than one p-essential hyperplane, then we use theorem 13.3.41 in order to calculate the blocks of Cpc>H<^.

Now recall that the Rouquier blocks of 7i^ are unions of the blocks of Opo'Hff, for all 0-bad prime ideals p of Z^- We distinguish again three cases.

• If the TT-cj belong to no essential hyperplane, then the Rouquier blocks of H^ are unions of the blocks of Ap/H for all </)-bad prime ideals p. We say that these are the Rouquier blocks associated with no essential hyperplane for W.

• If the ncj belong to exactly one essential hyperplane, corresponding to the essential monomial M, then the Rouquier blocks of Ti^ are unions of the blocks of A^^j^^^H, where qM '■= pA+ {M — 1)A, for all 0-bad prime ideals p (if M is not p-essential, then, by corollary 13.2.71 the blocks of Aqj^jH coincide with the blocks of Ap/H). We say that these are the Rouquier blocks associated with that essential hyperplane .

• If the ncj belong to more than one essential hyperplane, then the Rouquier blocks of H^ are unions of the Rouquier blocks associated with the essential hyperplanes to which the ucj belong.

Therefore, if we know the blocks of Ap/H and ^q„?^ for all p-essential monomials M, for all p, we know the Rouquier blocks of Ti,^ for any cyclo-tomic specialization 0.

In order to calculate the blocks of Ap/H (resp. of Aq^^Ti), we find a cyclotomic specialization (p : vq.j ^^ y^'^'^ such that the uc^j belong to no p-essential hyperplane (resp. the ricj belong to the p-essential hyperplane corresponding to M and no other) and we calculate the blocks of Opo'H^.

The algorithm presented in the next section uses some theorems proved in previous chapters in order to form a partition of Irr(Vr) into sets which are unions of blocks of OpoTi-^- These theorems are

12.4.181 An irreducible character x is a block by itself in Opo'H^ if and only

-11

14.3.51 If X, i^ belong to the same block of Opo'H^, then they are in the same ]9-block of W.

14.4.51 If x, "0 are in the same block of OpoH^, then a^^ + A^^ = a^^ + A^^.

13.2.61 Let C be a block of ApAH. If M is not a p-essential monomial for any X G C , then C is a block of Aqj^^H.

If the partition obtained is minimal, then it represents the blocks of OpoTi-^f,.

With the help of the package CHEVIE of GAP, we created a program that follows this algorithm to obtain the Rouquier blocks of all cyclotomic Hecke algebras of the groups G7, Gu, Gig, G26, G2& and G^2- We used Clifford theory (for more details, see Appendix) in order to obtain the Rouquier blocks for

• G4, Gs, Ge from G7,

• Gs, Gg, Gio, G12, Gi3, Gi4, Gi5 from Gu,

•

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