Cellular structures on Hecke algebras of type B

Abstract

The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type B. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.

Ordering Lusztig’s families in type B n

“…We can take N = 3 and write our bipartition as (4311, 320). The corresponding multiset is Z 3 1,1 (4311, 32) = {7, 5, 5, 3, 2, 1, 0}.…”

“…For all i such that δ i = 0, δ i+1 = 1 and u i+1 + a > u i , we set z i := u i+1 + a and z i+1 := u i . For all the remaining i, we set 3 1,1 (4311, 32) = {7,5, 5, 3,2, 1,0}, Z 3 1,1 (4321, 421) = {7, 6, 5, 3, 3, 1, 1}, where the hat indicates that δ i = 1. Note that, since u 2 = u 3 = 5, the sequence (δ i ) is not uniquely determined: we could take either δ 2 = 0, δ 3 = 1 or δ 2 = 1, δ 3 = 0.…”

“…The main results of this paper will deal with general choices of a, b. See also Bonnafé-Jacon [3] for a discussion of the combinatorics around the associated cellular structures in these cases.…”

Cell structures on the blob algebra

“…Bip 1 (10) KBip 1 (10) (10), (∅) (10), (∅) (9), (1) (9), (1) (8), (2) ((8, 1), (1)) (7), (3) ((7, 2), (1)) (6), (4) ((6, 3), (1)) (5), (5) ( (5,4), (1)) (4), (6) ((4, 2), (4)) (3), (7) ((5, 1), (4)) (2), (8) (6), (4) (1), (9) (7), (3) (∅), (10) (8), (2) e = 5, m = 3, s = (−2, 0) Bip 1 (10) KBip 1 (10) (10), (∅) (10), (∅) (9), (1) (9), (1) (8), (2) (8), (2) (7), (3) ((7, 1), (2)) (6), (4) ((6, 2), (2)) (5), (5) ((5, 3), (2)) (4), (6) ((4, 4), (2)) (3), (7) (4), (6) (2), (8) (5), (5) (1), (9) (6), (4) (∅), (10) (5) ((5, 2), (3)) (4), (6) ((4, 3), (3)) (3), (7) (3), (7) (2), (8) (4), (6) (1), (9) (5), (5) (∅), (10) (6), (4)…”

“…, (4) (5), (5) ((5, 1), (4)) (4), (6) ((4, 2), (4)) (3), (7) ((3, 3), (4)) (2), (8) (3), (7) (1), (9) (4), (6) (∅), (10) (5), (5) It can be seen that the correspondence between Bip 1 (10) and KBip 1 (10) works as the identity in the top m lines of all of these tables. This is a consequence of the following last result.…”