TY - JOUR

T1 - SKEW CELLULARITY OF THE HECKE ALGEBRAS OF TYPE G(l, p, n)

AU - Hu, Jun

AU - Mathas, Andrew

AU - Rostam, Salim

N1 - Publisher Copyright:
© 2023 American Mathematical Society

PY - 2023

Y1 - 2023

N2 - This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(l, p, n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(l, 2, n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(l, p, n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

AB - This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(l, p, n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(l, 2, n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(l, p, n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

KW - Cyclotomic quiver Hecke algebras

KW - cellular algebras

KW - complex reflection groups

KW - cyclotomic Hecke algebras

KW - diagrammatic Cherednik algebras

UR - http://www.scopus.com/inward/record.url?scp=85168600030&partnerID=8YFLogxK

U2 - 10.1090/ERT/646

DO - 10.1090/ERT/646

M3 - Article

AN - SCOPUS:85168600030

SN - 1088-4165

VL - 27

SP - 508

EP - 573

JO - Representation Theory

JF - Representation Theory

ER -