Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Abstract

Let ℋ = ℋ_{ℂ, −1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 − regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 − regular.

Key Words:

• Decomposition numbers
• Hecke algebras
• Specht modules

2010 Mathematics Subject Classification:

• 20C08

ACKNOWLEDGMENT

I would like to thank my supervisor Dr. Sinéad Lyle who brought this subject to my attention for her great help and advices. Also, I would like to thank the ministry of higher education in Saudi Arabia for granting me a scholarship to study a PhD degree in the UK. I would also like to thank the referee for their careful reading of this paper and constructive comments.

Notes

Communicated by K. Misra.