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The first chapter contains some known facts and some novel results on Commutative Algebra which are crucial for the proofs of the results of Chapters 3 and 4. The former are presented here without their proofs (with the exception of Theorem 8) for the convenience of the reader. In the first section of this chapter, we define the localization of a ring and give some main properties. The second section...
The second chapter focuses on the study of blocks of algebras of finite type and in particular, of symmetric algebras. In the first section, we introduce the notion of blocks and show how we can reduce, in many different ways, the problem of the determination of the blocks of an algebra to easier cases (mostly by changing the ring of definition). In the second section, we give the definition of a...
In this chapter we introduce the notion of “essential algebras”. These are symmetric algebras defined over a Laurent polynomial ring whose Schur elements are polynomials of a specific form (described by Definition 21). This form gives rise to the definition of the “essential monomials” for the algebra. As we have seen in the previous chapter, the Schur elements play an important role in the determination...
We will start this chapter by giving the definition and the classification of complex reflection groups. We will also define the braid group and the pure braid group associated to a complex reflection group. We will then introduce the generic Hecke algebra of a complex reflection group, which is a quotient of the group algebra of the associated braid group defined over a Laurent polynomial in a finite...
The aim of this chapter is the determination of the Rouquier blocks of the cyclotomic Hecke algebras of all irreducible complex reflection groups. In the previous chapter, we saw that the Rouquier blocks have the property of “semicontinuity”. This property allows us to obtain the Rouquier blocks for any cyclotomic Hecke algebra by actually calculating them in a small number of cases. Following the...
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