On Essential Algebras

Abstract

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 of the blocks of a symmetric algebra. In the following sections, we see how the form of the Schur elements affects the behavior of the blocks of an essential algebra when specialized via different types of morphisms (a morphism associated with a monomial in 3.2, an adapted morphism in 3.3, the morphism I ⁿ defined in 3.4). In particular, in the first two cases, we show that the blocks depend only on the essential monomials for the algebra. In the next chapter, we will see that the generic Hecke algebras of complex reflection groups are a particular case of essential algebras.