Definition of Irreducible Constituent in Isaac's Character Theory

Question

Going through Isaac's Character Theory book, the term "irreducible constituent" is, from what I see, first mentioned in exercise 2.8. But, I am not seeing a definition, or a mention of it, before this point. When I Google "irreducible constituent" or "constituent", I haven't been able to find a good definition (maybe I am not looking hard enough). I know these are mentioned later on during Clifford Theory, but can someone provide a "good definition" for chapter 2? Moreover, am I missing the definition somewhere?

Answer 1

An 'irreducible constituent' is a constituent that is irreducible. This is just standard English, so no need to have a specific definition for that. Looking at the index of the book, 'constituent' is defined on p17 (and it is, at the top), and 'irreducible' is defined on p15 for characters.

Indeed, if you had looked in the index, it actually says 'Irreducible constituent, see Constituent' in it.

In future, if you are confused about the definition of something in a book, try looking it up in the index of that book.

Answer 2

"Irreducible" or "simple" constituent is used often for groups, algebras and modules. For example, the Lie algebra $\mathfrak{f}_4$ of dimension $52$ is simple over a field of characteristic zero. However, this is no longer true for a field of characteristic $p=2$. A simple constituent of $\mathfrak{f}_4$ then is the ideal $J$ generated by the short roots. This is a simple modular Lie algebra of dimension $26$.