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Erschienen in:
2016 | OriginalPaper | Buchkapitel
6 The Grothendieck Ring
verfasst von : Benjamin Steinberg
Erschienen in: Representation Theory of Finite Monoids
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Abstract
In this chapter, we introduce the Grothendieck ring of a finite monoid M over a field \(\mathbb{k}\). The main result is that the Grothendieck ring of M is isomorphic to the direct product of the Grothendieck rings of its maximal subgroups (one per regular \(\mathcal{J}\)-class). This result was first proved by McAlister for \(\mathbb{k} = \mathbb{C}\) in the language of virtual characters [McA72]. In Chapter 7, the results of this chapter will be used to study the ring of characters and the character table of a finite monoid. Throughout this chapter we hold fixed a finite monoid M and a field \(\mathbb{k}\).