Minuscule Cocharacters Definition And Discussion In Lie Algebras

In the realm of algebraic groups and Lie theory, the concept of a minuscule cocharacter holds significant importance. This article aims to provide a comprehensive definition and discussion of minuscule cocharacters, particularly within the context of reductive groups. We will delve into the properties, significance, and applications of these intriguing mathematical objects. Understanding minuscule cocharacters requires navigating the intricate landscape of Lie algebras, algebraic groups, root systems, and reductive groups. Let's embark on this journey, unraveling the layers of abstraction and revealing the essence of minuscule cocharacters.

Defining Minuscule Cocharacters

Minuscule cocharacters, in the context of reductive groups, are special cocharacters that play a crucial role in representation theory and the geometry of flag varieties. To fully grasp their definition, we need to establish some foundational concepts. Let's consider a reductive group G defined over a field F. We assume that G is split, which simplifies the analysis by ensuring the existence of a maximal torus that splits over F. We fix a Borel subgroup B within G and a maximal torus T contained within B. These choices provide a framework for understanding the structure of G through its root system and Weyl group.

Root Systems and Weyl Groups

At the heart of the theory lies the root system Φ associated with G and T. The root system is a finite set of vectors in a Euclidean space, encoding the structure of the Lie algebra of G. These roots determine the weights of representations and govern the geometry of the group. Closely related to the root system is the Weyl group W, a finite group generated by reflections associated with the roots. The Weyl group acts on the torus T and its character and cocharacter lattices, providing crucial symmetries in the theory.

Cocharacters: Mapping the Multiplicative Group

A cocharacter of T is a homomorphism λ: Gm → T, where Gm is the multiplicative group. The cocharacters form a lattice, denoted by X*(T), which is dual to the character lattice X*(T), consisting of homomorphisms T → Gm. The interaction between characters and cocharacters is fundamental. When G is split, we can choose a maximal torus T that is isomorphic to a product of multiplicative groups, and the cocharacter lattice becomes a free abelian group of rank equal to the rank of G.

Dominant Cocharacters: A Special Class

Among all cocharacters, a special class emerges: the dominant cocharacters. A cocharacter λ is called dominant if <α, λ> ≥ 0 for all positive roots α, where < , > denotes the pairing between characters and cocharacters. The choice of the Borel subgroup B determines the set of positive roots, and thus the notion of dominance. Dominant cocharacters are pivotal in representation theory, as they parametrize irreducible representations of G. They also have a natural ordering, where λ ≥ μ if λ - μ is a non-negative linear combination of positive coroots. This ordering allows us to compare the