Centralizers of Representations

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tags: - colorclass/functional analysis ---see also: - Group Representation Theory

The concept of Centralizers of Representations and the center of the group algebra are integral to the study of group representations, particularly in understanding how certain elements of a group or its group algebra act on representations. These concepts are deeply intertwined with the insights provided by Schur’s Lemma, especially its second part, which elucidates the nature of endomorphisms of irreducible representations.

Centralizers of Representations

- Definition: The centralizer of a representation of a group $G$ on a vector space $V$ is the set of all linear operators on $V$ that commute with the action of $G$. Formally, if $\rho :G\to GL(V)$ is the representation, then the centralizer, denoted $Z(\rho )$ or sometimes $Com{m}_G(V)$, is defined as:

$$Z(\rho )=\{T\in End(V)\mid T\rho (g)=\rho (g)T\text{ for all }g\in G\},$$

where $End(V)$ is the set of all endomorphisms of $V$, and $GL(V)$ is the general linear group of $V$.

Center of the Group Algebra

- Group Algebra: For a group $G$ over a field $F$, the group algebra $F[G]$ is a vector space over $F$ with a basis consisting of the elements of $G$, equipped with a multiplication operation extended linearly from the group operation of $G$.

- Center Of The Group Algebra: The center of the group algebra, denoted $Z(F[G])$, consists of those elements in $F[G]$ that commute with all elements of $F[G]$. It is a subalgebra of $F[G]$.

Implications of Schur’s Lemma

- Scalars on Irreducible Representations: The second part of Schur’s Lemma states that any endomorphism of an irreducible representation $V$ of a group $G$ is a scalar multiple of the identity. This implies that for irreducible representations, the centralizer $Z(\rho )$ consists entirely of scalar transformations. Consequently, elements of $Z(F[G])$ act as scalars on irreducible representations of $G$.

- Structure of the Algebra: Understanding the centralizer and the center of the group algebra helps in dissecting the structure of representations. For example, the dimension of the center can give insights into the number and types of irreducible representations, as well as their degrees, contributing to the classification of representations of $G$.

- Character Theory: Elements of the center of the group algebra can be associated with class functions on $G$, which are constant on conjugacy classes. The behavior of these elements on irreducible representations is closely related to the characters of the representations, offering a pathway to studying the group’s representations through its character table.

The study of centralizers of representations and the center of the group algebra reveals the intricate relationship between the algebraic structure of groups and their representations. It underscores the unity between abstract algebra and linear algebra in the context of group theory, illuminating the fundamental role of symmetry in mathematical structures.