Center Of The Group Algebra

see also:

• Group Representation Theory
• Centralizers of Representations
• Group Algebra
• Group Theory
• Representation Theory

The center of the group algebra, $Z(F[G])$, plays a pivotal role in the study of group representations, reflecting the symmetry and structural properties inherent in the group $G$. Understanding $Z(F[G])$ provides insights into the nature of $G$ and its representations, particularly in terms of how these representations decompose and interact.

Definition and Properties

For a group $G$ and a field $F$, the group algebra $F[G]$ is formed by taking linear combinations of elements of $G$ with coefficients from $F$. The center $Z(F[G])$ of this algebra comprises elements that commute with every element in $F[G]$. Mathematically, $z\in Z(F[G])$ if and only if $zg=gz$ for all $g\in F[G]$. Elements of $Z(F[G])$ are significant because:

• Conjugacy Classes: Elements in $Z(F[G])$ often correspond to sums of elements in the same conjugacy class of $G$. A conjugacy class of $G$ is a set of elements that are conjugate to each other, that is, for any two elements $g$ and $h$ in the class, there exists an element $x$ in $G$ such that $g=xh{x}^{-1}$. The sum of elements in a conjugacy class forms a basis for $Z(F[G])$ when $G$ is finite.

• Representation Decomposition: The structure of $Z(F[G])$ can provide information about the irreducible representations of $G$. Specifically, the dimension of $Z(F[G])$ (as a vector space over $F$) equals the number of irreducible representations of $G$ over $F$. This is a consequence of the Artin-Wedderburn theorem in the case of finite groups and semisimple algebras.

• Character Theory: In the context of character theory, the center $Z(F[G])$ is directly related to class functions and characters of $G$. Characters of irreducible representations are class functions, and they can be used to study the structure of $Z(F[G])$.

Applications

• Invariant Theory: The study of invariants and symmetry properties of algebraic objects often relies on understanding the center of the group algebra, as it encapsulates the commutative aspects of the otherwise possibly non-commutative group action.

• Spectral Theory: In mathematics and physics, particularly in quantum mechanics, the elements of $Z(F[G])$ correspond to observables that are invariant under the group action, playing a crucial role in the spectral analysis of physical systems.

• Algebraic Number Theory: In the context of algebraic number theory, especially when studying extensions of number fields and Galois groups, the structure of the group algebra and its center can shed light on the Galois correspondence and the behavior of primes in extensions.

Example

Consider the group $G=S_3$, the symmetric group on three elements, and the field $F=\mathbb{C}$, the complex numbers. The group algebra $\mathbb{C}[S_3]$ consists of linear combinations of the six permutations that constitute $S_3$. The center $Z(\mathbb{C}[S_3])$ includes elements that are symmetric with respect to conjugation, such as the sum of all permutations in $S_3$ (which acts as a scalar in this algebra) and sums over conjugacy classes (e.g., the sum of all transpositions).

The study of $Z(F[G])$ not only reveals the deep algebraic structure of $G$ but also has practical implications in understanding the symmetries and invariant properties of systems modeled by the group $G$.