REGULAR REPRESENTATION Meaning and
Definition
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Regular representation refers to the mathematical concept that characterizes a group's algebraic structure by representing its elements as matrices acting on a linear space. It is a fundamental concept in group theory and representation theory, extensively used in various branches of mathematics and physics.
The regular representation of a group is constructed as follows: Let G be a group and V a vector space over a field F. The regular representation associates with each group element g an invertible linear transformation on V, denoted by ρ(g), such that all group axioms are preserved. The linear transformations ρ(g) form a representation of G, i.e., they encode the group multiplication and identity in a linear way.
In the regular representation, the group elements are represented by permutation matrices. Specifically, for each group element g, the matrix ρ(g) has ones in certain positions and zeros elsewhere. These permutation matrices form a group isomorphic to G and can be multiplied and inverted, just like group elements. Thus, the regular representation explicitly links the abstract group elements to concrete matrices acting on the vector space.
The regular representation is useful for understanding the group's structure and studying its properties. It enables manipulation and investigation of the group elements using matrix algebra and representation theory techniques. Regular representations often arise in the context of character theory, where they facilitate the study of group characters and their properties. Additionally, the regular representation is closely related to the group's subgroups, automorphisms, and isomorphisms, providing a powerful tool for exploring these important aspects of group theory.
Etymology of REGULAR REPRESENTATION
The word "regular" in the term "regular representation" comes from the Latin word "regula", which means "rule" or "straight". It originated in the early 15th century and was originally used to describe conformity to a rule or standard. In mathematics, the term "regular" often refers to things that are well-behaved or exhibit a certain level of symmetry.
The term "representation" in mathematics has its roots in Latin as well. It comes from the Latin word "representare", which means "to present" or "to show". In mathematics, a representation is a way of expressing an abstract mathematical structure or object in terms of simpler and more concrete objects or structures.
Therefore, the "regular representation" refers to a particular type of representation that is well-behaved and exhibits a certain type of symmetry. It is commonly used in algebra and group theory to study the structure and properties of groups.
