REPRESENTATION THEORY Meaning and
Definition
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Representation theory is a branch of mathematics that studies the ways in which mathematical objects, particularly groups, algebraic structures, and Lie algebras, can be represented by matrices or linear transformations. It seeks to understand how algebraic structures can be realized as groups of matrices or linear transformations acting on vector spaces, and conversely, how vector spaces can be decomposed into irreducible subspaces that correspond to these algebraic structures.
The main objective of representation theory is to study the symmetries and transformations of mathematical objects by associating them with linear transformations on vector spaces. By doing so, it enables us to better understand the structure and properties of these objects in terms of finite-dimensional linear algebra. For example, in the context of group representation theory, one can associate a group element with a matrix and study the properties and actions of the group through the corresponding matrix representations.
Representation theory has far-reaching applications in a wide range of fields, such as physics, chemistry, and computer science. In physics, for instance, representation theory plays a crucial role in the study of quantum mechanics and gauge theories. It helps describe particle symmetries and determine the behavior of particles under transformations.
Overall, representation theory provides a powerful tool for analyzing and understanding the structure and symmetries of mathematical objects, allowing us to bridge the gap between abstract algebraic structures and their concrete realizations as linear transformations.
Etymology of REPRESENTATION THEORY
The word "representation" comes from the Latin word "repraesentatio", which means "a making present again" or "a bringing back in the presence of". It is formed from the prefix "re-" (again), the verb "praesentare" (to present), and the suffix "-tion" (denoting an action, process, or result).
The word "theory" comes from the Greek word "theoria", which means "a looking at, viewing, or contemplation". It is derived from the verb "theoreo" (to look at, to observe) and the suffix "-ia" (denoting a state or condition).
Therefore, the term "representation theory" refers to the study and contemplation of making something present again or bringing it back in the presence of others.
