| Representation Theory |
Articles about Group Representation |
Website Links For Group |
Information AboutRepresentation Theory |
| CATEGORIES ABOUT GROUP REPRESENTATION | |
| representation theory of groups | |
|
Group representation theory is the branch of Mathematics that studies properties of abstract Groups via their representations as Linear Transformation s of Vector Space s. Representation theory is important because it enables many Group-theoretic problems to be reduced to problems in Linear Algebra , which is a very well-understood theory. It is also important in Physics because, for example, it is used to describe how the Symmetry Group of a physical system affects the solutions to that system. Representations can also be defined for other mathematical structures, such as Associative Algebras , and Lie or Hopf Algebra s; for the rest of this article ''representation'' and ''representation theory'' will refer only to representation of groups. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a Homomorphism from the group to the Automorphism Group of the object. If the object is a vector space we have a ''linear representation''. Some people use ''realization'' for the general notion and reserve the term ''representation'' for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations. BRANCHES OF REPRESENTATION THEORY Representation theory divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
Representation theory also depends heavily on the type of Vector Space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert Space , Banach Space , etc.). One must also consider the type of Field over which the vector space is defined. The most important case is the field of Complex Number s. The other important cases are the field of Real Numbers , Finite Field s, and fields of P-adic Number s. In general, Algebraically Closed fields are easier to handle than non-algebraically closed ones. The Characteristic of the field is also significant; many theorems for finite groups depend on the order of the group not dividing the characteristic of the field. DEFINITIONS A representation of a Group ''G'' on a Vector Space ''V'' over a Field ''K'' is a Group Homomorphism from ''G'' to GL(''V''), the General Linear Group on ''V''. That is, a representation is a map : such that : for all . ''V'' is called the representation space and the dimension of ''V'' is called the '''dimension''' of the representation. It is common practice to refer to ''V'' itself as the representation when the homomorphism is clear from context (and, often, even when it is not). In the case where ''V'' is of finite dimension ''n'' it is common to choose a Basis for ''V'' and identify GL(''V'') with GL (''n'', ''K'') the group of ''n''-by-''n'' Invertible Matrices . The kernel of a representation of a group ''G'' is defined as the normal subgroup of ''G'' whose image under is the identity transformation: : A faithful representation is one in which the homomorphism ''G'' → GL(''V'') is Injective ; in other words, one whose kernel is the trivial subgroup {''e''} consisting of just the group's identity element. Given two ''F'' vector spaces ''V'' and ''W'', two representations : and : are said to be equivalent or '''isomorphic''' if there exists a vector space isomorphism : so that for all ''g'' in ''G'' : EXAMPLES Consider the complex number ''u'' = e2πi / 3 which has the property ''u''3 = 1. The Cyclic Group ''C''3 = {1, ''u'', ''u''2} has a representation ρ on C2 given by: : (the three matrices are ρ(1), ρ(''u'') and ρ(''u''2) respectively). This representation is faithful because ρ is a One-to-one Map . An isomorphic representation for ''C''3 is : REDUCIBILITY A subspace ''W'' of ''V'' that is fixed under the group action is called a ''subrepresentation''. If ''V'' has a non-zero proper subrepresentation, the representation is said to be ''reducible''. Otherwise, it is said to be ''irreducible''. Under a certain assumption, representations of Finite Group s can be decomposed into a Direct Sum of irreducible subrepresentations (see Maschke's Theorem ). The required assumption is that the Characteristic of the field K does not divide the size of the group. This is true for representations over the Complex Numbers . In the example above, the representation given is decomposable into two 1-dimensional subrepresentations (given by span{(1,0) } and span{(0,1)}). GENERALIZATIONS Set-theoretical representations A ''set-theoretic representation'' (also known as a Group Action or ''permutation representation'') of a Group ''G'' on a Set ''X'' is given by a Function ρ from ''G'' to ''X''''X'', the Set of Function s from ''X'' to ''X'', such that for all ''g''1, ''g''2 in ''G'' and all ''x'' in ''X'': : : This condition and the axioms for a group imply that ρ(''g'') is a Bijection (or Permutation ) for all ''g'' in ''G''. Thus we may equivalently define a permutation representation to be a Group Homomorphism from G to the Symmetric Group S''X'' of ''X''. For more information on this topic see the article on Group Action . Representations in other categories Every group ''G'' can be viewed as a Category with a single object; Morphism s in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a Functor from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the Automorphism Group of ''X''. In the case where ''C'' is Vect''K'', the Category Of Vector Spaces over a field ''K'', this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the Category Of Sets . For another example consider the Category Of Topological Spaces , Top. Representations in Top are homomorphisms from ''G'' to the Homeomorphism group of a topological space ''X''. Two types of representations closely related to linear representations are:
SEE ALSO |
|
|