General linear groups, Permutation groups & representation theory

General linear group: GL(n, R/C)

• GL(n, R/C) denotes the set of all n x n invertible matrices with real/complex coefficients.

• Operation: matrix multiplication • Identity element: the identity matrix• Inverse of an element: matrix inverse • Not commutative (not Abelian)

More on GL(n, R/C)

• GL(n, R/C) has two disconnected subgroups – GL+(n, R/C), GL-(n, R/C)

• Each element of GL(n, R) is a linear map.– They do not commute.– The matrices of rotations make a sub-group

Permutation group:

• is the set of all permutations on n distinct elements.

• Binary operation: composition

Representation Theory

• Representation theory studies how any given abstract group can be realized as a group of matrices.

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“quality” of the representationhttp://en.wikipedia.org/wiki/Group_representation

• If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application Φ : G × V → V defined by Φ(g, v) = ρ(g)(v) is continuous.

• Faithful

• Isomorphism