quotient representations

We assume that all representations (G-modules) are finite-dimensional.

Definition 1

If N1 and N2 are G-modules over a field k (i.e. representations of G in N1 and N2), then a map φ:N1N2 is a G-map if φ is k-linear and preserves the G-action, i.e. if


G-maps have subrepresentations, also called G-submodules, as their kernel and image. To see this, let φ:N1N2 be a G-map; let M1N1 and M2N2 be the kernel and image respectively of φ. M1 is a submodule of N1 if it is stable under the action of G, but


M2 is a submodule of N2 if it is stable under the action of G, but


Finally, we define the intuitive concept of a quotient G-module. Suppose NN is a G-submodule. Then N/N is a finite-dimensional vector spaceMathworldPlanetmath. We can define an action of G on N/N via σ(n+N)=σ(n)+σ(N)=σ(n)+N, so that n+N is well-defined under the action and N/N is a G-module.

Title quotient representations
Canonical name QuotientRepresentations
Date of creation 2013-03-22 16:37:59
Last modified on 2013-03-22 16:37:59
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Definition
Classification msc 20C99