example of reducible and irreducible G-modules

Let G=Sr, the permutation groupMathworldPlanetmath on r elements, and N=kr where k is an arbitrary field. Consider the permutation representation of G on N given by


If r>1, we can define two submodules of N, called the trace and augmentation, as


Clearly both N and N′′ are stable under the action of G and thus in fact form submodules of N.

If the characteristic of k divides r, then obviously N′′N. Otherwise, N′′ is a simple (irreducible) G-module. For suppose N′′ has a nontrivial submodule M, and choose a nonzero uM. Then some pair of coordinates of u are unequal, for if not, then u=(a,,a) and then uN′′ because of the restrictionPlanetmathPlanetmathPlanetmath on the characteristic of k forces ra0. So apply a suitable element of G to get another element of M, v=(b1,b2,,br) where b1b2 (note here that we use the fact that M is a submodule and thus is stable under the action of G).

But now (12)v-ev=(b1-b2,b2-b1,0,,0) is also in M, so w=(1,-1,0,,0)M. It is obvious that by multiplying w by elements of k and by permuting, we can obtain any element of N′′ and thus M=N′′. Thus N′′ is simple.

It is also obvious that N=NN′′.

Title example of reducible and irreducible G-modules
Canonical name ExampleOfReducibleAndIrreducibleGmodules
Date of creation 2013-03-22 16:37:50
Last modified on 2013-03-22 16:37:50
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Example
Classification msc 16D60
Defines augmentation