Thread: Representation of a group acting on finite sets?

1. Representation of a group acting on finite sets?

Hi, I'm just doing a course in the Representation of Finite Groups, and in the notes, we are given the following definition of a representation which is not actually given a name:

Let G be a finite group, M a finite set, and let G act on M. For F some field we can construct a representation of G on $FM = \{\sum c_m e_m : c_m \in F, m \in M\}$ , FM having a basis $e_m$ by $\rho(g)e_m = e_{g.m}$ , extended to all of FM by linearity.

Several examples follow, e.g. where G is a group and M is a 1-element set, then FM is the trivial representation.

My notes are a little vague and I'm not exactly clear on the topic, so does anyone know what exactly this sort of representation is called so I can look at it in more depth online, or better yet have links to a source where I can research it?

Thanks

2. Re: Representation of a group acting on finite sets?

It appears that FM is the free vector space (equivalently, the free F-module) generated by M (we just declare each element of M to be a basis, and take all formal F-linear combinations).

Since G acts on M, each element g of G induces a permutation of the basis vectors, which is necessarily an invertible linear transformation (a change-of-basis). I'm not aware of a name for this construction, however if M = G, we obtain (via the action of left-multiplication) the left regular representation for G.

Basically, what you are doing is "extending" an action of G on M to a representation of G over FM. For example, suppose G = (Z3,+), a cyclic group of order 3, and M = {a,b,c}, and F is the field of rational numbers. Then:

FM = {pa + qb + rc: p,q,r in Q}, which is isomorphic to Q3.

One possible action of Z3 on M is:

0.b = a, 0.b = b, 0.c = c
1.a = b, 1.b = c, 1.c = a
2.a = c, 2.b = a, 2.c = b

(we let Z3 act on M by permuting the elements cyclically).

the representation over FM in the basis {a,b,c} is then given by the linear transformations:

0 --> I (the identity 3x3 matrix in the basis {a,b,c})
1 -->
[0 1 0]
[0 0 1]
[1 0 0], which sends pa + qb + rc --> qa + rb + pc
2-->
[0 0 1]
[1 0 0]
[0 1 0], which sends pa + qb + rc --> ra + pb + qc

In actual practice, since only the cardinality of M really matters, it is customary to work in the vector space F|M|, with the standard basis vectors, and consider the matrices of the representation as a subgroup of the embedding of the symmetric group into GL(F|M|) as standard permutation matrices.

3. Re: Representation of a group acting on finite sets?

Originally Posted by Conn
Hi, I'm just doing a course in the Representation of Finite Groups, and in the notes, we are given the following definition of a representation which is not actually given a name:

Let G be a finite group, M a finite set, and let G act on M. For F some field we can construct a representation of G on $FM = \{\sum c_m e_m : c_m \in F, m \in M\}$ , FM having a basis $e_m$ by $\rho(g)e_m = e_{g.m}$ , extended to all of FM by linearity.

Several examples follow, e.g. where G is a group and M is a 1-element set, then FM is the trivial representation.

My notes are a little vague and I'm not exactly clear on the topic, so does anyone know what exactly this sort of representation is called so I can look at it in more depth online, or better yet have links to a source where I can research it?

Thanks
The name is permutation representation.