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Math Help - Representation of a group acting on finite sets?

  1. #1
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    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
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  2. #2
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    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.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Re: Representation of a group acting on finite sets?

    Quote Originally Posted by Conn View Post
    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.
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