Permutation representation

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In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.

Abstract permutation representation

A permutation representation of a group G on a set X is a homomorphism from G to the symmetric group of X:

ρ:GSym(X).

The image ρ(G)\subSym(X) is a permutation group and the elements of G are represented as permutations of X.[1] A permutation representation is equivalent to an action of G on the set X:

G×XX.

See the article on group action for further details.

Linear permutation representation

If G is a permutation group of degree n, then the permutation representation of G is the linear representation of G

ρ:GGLn(K)

which maps gG to the corresponding permutation matrix (here K is an arbitrary field).[2] That is, G acts on Kn by permuting the standard basis vectors.

This notion of a permutation representation can, of course, be composed with the previous one to represent an arbitrary abstract group G as a group of permutation matrices. One first represents G as a permutation group and then maps each permutation to the corresponding matrix. Representing G as a permutation group acting on itself by translation, one obtains the regular representation.

Character of the permutation representation

Given a group G and a finite set X with G acting on the set X then the character χ of the permutation representation is exactly the number of fixed points of X under the action of ρ(g) on X. That is χ(g)= the number of points of X fixed by ρ(g).

This follows since, if we represent the map ρ(g) with a matrix with basis defined by the elements of X we get a permutation matrix of X. Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in X is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of X.

For example, if G=S3 and X={1,2,3} the character of the permutation representation can be computed with the formula χ(g)= the number of points of X fixed by g. So

χ((12))=tr([010100001])=1 as only 3 is fixed
χ((123))=tr([010001100])=0 as no elements of X are fixed, and
χ(1)=tr([100010001])=3 as every element of X is fixed.

References

  1. Dixon, John D.; Mortimer, Brian (2012-12-06) (in en). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313. https://books.google.com/books?id=1SPjBwAAQBAJ. 
  2. Robinson, Derek J. S. (2012-12-06) (in en). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288. https://books.google.com/books?id=BFrTBwAAQBAJ. 

External links




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