Permutation representation explained
can refer to either of two closely related notions: a
representation of
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
on a
set
is a
homomorphism from
to the
symmetric group of
:
\rho\colonG\to\operatorname{Sym}(X).
The image
\rho(G)\sub\operatorname{Sym}(X)
is a
permutation group and the elements of
are represented as permutations of
.
[1] A permutation representation is equivalent to an
action of
on the set
:
See the article on group action for further details.
Linear permutation representation
If
is a
permutation group of degree
, then the
permutation representation of
is the
linear representation of
\rho\colonG\to\operatorname{GL}n(K)
which maps
to the corresponding
permutation matrix (here
is an arbitrary
field).
[2] That is,
acts on
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
as a group of permutation matrices. One first represents
as a permutation group and then maps each permutation to the corresponding matrix. Representing
as a permutation group acting on itself by translation, one obtains the
regular representation.
Character of the permutation representation
Given a group
and a finite set
with
acting on the set
then the
character
of the permutation representation is exactly the number of fixed points of
under the action of
on
. That is
the number of points of
fixed by
.
This follows since, if we represent the map
with a matrix with basis defined by the elements of
we get a permutation matrix of
. 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
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
.
For example, if
and
the character of the permutation representation can be computed with the formula
the number of points of
fixed by
.So
\chi((12))=\operatorname{tr}(\begin{bmatrix}0&1&0\ 1&0&0\ 0&0&1\end{bmatrix})=1
as only 3 is fixed
\chi((123))=\operatorname{tr}(\begin{bmatrix}0&1&0\ 0&0&1\ 1&0&0\end{bmatrix})=0
as no elements of
are fixed, and
\chi(1)=\operatorname{tr}(\begin{bmatrix}1&0&0\ 0&1&0\ 0&0&1\end{bmatrix})=3
as every element of
is fixed.
External links
- https://mathoverflow.net/questions/286393/how-do-i-know-if-an-irreducible-representation-is-a-permutation-representation
Notes and References
- Book: Dixon, John D.. Permutation Groups. Mortimer. Brian. 2012-12-06. Springer Science & Business Media. 9781461207313. 5 - 6. en.
- Book: Robinson, Derek J. S.. A Course in the Theory of Groups. 2012-12-06. Springer Science & Business Media. 9781468401288. en.
