Characters of Permutation Representations
Let’s take to be a permutation representation coming from a group action on a finite set
that we’ll also call
. It’s straightforward to calculate the character of this representation.
Indeed, the standard basis that comes from the elements of gives us a nice matrix representation:
On the left is the matrix of the action on
, while on the right it’s the group action on the set
. Hopefully this won’t be too confusing. The matrix entry in row
and column
is
if
sends
to
, and it’s
otherwise.
So what’s the character ? It’s the trace of the matrix
, which is the sum of all the diagonal elements:
This sum counts up for each point
that
sends back to itself, and
otherwise. That is, it counts the number of fixed points of the permutation
.
As a special case, we can consider the defining representation of the symmetric group
. The character
counts the number of fixed points of any given permutation. For instance, in the case
we calculate:
In particular, the character takes the value on the identity element
, and the degree of the representation is
as well. This is no coincidence;
will always be the degree of the representation in question, since any matrix representation of degree
must send
to the
identity matrix, whose trace is
. This holds both for permutation representations and for any other representation.
[…] pass from representations to their characters. Of course, this isn’t much of a stretch, since we saw that the character of a representation includes information about the dimension: […]
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[…] The nice thing here is that it’s a permutation representation, and that means we have a shortcut to calculating its character: is the number of fixed point of the action of on the standard basis […]
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[…] we’ve constructed. Since these come from actions of on various sets, we have our usual shortcut to calculate their characters: count fixed […]
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thank you