## 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

Comment by خرید کریو | December 17, 2014 |