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I'd like to know a reference for a simple proof that $\{c_\mu\mid \mu\vdash n\}$ is a basis for the centre of the symmetric group algebra $\mathbb{C}\mathfrak{S}_n$, where $c_\mu$ is the sum of all elements of a given cycle type (i.e. the sum of every element in the conjugacy class corresponding to $\mu$).

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This is XVIII, Proposition 4.2 in Lang's Algebra. It follows easily from some projection formulas that are in every book (e.g. the friendlier book by Fulton and Harris) but I'm too lazy to write that up at present. – Dylan Moreland Jul 21 '11 at 1:35
up vote 7 down vote accepted

The corresponding result (sums over conjugacy classes) is true with $S_n$ replaced by any group (although for infinite groups we only sum over the finite conjugacy classes), and it can probably be found in any book on representation theory of finite groups; for example, this is Proposition 12.22 in James and Liebeck's Representations and characters of groups.

In fact the following even more general statement is true.

Proposition: Let $G$ be a group acting on a set $X$. The invariant subspace of the free vector space on $X$, over an arbitrary field $k$, is spanned by elements of the form $s_O = \sum_{x \in O} x$ where $O$ ranges over all finite orbits of the action of $G$ on $X$.

Proof. Clearly any such element is invariant. In the other direction, if a vector $v = \sum c_x x$ lies in the invariant subspace, then by invariance $c_{gx} = c_x$ for all $g \in G$, hence $c_x = c_y$ whenever $x, y$ lie in the same (necessarily finite) orbit.

In this case $G = G, X = G$ and $G$ acts on $X$ by conjugation. The center of the group algebra is precisely the invariant subspace of the group algebra under conjugation, and the conclusion follows.

Note that if $G$ and $X$ are both finite and $k = \mathbb{C}$, the above gives an elegant proof of Burnside's lemma using the fact that the invariant subspace of a representation with character $\chi$ has dimension $\frac{1}{|G|} \sum_{g \in G} \chi(g)$.

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