Linear representation theory of alternating groups

This article discusses the linear representation theory of the alternating group $$A_n$$ of finite degree $$n$$. Note that for $$n = 0,1$$, the alternating group coincides with the symmetric group, and for $$n = 2$$, it is trivial, so the interesting behavior begins from $$n = 3$$.

The article builds heavily on the linear representation theory of symmetric groups.

See also element structure of alternating groups.

Split conjugacy classes
We have canonical bijections:

Conjugacy classes from $$S_n$$ that split in $$A_n$$ $$\leftrightarrow$$ Partitions of $$n$$ into distinct odd parts (via splitting criterion) $$\leftrightarrow$$ Self-conjugate unordered integer partitions of $$n$$ $$\leftrightarrow$$ Irreducible representations of $$S_n$$ that split in $$A_n$$

For more on these bijections, see set of self-conjugate unordered integer partitions

Non-split conjugacy classes
We have equalities:

Number of conjugacy classes of even permutations in $$S_n$$ that don't split in $$A_n$$ = Number of conjugacy classes of odd permutations in $$S_n$$ = Number of (conjugate) pairs of non-self-conjugate partitions of $$n$$ = Number of pairs (in the sense of restricting to the same thing on $$A_n$$) of irreducible representations of $$S_n$$ that don't split in $$A_n$$.

Totals
If $$A$$ equals the number of conjugacy classes of even permutations from $$S_n$$ that split in $$A_n$$, and $$B$$ equals the number of conjugacy classes of even permutations in $$S_n$$ that don't split i $$A_n$$, then:


 * The number of conjugacy classes in $$A_n$$ is $$2A + B$$
 * The number of conjugacy classes in $$S_n$$ is $$A + 2B$$

Field generated by character values
Below is information on the field generated by character values for the alternating group of degree $$n$$ for small values of $$n$$, all in characteristic zero. It is the same as the unique minimal splitting field, because all the irreducible representations have Schur index one.

Note that for each irreducible representation, the extension over the rationals for its character values, if nontrivial, is a quadratic extension. The overall field generated by character values is thus a Galois extension of the rationals with Galois group an elementary abelian 2-group and extension degree a power of 2.

Criterion for the field generated by character values to be real
For a finite group, the field generated by character values is real if and only if the group is an ambivalent group, i.e., every element is conjugate to its inverse. For alternating groups, using the criterion for element of alternating group to be real, we can obtain a classification of ambivalent alternating groups, which basically gives us that the alternating group of degree $$n$$ is ambivalent iff $$n \in \{ 1,2,5,6,10,14 \}$$.