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.

Particular cases

$n$	Alternating group	Order	Degrees of irreducible representations	Number of irreducible representations	Linear representation theory page
3	cyclic group:Z3	3	1,1,1	3	linear representation theory of cyclic group:Z3
4	alternating group:A4	12	1,1,1,3	4	linear representation theory of alternating group:A4
5	alternating group:A5	60	1,3,3,4,5	5	linear representation theory of alternating group:A5
6	alternating group:A6	360	1,5,5,8,8,9,10	7	linear representation theory of alternating group:A6
7	alternating group:A7	2520	1,6,10,10,14,14,15,21,35	9	linear representation theory of alternating group:A7
8	alternating group:A8	20160	1,7,14,20,21,21,21,28,35,45,45,56,64,70	14	linear representation theory of alternating group:A8
9	alternating group:A9	181440			linear representation theory of alternating group:A9
10	alternating group:A10	1814400			linear representation theory of alternating group:A10

Combinatorics

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 $S_{n}$ = 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 from $S_{n}$ that split in $A_{n}$ , and $B$ equals the number that don't, then:

The number of conjugacy classes in $A_{n}$ is $2 A + B$
The number of conjugacy classes in $S_{n}$ is $A + 2 B$

$n$	Alternating group $A_{n}$	Order	$A =$ number of irreducible representations from $S_{n}$ that split in $A_{n}$	$B =$ number of irreducible representations from $S_{n}$ that don't split in $A_{n}$	$2 A + B =$ number of irreducible representations in $A_{n}$	$A + 2 B =$ number of irreducible representations in $S_{n}$	$C =$ number of irreducible representations from $S_{n}$ which, even after splitting, remain real	$2 C + B =$ number of conjugacy classes of real elements in $A_{n}$	$A + B + C =$ number of equivalence classes under real conjugacy in $A_{n}$
3	cyclic group:Z3	3	1	1	3	3	0	1	2
4	alternating group:A4	12	1	2	4	5	0	2	3
5	alternating group:A5	60	1	3	5	7	1	5	5
6	alternating group:A6	360	1	5	7	11	1	7	7
7	alternating group:A7	2520	1	7	9	15	0	7	8
8	alternating group:A8	20160	2	10	14	22	0	10	12
9	alternating group:A9	181440	2	14	18	30	1	16	17
10	alternating group:A10	1814400	2	20	24	42	2	24	24