Characters of the Alternating Group

There is a special routine which computes the character table of the alternating group, as well as routines which compute values of alternating characters and the characters themselves. These routines make use of the fact that in most cases the irreducible characters of the symmetric group, which can be computed quickly, are also irreducible in the alternating group. So the irreducible characters of the alternating group may be indexed by partitions in the same way as those of the symmetric group. As the restriction of the irreducible character indexed by the partition \(\lambda\) is equal to the restriction of the character indexed by the conjugate partition, we only need to take one from each of these pairs of partitions to form a full set of irreducible characters. When a partition is conjugate to itself the character of the symmetric group indexed by that partition is no longer irreducible but is the sum of two irreducibles. This method is described in [James and Kerber, 1981].

Single Values

AlternatingCharacterValue(pa, pe): SeqEnum, GrpPermElt -> RngElt

Return the value of the character of the alternating group of degree \(n\) indexed by the partition \(pa\) of weight \(n\) on the permutation \(pe\). The partition \(pa\) and its conjugate should be distinct.

AlternatingCharacterValue(pa, i, pe): SeqEnum, RngIntElt, GrpPermElt -> RngElt

Return the value of the \(i\)th character of the alternating group of degree \(n\) indexed by the self conjugate partition \(pa\) of weight \(n\) on the permutation \(pe\). Since there are two possible irreducible characters indexed by such partitions \(i\) must be either \(1\) or \(2\).

Irreducible Characters

AlternatingCharacter(pa): SeqEnum -> AlgChtrElt

Return the character of the alternating group of degree \(n\) indexed by the partition \(pa\) of weight \(n\). The partition \(pa\) and its conjugate should be distinct.

AlternatingCharacter(pa, i): SeqEnum, RngIntElt -> AlgChtrElt

Return the \(i\)th character of the alternating group of degree \(n\) indexed by the self conjugate partition \(pa\) of weight \(n\). Since there are two possible irreducible characters indexed by such partitions \(i\) must be either \(1\) or \(2\).

Character Table

AlternatingCharacterTable(d): RngIntElt -> SeqEnum

Returns the character table of the alternating group of degree \(d\).