On conjugacy classes of S_n containing all irreducibles · arXivDesk
1607.08581Jul 28, 201616 pages, 22 figures. Minor corrections in proofs. Added Theorem 5.1, which completely characterises the global conjugacy classes in $S_n$ (Conjecture 2 of original paper was subsequently proved by J. Swanson, and is now Proposition 5.2)
On conjugacy classes of Sn containing all irreducibles
It is shown that for the conjugation action of the symmetric group Sn, when n=6 or n≥8, all Sn
Nearby in the stack
-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition
λ
of
n
with at least two parts, whose parts are all distinct and taken from the set of odd primes and 1. The following simple characterisation of conjugacy classes containing all irreducibles is proved: If
n=4,8,
the partition
λ
of
n
indexes a global conjugacy class for
Sn
if and only if it has at least two parts, and all its parts are odd and distinct.