Here are a few small observations on permutation characters motivated by discussions at the Herstmonceux conference Algebraic combinatorics and group actions. Let denote the irreducible character of the symmetric group labelled by the partition of . Let denote the permutation character of acting on the cosets of the Young subgroup .
- Since where is an integral linear combination of characters for partitions dominating , any irreducible character of is an integral linear combination of the .
This can be made more explicit as follows. Let . For , let , where acts on by place permutation. The Jacobi–Trudi formula states that
- It is a special property of symmetric groups that any character is a difference of permutation characters. For example, the cyclic group has (the number of divisors of ) distinct permutation characters, but irreducible characters, so unless , the additive group generated by the permutation characters is a proper subgroup of the integral character ring. And clearly no irreducible character of a finite group taking a non-rational value can be an integral linear combination of permutation characters.
- Since a transitive permutation character contains the trivial character exactly once, the trivial character is not a difference of two transitive permutation characters. A further example is given below.
Claim. is not a difference of two transitive permutation characters of .
Proof. Let denote the permutation character of acting on the subgroup of . Note that the sign character, , appears in if and only if .
Let be the number of orbits of on . Since
it follows from Frobenius reciprocity and Mackey’s Formula that . Consider . Multiplying the displayed equation above by , we get
By Frobenius reciprocity
By Mackey’s Formula, the right-hand side is
where the sum is over distinct double cosets . It follows that , with equality when .
Suppose that and are subgroups of such that contains and . From we see that and . From , we see that . By the previous paragraph,
Hence also appears in . Therefore
is not a difference of two transitive permutation characters.
- Since for (this is a special case of the Jacobi—Trudi formula) each is a difference of two transitive permutation characters. Moreover, . Exhausting over all subgroups of by computer algebra supports the conjecture that, provided , these are the only irreducible characters of that are the difference of two transitive permutation characters.
- The behaviour of is interesting: all irreducible characters except for are the difference of two transitive permutation characters. This is explained in part by the outer automorphism that exists in this case.