Some notes on pólya’s theorem, kostka numbers and the rsk correspondence

Definition 1.1. Associated with the partition λ of n is an irreducible representation of Sn. Let χ λ be the character of this representation. Define Kλ,G := 〈χ, Indn G 1G〉Sn = 〈Res Sn G χ ,1G〉G = dimC(χ) In other words, Kλ,G is the dimension of theG-invariant subspace when one restricts the irreducible χ from Sn to G. Note that this immediately implies that for any permutation group G, one has Kλ,G ≤ dimC χ = f, where f is the number of standard Young tableaux T of shape λ. Furthermore, when G = Sν = Sν1×Sν2×· · ·×Sν` , a Young subgroup of Sn, then Kλ,G = Kλ,ν , the well-known Kostka number counting column-strict tableaux of shape λ and content ν, that is, having ν1 ones, ν2 twos, etc. So Kλ,G generalizes Kostka numbers. Problem 1.2. For a permutation group G, interpret Kλ,G as counting some subset of standard Young tableaux of shape λ.