Relation between characters of symmetric group and general linear group

Question

What is the relation between characters of the symmetric group ($S_n$) and characters of general linear group ($GL(n)$), if any? Can the character of $S_n$ be expanded in terms of characters of $GL(n)$. Thanks.

Answer 1

There is a link between irreducible represetations kwonn as Schur-Weyl duality. https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality

Answer 2

We posted a paper on the arXiv a few years ago that defines the characters of $S_n$ (embedded in $Gl(n)$ as permutation matrices) as symmetric functions which can then be expanded in the Schur basis (irreducible $Gl(n)$ characters): https://arxiv.org/abs/1605.06672

There isn't an explicit formula for computing the Schur expansions of these symmetric functions, but since the characters of $S_n$ form a basis, the Schur expansion can be computed.

There is code in Sage to do this for specific examples. sage: s = SymmetricFunctions(QQ).s(); s Symmetric Functions over Rational Field in the Schur basis sage: st = SymmetricFunctions(QQ).st(); st Symmetric Functions over Rational Field in the irreducible symmetric group character basis sage: s(st[2,1]) # expand irreducible character indexed by (n-3,2,1) in Schur basis 3*s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1]

Sami Assaf and David Speyer have an unpublished proof that this expansion is alternating in sign by degree.