The Frobenius transform of a symmetric function

Abstract

We define an abelian group homomorphism $\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\mathscr{F}$ in the Schur basis are the restriction coefficients $r_λ^μ= \dim \operatorname{Hom}_{\mathfrak{S}_n}(V_μ, \mathbb{S}^λ\mathbb{C}^n)$, which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity $\mathscr{F}\{fg\} = \mathscr{F}\{f\} \ast \mathscr{F}\{g\}$, where $\ast$ is the Kronecker product. We prove for all symmetric functions $f$ that $\mathscr{F}\{f\} = \mathscr{F}_{\mathrm{Sur}}\{f\} \cdot (1 + h_1 + h_2 + \cdots)$, where $\mathscr{F}_{\mathrm{Sur}}\{f\}$ is a symmetric function with the same degree and leading term as $f$. Then, we compute the matrix entries of $\mathscr{F}_{\mathrm{Sur}}\{f\}$ in the complete homogeneous, elementary, and power sum bases and of $\mathscr{F}^{-1}_{\mathrm{Sur}}\{f\}$ in the complete homogeneous and elementary bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of $\mathscr{F}^{-1}_{\mathrm{Sur}}\{f\}$ in the elementary basis count words with a constraint on their Lyndon factorization. As an example application of our main results, we prove that $r_λ^μ= 0$ if $|λ\cap \hatμ| < 2|\hatμ| - |λ|$, where $\hatμ$ is the partition formed by removing the first part of $μ$. We also prove that $r_λ^μ= 0$ if the Young diagram of $μ$ contains a square of side length greater than $2^{λ_1 - 1}$, and this inequality is tight.