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 $r_\lambda^\mu = \dim \operatorname{Hom}{\mathfrak{S}_n}(V\mu, \mathbb{S}^\lambda \mathbb{C}^n)$, 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 $r_\lambda^\mu = 0$ if $|\lambda \cap \hat\mu| < 2|\hat\mu| - |\lambda|$, where $\hat\mu$ is the partition formed by removing the first part of $\mu$. We also prove that $r_\lambda^\mu = 0$ if the Young diagram of $\mu$ contains a square of side length greater than $2^{\lambda_1 - 1}$, and this inequality is tight.