The Frobenius transform of a symmetric function

TL;DR

This paper introduces the Frobenius transform, an algebraic map on symmetric functions, providing new formulas, combinatorial interpretations, and bounds for restriction coefficients related to symmetric group representations.

Contribution

It defines the Frobenius transform, explores its properties, computes its matrix entries in various bases, and offers combinatorial interpretations and bounds for restriction coefficients.

Findings

The Frobenius transform satisfies a multiplicative identity with respect to the Kronecker product.

Explicit formulas for the matrix entries of the transform in different bases are derived.

New bounds and combinatorial interpretations for the restriction coefficients are established.

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.