Species, Symmetric Functions, and Kronecker Product

TL;DR

This paper introduces two new symmetric function families from species theory, providing bases, explicit formulas, and a combinatorial framework for the Kronecker product with nonnegative coefficients.

Contribution

It constructs novel bases of symmetric functions from species theory, generalizes classical Kronecker product results, and offers a new combinatorial perspective.

Findings

Established explicit cycle-index formulas for the new bases.

Proved the bases are closed under the Kronecker product.

Showed the product expands with nonnegative integer coefficients.

Abstract

We study two new families of symmetric functions arising from a species-theoretic construction motivated by cycle structure. For each partition of $n$, we define two combinatorial species that decompose into molecules indexed by the same partition, giving rise to two corresponding basis of the homogeneous symmetric functions of degree $n$. We prove that each of these families forms a basis by exhibiting explicit cycle-index formulas and triangular transition matrices to the power-sum basis. Using these constructions, we generalize a classical result describing the Kronecker (Hadamard) product in the homogeneous basis to the two new settings. In particular, we show that the categories generated by these species are closed under the Kronecker product, and that the product of two basis elements expands with nonnegative integer coefficients. Our results provide a new combinatorial framework for studying the Kronecker product and suggest avenues toward interpreting its structure constants