2023/07/13 by Mitchell Lee, Lee, Mitchell · 2 citations
Computer Science · Mathematics · #05E05 #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2307.06678
openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define an abelian group homomorphism \mathscrF, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of \mathscrF in the Schur basis are the restriction coefficients rλμ= dim Hom_\mathfrakSn(Vμ, \mathbbSλℂn), which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity \mathscrF\fg\ = \mathscrF\f\ ∗ \mathscrF\g\, where ∗ is the Kronecker product. We prove for all symmetric functions f that \mathscrF\f\ = \mathscrFSur\f\ ⋅ (1 + h1 + h2 + ⋯), where \mathscrFSur\f\ is a symmetric function with the same degree and leading term as f. Then, we compute the matrix entries of \mathscrFSur\f\ in the complete homogeneous, elementary, and power sum bases and of \mathscrF-1Sur\f\ in the complete homogeneous and elementary bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of \mathscrF-1Sur\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 rλμ= 0 if |λ∩ μ| < 2|μ| - |λ|, where μ is the partition formed by removing the first part of μ. We also prove that rλμ= 0 if the Young diagram of μ contains a square of side length greater than 2λ1 - 1, and this inequality is tight.