Séminaire Lotharingien de Combinatoire, 93B.85 (2025), 12 pp.

Mitchell Lee

The Restriction Problem and the Frobenius Transform

Abstract. We define an abelian group homomorphism 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 F in the Schur basis are the restriction coefficients r_λ^μ = dim Hom_{S_n(V_μ,S^λCⁿ}), which are known to be nonnegative integers but have no known combinatorial interpretation.

We compute F{f} when f is an elementary, complete homogeneous, or power sum symmetric function. As a consequence, we prove that r_λ^μ = 0 if |λ ∩ μ^| < 2|μ^| - |λ|, where μ^ 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^{λ₁ - 1}, and this inequality is tight.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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