3-Plethysms of homogeneous and elementary symmetric functions

2022/09/29 by Florence Maas-Gariépy, Maas-Gariépy, Florence, Étienne Tétreault +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2209.14942

Abstract

We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms s_ν[h_λ] and s_ν[e_λ], for any partitions λ and ν. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where ν is a partition of 3 and λ has one part. We then use a Kronecker map to extend these results to any partition λ.

Related

• Splitting the square of homogeneous and elementary functions into their symmetric and anti-symmetric parts
• Plethysms of symmetric functions and highest weight representations
• Two stability theorems on plethysms of Schur functions
• A new order on integer partitions
• On partitions associated with elementary symmetric polynomials