Diagonal invariants and the refined multimahonian distribution

TL;DR

This paper explores multivariate diagonal invariants of the symmetric group, introduces a multivariate Robinson-Schensted extension, and provides combinatorial algorithms and symmetry results related to permutation enumeration.

Contribution

It presents a novel multivariate extension of the Robinson-Schensted correspondence and new combinatorial algorithms for tensor product decomposition.

Findings

Established the existence of a multivariate Robinson-Schensted correspondence

Developed a combinatorial algorithm for tensor product decomposition

Discovered new symmetry properties in permutation enumeration

Abstract

Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.