Robinson-Schensted Algorithms Obtained from Tableau Recursions

TL;DR

This paper derives Robinson-Schensted algorithms from tableau recursions, providing combinatorial algorithms that give bijective proofs of the dimension formula for symmetric group representations.

Contribution

It introduces a family of algorithms based on tableau recursions, including classical Robinson-Schensted algorithms, with new combinatorial proofs of key algebraic identities.

Findings

Derivation of Robinson-Schensted algorithms from tableau recursions

Development of new combinatorial algorithms with bijective proofs

Connection between tableau recursions and representation theory

Abstract

The numbers $f_\lambda$ of standard tableaux of shape $\lambda\vdash n$ satisfy 2 fundamental recursions: $f_\lambda = \sum f_{\lambda^-}$ and $(n + 1)f_\lambda=\sum f_{\lambda^+}$, where $\lambda^-$ and $\lambda^+$ run over all shapes obtained from $\lambda$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_\lambda^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, among which are the classical Robinson-Schensted row and column insertion algorithms. Each of our algorithms yields a bijective proof of the dimension formula.