A symmetry property for q-weighted Robinson-Schensted algorithms and other branching insertion algorithms

TL;DR

This paper proves a symmetry property for a q-weighted Robinson-Schensted algorithm, extending the classical symmetry to a broader class of branching insertion algorithms using growth graph techniques.

Contribution

It establishes a symmetry property for the q-weighted Robinson-Schensted algorithm and generalizes the growth diagram approach to other branching insertion algorithms.

Findings

Proves symmetry property for q-weighted Robinson-Schensted algorithm

Extends growth diagram approach to a wider class of algorithms

Applies to recent q-weighted Robinson-Schensted variants

Abstract

In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently been introduced by Borodin-Petrov(2013).