Insertion algorithms for Gelfand $S_n$-graphs

TL;DR

This paper introduces a novel column-based insertion algorithm for Gelfand $S_n$-graphs that provides a new bijection between involutions and standard tableaux, revealing connections to representation theory.

Contribution

It presents a modified insertion algorithm adding cells at columns instead of rows, establishing a new bijection and linking to the structure of $W$-graphs in symmetric group representations.

Findings

New column-based insertion algorithm for involutions

Connection between insertion algorithms and $W$-graph molecules

Potential insights into symmetric group representation theory

Abstract

The two tableaux assigned by the Robinson--Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard tableaux. Beissinger found a concise way of formulating this restricted map, which involves adding an extra cell at the end of a row after a Schensted insertion process. We show that by changing this algorithm slightly to add cells at the end of columns rather than rows, one obtains a different bijection from involutions to standard tableaux. Both maps have an interesting connection to representation theory. Specifically, our insertion algorithms classify the molecules (and conjecturally the cells) in the pair of $W$-graphs associated to the unique equivalence class of perfect models for a generic symmetric group.