Fomin-Greene monoids and Pieri operations

TL;DR

This paper investigates monoids generated by operators on infinite partial orders, extending Fomin-Greene monoids, and studies the commutation and structure constants of associated non-commutative functions.

Contribution

It introduces generalized monoids beyond Fomin-Greene relations and analyzes their properties and symmetric functions, expanding the understanding of algebraic structures in combinatorics.

Findings

Operators generate commuting non-commutative functions

Symmetric functions encode structure constants

Extension of Fomin-Greene monoid relations

Abstract

We explore monoids generated by operators on certain infinite partial orders. Our starting point is the work of Fomin and Greene on monoids satisfying the relations $(\u{r}+\u{r+1})\u{r+1}\u{r}=\u{r+1}\u{r}(\u{r}+\u{r+1})$ and $\u{r}\u{t}=\u{s}\u{r}$ if $|r-t|>1.$ Given such a monoid, the non-commutative functions in the variables $\u{}$ are shown to commute. Symmetric functions in these operators often encode interesting structure constants. Our aim is to introduce similar results for more general monoids not satisfying the relations of Fomin and Greene. This paper is an extension of a talk by the second author at the workshop on algebraic monoids, group embeddings and algebraic combinatorics at The Fields Institute in 2012.