A non-crossing standard monomial theory

TL;DR

This paper develops a non-crossing analogue of standard monomial theory, relating non-crossing tableaux to Gelfand-Tsetlin patterns and connecting them to weakly separated sets, offering new insights into combinatorial structures.

Contribution

It introduces non-crossing tableaux and establishes their relation to Gelfand-Tsetlin patterns and weakly separated sets, expanding the combinatorial framework of standard monomial theory.

Findings

Non-crossing tableaux are related to Gelfand-Tsetlin patterns.

Non-crossing tableaux include weakly separated sets as special cases.

The non-crossing analogue of standard monomial theory is developed.

Abstract

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.