On the greatest and least elements in the set of semistandard tableaux of given shape and weight

TL;DR

This paper introduces three algorithms for constructing semistandard tableaux with specified shape and weight, and proves the existence of unique greatest and least elements under a natural partial order.

Contribution

It provides new algorithms to explicitly find the extremal semistandard tableaux for arbitrary weights and establishes their uniqueness within the set.

Findings

Existence of unique greatest and least semistandard tableaux

Two algorithms explicitly construct these extremal tableaux

The set of tableaux is well-ordered under a natural partial order

Abstract

We give three algorithms to construct a semistandard tableau of given shape and weight, where the weight is a composition which is not necessarily a partition. With respect to a natural partial order on the set of semistandard tableaux, we show that the set of semistandard tableaux of given shape and weight has a unique greatest element and a unique least element. Two of our algorithms give each of these elements.