Inner tableau translation property of the weak order and related results

Abstract

Let $SYT_{n}$ be the set of all standard Young tableaux with $n$ cells and $\leq_{weak}$ be Melnikov's the weak order on $SYT_n$. The aim of this paper is to introduce a conjecture, called the {\it property of inner tableau translation} and discuss its significance in the studies of the weak order. We will prove the conjecture for some specific cases