Inner tableau translation property of the weak order and related results

Abstract

Let $SY{T}_n$ be the set of all standard Young tableaux with $n$ cells and ${\le }_{weak}$ be Melnikov's the weak order on $SY{T}_n$. 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