Inner tableau translation property of the weak order and related results

TL;DR

This paper introduces the inner tableau translation property within the weak order on standard Young tableaux, explores its significance, and proves it for specific cases, advancing understanding of tableau combinatorics.

Contribution

It proposes the conjecture of inner tableau translation property and provides proofs for particular instances, contributing new insights to the structure of the weak order.

Findings

Conjecture of inner tableau translation property introduced

Proved for specific cases within the weak order

Highlights significance in tableau combinatorics

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