A non-iterative formula for straightening fillings of Young diagrams

TL;DR

This paper introduces a direct, non-iterative formula for straightening fillings of Young diagrams, simplifying a fundamental process in combinatorics with applications in representation theory and algebraic geometry.

Contribution

It provides the first explicit non-iterative formula for straightening Young diagram fillings, advancing the combinatorial methods used in related mathematical fields.

Findings

Derived a non-iterative straightening formula

Generalized a theorem of Gonciulea and Lakshmibai

Simplified the process of expressing fillings as sums of semistandard tableaux

Abstract

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai.