A closed non-iterative formula for straightening fillings of Young diagrams

Reuven Hodges

Published 2017-10-14Version 1

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, due to Alfred Young, that expresses a filling of a Young diagram as a sum of semistandard tableau subject to certain relations. It has been a long-standing open problem to give a non-iterative, closed formula for this straightening process. This paper gives such a formula, as well as a simple combinatorial description of the coefficients that arise. Moreover, an interpretation of these coefficients in terms of paths in a directed graph is provided.