Youngdiagram

A Young diagram is a graphical representation of a partition of a positive integer. In a partition, an integer n is expressed as a sum of positive integers arranged in non-increasing order. Each part of the partition corresponds to a row of boxes in the diagram, with the largest part forming the top row. For example, the partition 5 = 3 + 2 is depicted as three boxes in the first row and two boxes in the second row. The diagram is usually drawn with rows aligned to the left and arranged from top to bottom, each row shorter or equal in length to the one above it.

Young diagrams are used extensively in representation theory, combinatorics, and the study of symmetric functions. They provide a visual tool for understanding the structure of symmetric group representations and the combinatorics of Schur functions. The diagram also plays a role in the study of crystal bases and the theory of Macdonald polynomials. Two diagrams are considered equivalent if they share the same partition, and operations like adding or removing a box correspond to adding or removing a part from the partition.

Beyond mathematics, Young diagrams have applications in physics, particularly in the context of quantum many-body systems and the study of entanglement patterns. In algebraic geometry, they are related to the classification of certain vector bundles via the representation theory of general linear groups. Their utility as a unifying visual language across several domains highlights their importance in both pure and applied mathematical contexts.