Young diagrams are used to describe many objects in algebra and combinatorics, including:
17 = 5 + 4 + 4 + 2 + 1 + 1
is drawn as the Young diagram
conjugacy classes in .
irreducible representations of the symmetric groups over any field of characteristic zero
irreducible algebraic representations of the special linear groups
irreducible unitary representations of the special unitary groups
elementary symmetric functions
basis vectors for the free lambda-ring on one generator,
finite-dimensional C-algebras: any such algebra is of the form for some unique list of natural numbers .
finite abelian -groups: any such group is of the form for some unique list of natural numbers .
finite commutative semisimple algebras over : any such algebra is of the form for some unique list of natural numbers .
the trace of the category of finite sets has isomorphism classes of objects corresponding to Young diagrams.
A Young diagram , also called Ferrers diagram, is a graphical representation of an unordered integer partition ). If is a partition of then the Young diagram has boxes. A partition can be addressed as a multiset over .
There are two widely used such representations. The English one uses matrix-like indices, and the French one uses Cartesian coordinate-like indices for the boxes in the diagram .
In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the ‘y’-axis being downward oriented. For instance the diagram representing the partition of is given in the English representation as:
Let be the set of Young diagrams. Important functions on Young diagrams include:
A filling of a Young diagram with elements from a set is called a Young tableau?.
A generalization of a Young diagram is a skew Young diagram. Let be two partitions, and let be defined as (possibly adding trailing zeros). The skew Young diagram is given by the Young diagram with all boxes belonging to when superimposed removed. If and then looks like:
For a quick online introduction to Young diagrams, try:
A nice introduction to Young diagrams can be found here:
A more detailed reference is:
A connection to algebraic geometry: