Young Tableau - Applications in Representation Theory

Applications in Representation Theory

Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers. They provide a convenient way of specifying the Young symmetrizers from which the irreducible representations are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram.

Young diagrams also parametrize the irreducible polynomial representations of the general linear group GL_n (when they have at most n nonempty rows), or the irreducible representations of the special linear group SL_n (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SU_n (again when they have at most n − 1 nonempty rows). In these case semistandard tableaux with entries up to n play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.