nLab representation theory of the special unitary group

Idea

The representation theory of the special unitary group.

Properties

Irreps

The irreps of $SU(n)$ are those polynomial irreps of GL(n,C), hence those irreps of $SL(n,\mathbb{C})$, which are labeled by partitions/Young diagrams $\lambda \in Part(n)$ with $rows(\lambda) \leq n - 1$.

(e.g. Peluse 14, p. 14))

Dimension of irreps and hook/content

hook length formula	hook-content formula
number of standard Young tableaux	number of semistandard Young tableaux
dimension of irreps of Sym(n)	dimension of irreps of SL(n)

Related concepts

• hook-content formula

• representation theory of the symmetric group

• representation theory of the general linear group

References

• Shlomo Sternberg, Section 5 of: Group Theory and Physics, Cambridge University Press 1994

• Sarah Peluse, Irreducible representations of $SU(n)$ with prime power degree, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d (pdf)