nLab representation theory of the general linear group

Idea

The representation theory of the general linear groups.

Properties

Polynomial irreps

The polynomial irreps of $GL(n,\mathbb{C})$ are labeled by Young diagrams $\lambda \in Part(n)$ (e.g. Fulton 97, Thm. 2 on p. 114).

These are equivalently the irreps of the special linear group (e.g. Sternberg 94, Sec. 5.8).

And for $rows(\lambda) \leq n-1$ these are also the irreps of the special unitary group (e.g. Peluse 14, p. 2).

Dimension of irreps and hook-content formula

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

• Schur-Weyl duality

• representation theory of the symmetric group

• representation theory of the special unitary group

References

• Shlomo Sternberg, Section 5 and Appendix C.7 of: Group Theory and Physics, Cambridge University Press 1994 (ISBN:9780521558853)

• William Fulton, Section 8 of: Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997 (doi:10.1017/CBO9780511626241)

• Amritanshu Prasad: Representations of General Linear Groups [doi:10.1017/CBO9781139976824.007], Chapter 6 in: Representation theory – A Combinatorial Viewpoint, Cambridge University Press (2014) [doi:10.1017/CBO9781139976824]