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representation theory of the general linear group

Contents

Contents

Idea

The representation theory of the general linear groups.

Properties

Polynomial irreps

The polynomial irreps of GL(n,)GL(n,\mathbb{C}) are labeled by Young diagrams λPart(n)\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(λ)n1rows(\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 formulahook-content formula
number of standard Young tableauxnumber of semistandard Young tableaux
dimension of irreps of Sym(n)dimension of irreps of SL(n)

References

Last revised on May 17, 2021 at 04:21:09. See the history of this page for a list of all contributions to it.