Complex general linear group

Tensor representation

Let be the defining representation of . A tensor representation of is a subrepresentation of for some . lie Weyl’s construction associates every tensor irrep for fixed to a Young diagram.

Weyl’s construction

Let , act on by and act on by the permutation representation . In addition let act by the corresponding ∗-representation.

Let be the set of Young diagrams of boxes and at most rows. For each let be the corresponding minimal left ideal with basis . For a given , either vanish or transform in the irrep (see below). Let be a complete¹ set of tensors such that each is unique. Then

form an irreducible orthonormal basis under both and , where

is a -dimensional irreducible invariant subspace under and

is a -dimensional irreducible invariant subspace transforming under in an irrep henceforth labeled . Thus

where is given by the Hook length formula and is given by Stanley’s hook content formula.

Proof of vanishing property

For there exists some s.t. . Then for since , giving invariance. Clearly is a basis of . We define a matrix representation by

with summation convention. Then

as required.

Proof

proof See 07 Tensor method for constructing irreps of GL(n) and subgroups for discussion.

Tensor representations of and

Every irrep of GLₙ(ℂ) is an irrep of U(n) and SU(n), so tensor irreps given above are also tensor irreps for these subgroups. However, since each column of length corresponds to the determinant representation, which is trivial for , such columns may be removed without changing the representation up to equivalence.

Properties

• Product of tensor representations
• Conjugate representations of are given by completing every column to be of length , and rotating the added shape 90°.

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Footnotes

1. In the sense that all such that all such subspaces are generated. ↩