Exterior-power representation

Core idea

Let $\rho:G\to\operatorname{GL}(V)$ be a representation of a Lie group and let $k\geq0$ . Its $k$ -th exterior-power representation is the representation on the degree- $k$ part of the exterior algebra given by

(\Lambda^k\rho)(g)(v_1\wedge\cdots\wedge v_k) =\rho(g)v_1\wedge\cdots\wedge\rho(g)v_k.

If $d\rho:\mathfrak g\to\mathfrak{gl}(V)$ is a Lie-algebra representation, the induced action is

(\Lambda^k d\rho)(X)(v_1\wedge\cdots\wedge v_k) =\sum_{j=1}^k v_1\wedge\cdots\wedge d\rho(X)v_j\wedge\cdots\wedge v_k.

Thus the Lie algebra acts as a degree-zero derivation, while a group element acts by an algebra automorphism.

Basic cases

The zeroth exterior power $\Lambda^0V$ is the one-dimensional trivial representation, and $\Lambda^1V=V$ . If $V$ has dimension $n$ , then $\Lambda^nV$ is one-dimensional and carries the determinant character:

\Lambda^n\rho(g)=\det(\rho(g)).

Exterior powers above degree $n$ vanish.

For the defining representation of $\mathfrak{sl}_n(\mathbb C)$ , the representations $\Lambda^k\mathbb C^n$ , $1\leq k\leq n-1$ , are the fundamental representations.

Weights

Suppose $V$ has a basis of weight vectors $v_1,\ldots,v_n$ with weights $\lambda_1,\ldots,\lambda_n$ . Then a nonzero wedge

v_{i_1}\wedge\cdots\wedge v_{i_k}

has weight $\lambda_{i_1}+\cdots+\lambda_{i_k}$ . Antisymmetry forces repeated basis vectors to vanish, which is the representation-theoretic difference between exterior and tensor powers.

Group versus Lie algebra

Differentiating the group formula gives the displayed derivation formula. Conversely, an exterior power of a Lie-algebra representation integrates whenever the original representation does, but the global representation still belongs naturally to the same covering group as the original. Passing to a quotient group requires its kernel to act trivially on $\Lambda^kV$ , which can occur even when it does not act trivially on $V$ .

References

William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§6 and 15. Publisher record.
Roe Goodman and Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, 2009, §3.1. Publisher record.