corresponding to the linear map
ei 1 ⊗···⊗eij→α 1 (ei 1 )···αj(eij)Antisymmetric bilinear forms lie in Λ^2 (V∗)⊂V∗⊗V∗and correspond to
antisymmetric matrices. A multiplication (called the “wedge product”) can be
defined onV∗that takes values in Λ^2 (V∗) by
(α 1 ,α 2 )∈V∗×V∗→α 1 ∧α 2 =1
2
(α 1 ⊗α 2 −α 2 ⊗α 1 )∈Λ^2 (V∗) (9.5)This multiplication can be extended to a product on the space
Λ∗(V∗) =⊕nΛn(V∗)(called the space of antisymmetric multilinear forms) by defining
(α 1 ⊗···⊗αj)∧(αj+1⊗···⊗αn) =P−(α 1 ⊗···⊗αn)≡1
n!∑
σ∈Sn(−1)|σ|ασ(1)⊗···⊗ασ(n) (9.6)This can be used to get a product on the space of antisymmetric multilinear
forms of different degrees, giving something in many ways analogous to the
algebra of polynomials (although without a notion of evaluation at a pointv).
This plays a role in the description of fermions and will be considered in more
detail in chapter 30. Much like in the symmetric case, there is an isomorphism
between Λ∗(V∗) and (Λ∗(V))∗.
9.7 For further reading
For more about the tensor product and tensor product of representations, see
section 6 of [95], or appendix B of [84]. Almost every quantum mechanics text-
book will contain an extensive discussion of the Clebsch-Gordan decomposition
for the tensor product of two irreducibleSU(2) representations.
A complete discussion of bilinear forms, together with the algebra of sym-
metric and antisymmetric multilinear forms, can be found in [36].