f = f ◦ i,
as in the diagram below:
V
i /
f
$❍
❍
❍
❍
❍
❍
❍
❍
❍
❍
Sym(V )
f
A
Remark: If E is finite-dimensional, recall the isomorphism, µ : Symn(E∗) −→ Sn(E, K),
defined as the linear extension of the map given by
µ(v∗1
· · ·
v∗n)(u1, . . . , un) =
v∗σ(1)(u1) · · · v∗σ(n)(un),
σ∈Sn
Now, we have also a multiplication operation, Symm(E∗)×Symn(E∗) −→ Symm+n(E∗). The
following question then arises:
Can we define a multiplication, Sm(E, K) × Sn(E, K) −→ Sm+n(E, K), directly on sym-
metric multilinear forms, so that the following diagram commutes:
Symm(E∗) × Symn(E∗)
/ Symm+n(E∗)
µ×µ
µ
Sm(E, K) × Sn(E, K)
·
/ Sm+n(E, K).
The answer is yes! The solution is to define this multiplication such that, for f ∈ Sm(E, K)
and g ∈ Sn(E, K),
(f · g)(u1, . . . , um+n) =
f (uσ(1), . . . , uσ(m))g(uσ(m+1), . . . , uσ(m+n)),
σ∈shuffle(m,n)
where shuffle(m, n) consists of all (m, n)-“shuffles”, that is, permutations, σ, of {1, . . . m+n},
such that σ(1) < · · · < σ(m) and σ(m + 1) < · · · < σ(m + n). We urge the reader to check
this fact.
Another useful canonical isomorphim (of K-algebras) is
Sym(E ⊕ F ) ∼
= Sym(E) ⊗ Sym(F ).
634
CHAPTER 23. TENSOR ALGEBRAS
23.11
Exterior Tensor Powers
We now consider alternating (also called skew-symmetric) multilinear maps and exterior
tensor powers (also called alternating tensor powers), denoted
n(E). In many respect,
alternating multilinear maps and exterior tensor powers can be treated much like symmetric
tensor powers except that the sign, sgn(σ), needs to be inserted in front of the formulae valid
for symmetric powers. Roughly speaking, we are now in the world of determinants rather
than in the world of permanents. However, there are also some fundamental differences, one
of which being that the exterior tensor power,
n(E), is the trivial vector space, (0), when
E is finite-dimensional and when n > dim(E). As in the case of symmetric tensor powers,
since we already have the tensor algebra, T (V ), we can proceed rather quickly. But first, let
us review some basic definitions and facts.
Definition 23.7. Let f : En → F be a multilinear map. We say that f alternating iff
f (u1, . . . , un) = 0 whenever ui = ui+1, for some i with 1 ≤ i ≤ n − 1, for all ui ∈ E,
that is, f (u1, . . . , un) = 0 whenever two adjacent arguments are identical. We say that f is
skew-symmetric (or anti-symmetric) iff
f (uσ(1), . . . , uσ(n)) = sgn(σ)f(u1, . . . , un),
for every permutation, σ ∈ Sn, and all ui ∈ E.
For n = 1, we agree that every linear map, f : E → F , is alternating. The vector
space of all multilinear alternating maps, f : En → F , is denoted Altn(E; F ). Note that
Alt1(E; F ) = Hom(E, F ). The following basic proposition shows the relationship between
alternation and skew-symmetry.
Proposition 23.15. Let f : En → F be a multilinear map. If f is alternating, then the
following properties hold:
(1) For all i, with 1 ≤ i ≤ n − 1,
f (. . . , ui, ui+1, . . .) = −f(. . . , ui+1, ui, . . .).
(2) For every permutation, σ ∈ Sn,
f (uσ(1), . . . , uσ(n)) = sgn(σ)f(u1, . . . , un).
(3) For all i, j, with 1 ≤ i < j ≤ n,
f (. . . , ui, . . . uj, . . .) = 0 whenever ui = uj.
Moreover, if our field, K, has characteristic different from 2, then every skew-symmetric
multilinear map is alternating.
23.11. EXTERIOR TENSOR POWERS
635
Proof. (i) By multilinearity applied twice, we have
f (. . . , ui + ui+1, ui + ui+1, . . .) = f (. . . , ui, ui, . . .) + f (. . . , ui, ui+1, . . .)
+ f (. . . , ui+1, ui, . . .) + f (. . . , ui+1, ui+1, . . .).
Since f is alternating, we get
0 = f (. . . , ui, ui+1, . . .) + f (. . . , ui+1, ui, . . .),
that is, f (. . . , ui, ui+1, . . .) = −f(. . . , ui+1, ui, . . .).
(ii) Clearly, the symmetric group, Sn, acts on Altn(E; F ) on the left, via
σ · f(u1, . . . , un) = f(uσ(1), . . . , uσ(n)).
Consequently, as Sn is generated by the transpositions (permutations that swap exactly two
elements), since for a transposition, (ii) is simply (i), we deduce (ii) by induction on the
number of transpositions in σ.
(iii) There is a permutation, σ, that sends ui and uj respectively to u1 and u2. As f is
alternating,
f (uσ(1), . . . , uσ(n)) = 0.
However, by (ii),
f (u1, . . . , un) = sgn(σ)f (uσ(1), . . . , uσ(n)) = 0.
Now, when f is skew-symmetric, if σ is the transposition swapping ui and ui+1 = ui, as
sgn(σ) = −1, we get
f (. . . , ui, ui, . . .) = −f(. . . , ui, ui, . . .),
so that
2f (. . . , ui, ui, . . .) = 0,
and in every characteristic except 2, we conclude that f (. . . , ui, ui, . . .) = 0, namely, f is
alternating.
Proposition 23.15 shows that in every characteristic except 2, alternating and skew-
symmetric multilinear maps are identical. Using Proposition 23.15 we easily deduce the
following crucial fact:
Proposition 23.16. Let f : En → F be an alternating multilinear map. For any families of
vectors, (u1, . . . , un) and (v1, . . . , vn), with ui, vi ∈ E, if
n
vj =
aijui,
1 ≤ j ≤ n,
i=1
then
f (v1, . . . , vn) =
sgn(σ) aσ(1),1 · · · aσ(n),n f(u1, . . . , un) = det(A)f(u1, . . . , un),
σ∈Sn
where A is the n × n matrix, A = (aij).
636
CHAPTER 23. TENSOR ALGEBRAS
Proof. Use property (ii) of Proposition 23.15.
We are now ready to define and construct exterior tensor powers.
Definition 23.8. An n-th exterior tensor power of a vector space, E, where n ≥ 1, is a
vector space, A, together with an alternating multilinear map, ϕ : En → A, such that, for
every vector space, F , and for every alternating multilinear map, f : En → F , there is a
unique linear map, f∧ : A → F , with
f (u1, . . . , un) = f∧(ϕ(u1, . . . , un)),
for all u1, . . . , un ∈ E, or for short
f = f∧ ◦ ϕ.
Equivalently, there is a unique linear map f∧ such that the following diagram commutes:
ϕ
En
/ A
f∧
f
!❈
❈
❈
❈
❈
❈
❈
❈
F
First, we show that any two n-th exterior tensor powers (A1, ϕ1) and (A2, ϕ2) for E, are
isomorphic.
Proposition 23.17. Given any two n-th exterior tensor powers (A1, ϕ1) and (A2, ϕ2) for
E, there is an isomorphism h : A1 → A2 such that
ϕ2 = h ◦ ϕ1.
Proof. Replace tensor product by n exterior tensor power in the proof of Proposition 23.4.
We now give a construction that produces an n-th exterior tensor power of a vector space
E.
Theorem 23.18. Given a vector space E, an n-th exterior tensor power ( n(E), ϕ) for E
can be constructed (n ≥ 1). Furthermore, denoting ϕ(u1, . . . , un) as u1 ∧· · ·∧un, the exterior
tensor power
n(E) is generated by the vectors u1 ∧ · · · ∧ un, where u1, . . . , un ∈ E, and for
every alternating multilinear map f : En → F , the unique linear map f∧ :
n(E) → F such
that f = f∧ ◦ ϕ, is defined by
f∧(u1 ∧ · · · ∧ un) = f(u1, . . . , un),
on the generators u1 ∧ · · · ∧ un of
n(E).
23.11. EXTERIOR TENSOR POWERS
637
Proof sketch. We can give a quick proof using the tensor algebra, T (E). let Ia be the two-
sided ideal of T (E) generated by all tensors of the form u ⊗ u ∈ E⊗2. Then, let
n
(E) = E⊗n/(Ia ∩ E⊗n)
and let π be the projection, π : E⊗n → n(E). If we let u1 ∧ · · · ∧ un = π(u1 ⊗ · · · ⊗ un), it
is easy to check that ( n(E), ∧) satisfies the conditions of Theorem 23.18.
Remark: We can also define
n
(E) = T (E)/Ia =
(E),
n≥0
the exterior algebra of E. This is the skew-symmetric counterpart of Sym(E) and we will
study it a little later.
For simplicity of notation, we may write
n E for n(E). We also abbreviate “exterior
tensor power” as “exterior power”. Clearly,
1(E) ∼
= E and it is convenient to set
0(E) =
K.
The fact that the map ϕ : En → n(E) is alternating and multinear, can also be expressed
as follows:
u1 ∧ · · · ∧ (ui + vi) ∧ · · · ∧ un = (u1 ∧ · · · ∧ ui ∧ · · · ∧ un)
+ (u1 ∧ · · · ∧ vi ∧ · · · ∧ un),
u1 ∧ · · · ∧ (λui) ∧ · · · ∧ un = λ(u1 ∧ · · · ∧ ui ∧ · · · ∧ un),
uσ(1) ∧ · · · ∧ uσ(n) = sgn(σ) u1 ∧ · · · ∧ un,
for all σ ∈ Sn.
Theorem 23.18 yields a canonical isomorphism
n
Hom(
(E), F ) ∼
= Altn(E; F ),
between the vector space of linear maps Hom( n(E), F ), and the vector space of alternating
multilinear maps Altn(E; F ), via the linear map − ◦ ϕ defined by
h → h ◦ ϕ,
where h ∈ Hom( n(E), F ). In particular, when F = K, we get a canonical isomorphism
n
∗
(E)
∼
= Altn(E; K).
638
CHAPTER 23. TENSOR ALGEBRAS
Tensors α ∈
n(E) are called alternating n-tensors or alternating tensors of degree n
and we write deg(α) = n. Tensors of the form u1 ∧ · · · ∧ un, where ui ∈ E, are called simple
(or decomposable) alternating n-tensors. Those alternating n-tensors that are not simple are
often called compound alternating n-tensors. Simple tensors u1 ∧ · · · ∧ un ∈ n(E) are also
called n-vectors and tensors in
n(E∗) are often called (alternating) n-forms.
Given two linear maps f : E → E and g : E → E , we can define h: E × E → 2(E ) by
h(u, v) = f (u) ∧ g(v).
It is immediately verified that h is alternating bilinear, and thus, it induces a unique linear
map
2
2
f ∧ g :
(E) →
(E ),
such that
(f ∧ g)(u ∧ v) = f(u) ∧ g(u).
If we also have linear maps f : E → E and g : E → E , we can easily verify that
(f ◦ f) ∧ (g ◦ g) = (f ∧ g ) ◦ (f ∧ g).
The generalization to the alternating product f1∧· · ·∧fn of n ≥ 3 linear maps fi : E → E
is immediate, and left to the reader.
23.12
Bases of Exterior Powers
Let E be any vector space. For any basis, (ui)i∈Σ, for E, we assume that some total ordering,
≤, on Σ, has been chosen. Call the pair ((ui)i∈Σ, ≤) an ordered basis. Then, for any nonempty
finite subset, I ⊆ Σ, let
uI = ui ∧ · · · ∧ u ,
1
im
where I = {i1, . . . , im}, with i1 < · · · < im.
Since
n(E) is generated by the tensors of the form v1 ∧ · · · ∧ vn, with vi ∈ E, in view of
skew-symmetry, it is clear that the tensors uI, with |I| = n, generate n(E). Actually, they
form a basis.
Proposition 23.19. Given any vector space, E, if E has finite dimension, d = dim(E),
then for all n > d, the exterior power
n(E) is trivial, that is
n(E) = (0). Otherwise,
for every ordered basis, ((ui)i∈Σ, ≤), the family, (uI), is basis of
n(E), where I ranges over
finite nonempty subsets of Σ of size |I| = n.
23.12. BASES OF EXTERIOR POWERS
639
Proof. First, assume that E has finite dimension, d = dim(E) and that n > d. We know
that
n(E) is generated by the tensors of the form v1 ∧ · · · ∧ vn, with vi ∈ E. If u1, . . . , ud
is a basis of E, as every vi is a linear combination of the uj, when we expand v1 ∧ · · · ∧ vn
using multilinearity, we get a linear combination of the form
v1 ∧ · · · ∧ vn =
λ(j
∧ · · · ∧ u ,
1,...,jn) uj1
jn
(j1,...,jn)
where each (j1, . . . , jn) is some sequence of integers jk ∈ {1, . . . , d}. As n > d, each sequence
(j1, . . . , jn) must contain two identical elements. By alternation, uj ∧ · · · ∧ u = 0 and so,
1
jn
v1 ∧ · · · ∧ vn = 0. It follows that
n(E) = (0).
Now, assume that either dim(E) = d and that n ≤ d or that E is infinite dimensional.
The argument below shows that the uI are nonzero and linearly independent. As usual, let
u∗i ∈ E∗ be the linear form given by
u∗i(uj) = δij.
For any nonempty subset, I = {i1, . . . , in} ⊆ Σ, with i1 < · · · < in, let lI be the map given
by
lI(v1, . . . , vn) = det(u∗i (vk)),
j
for all vk ∈ E. As lI is alternating multilinear, it induces a linear map, LI :
n(E) → K.
Observe that for any nonempty finite subset, J ⊆ Σ, with |J| = n, we have
1 if I = J
LI(uJ) =
0 if I = J.
Note that when dim(E) = d and n ≤ d, the forms u∗i , . . . , u∗ are all distinct so, the above
1
in
does hold. Since LI(uI) = 1, we conclude that uI = 0. Now, if we have a linear combination,
λIuI = 0,
I
where the above sum is finite and involves nonempty finite subset, I ⊆ Σ, with |I| = n, for
every such I, when we apply LI we get
λI = 0,
proving linear independence.
As a corollary, if E is finite dimensional, say dim(E) = d and if 1 ≤ n ≤ d, then we have
n
n
dim(
(E)) =
d
640
CHAPTER 23. TENSOR ALGEBRAS
and if n > d, then dim( n(E)) = 0.
Remark: When n = 0, if we set u∅ = 1, then (u∅) = (1) is a basis of
0(V ) = K.
It follows from Proposition 23.19 that the family, (uI)I, where I ⊆ Σ ranges over finite
subsets of Σ is a basis of
(V ) =
n(V ).
n≥0
As a corollary of Proposition 23.19 we obtain the following useful criterion for linear
independence:
Proposition 23.20. For any vector space, E, the vectors, u1, . . . , un ∈ E, are linearly
independent iff u1 ∧ · · · ∧ un = 0.
Proof. If u1 ∧ · · · ∧ un = 0, then u1, . . . , un must be linearly independent. Otherwise, some
ui would be a linear combination of the other uj’s (with j = i) and then, as in the proof
of Proposition 23.19, u1 ∧ · · · ∧ un would be a linear combination of wedges in which two
vectors are identical and thus, zero.
Conversely, assume that u1, . . . , un are linearly independent. Then, we have the linear
forms, u∗i ∈ E∗, such that
u∗i(uj) = δi,j
1 ≤ i, j ≤ n.
As in the proof of Proposition 23.19, we have a linear map, Lu
:
n(E) → K, given by
1,...,un
Lu
(v
1,...,un
1 ∧ · · · ∧ vn) = det(u∗j(vi)),
for all v1 ∧ · · · ∧ vn ∈ n(E). As,
Lu
(u
1,...,un
1 ∧ · · · ∧ un) = 1,
we conclude that u1 ∧ · · · ∧ un = 0.
Proposition 23.20 shows that, geometrically, every nonzero wedge, u1 ∧ · · · ∧ un, corre-
sponds to some oriented version of an n-dimensional subspace of E.
23.13
Some Useful Isomorphisms for Exterior Powers
We can show the following property of the exterior tensor product, using the proof technique
of Proposition 23.7:
n
n
k
n−k
(E ⊕ F ) ∼
=
(E) ⊗
(F ).
k=0
23.14. DUALITY FOR EXTERIOR POWERS
641
23.14
Duality for Exterior Powers
In this section, all vector spaces are assumed to have finite dimension. We define a nonde-
generate pairing,
n(E∗) × n(E) −→ K, as follows: Consider the multilinear map,
(E∗)n × En −→ K,
given by
(v∗1, . . . , v∗n, u1, . . . , un) →
sgn(σ) v∗σ(1)(u1) · · · v∗σ(n)(un) = det(v∗j(ui)).
σ∈Sn
It is easily checked that this expression is alternating w.r.t. the ui’s and also w.r.t. the v∗j.
For any fixed (v∗1, . . . , v∗n) ∈ (E∗)n, we get an alternating multinear map,
lv∗,...,v∗ : (u1, . . . , un) → det(v∗
1
n
j (ui)),
from En to K. By the argument used in the symmetric case, we get a bilinear map,
n
n
(E∗) ×
(E) −→ K.
Now, this pairing in nondegenerate. This can be done using bases and we leave it as an
exercise to the reader. Therefore, we get a canonical isomorphism,
n
n
(
(E))∗ ∼
=
(E∗).
Since we also have a canonical isomorphism
n
(
(E))∗ ∼
= Altn(E; K),
we get a canonical isomorphism
n
(E∗) ∼
= Altn(E; K)
which allows us to interpret alternating tensors over E∗ as alternating multilinear maps.
The isomorphism, µ :
n(E∗) ∼
= Altn(E; K), discussed above can be described explicity
as the linear extension of the map given by
µ(v∗1 ∧ · · · ∧ v∗n)(u1, . . . , un) = det(v∗j(ui)).
Remark: Variants of our isomorphism, µ, are found in the literature. For example, there
is a version, µ , where
1
µ =
µ,
n!
642
CHAPTER 23. TENSOR ALGEBRAS
with the factor 1 added in front of the determinant. Each version has its its own merits
n!
and inconvenients. Morita [80] uses µ because it is more convenient than µ when dealing
with characteristic classes. On the other hand, when using µ , some extra factor is needed
in defining the wedge operation of alternating multilinear forms (see Section 23.15) and for
exterior differentiation. The version µ is the one adopted by Warner [109], Knapp [62],
Fulton and Harris [40] and Cartan [18, 19].
If f : E → F is any linear map, by transposition we get a linear map, f : F ∗ → E∗,
given by
f (v∗) = v∗ ◦ f,
v∗ ∈ F ∗.
Consequently, we have
f (v∗)(u) = v∗(f (u)),
for all u ∈ E and all v∗ ∈ F ∗.
For any p ≥ 1, the map,
(u1, . . . , up) → f(u1) ∧ · · · ∧ f(up),
from En to
p F is multilinear alternating, so it induces a linear map, p f :
p E → p F,
defined on generators by
p
f (u1 ∧ · · · ∧ up) = f(u1) ∧ · · · ∧ f(up).
Combining
p and duality, we get a linear map, p f :
p F ∗ → p E∗, defined on gener-
ators by
p
f
(v∗1 ∧ · · · ∧ v∗p) = f (v∗1) ∧ · · · ∧ f (v∗p).
Proposition 23.21. If f : E → F is any linear map between two finite-dimensional vector
spaces, E and F , then
p
p
µ
f
(ω) (u1, . . . , up) = µ(ω)(f (u1), . . . , f (up)),
ω ∈
F ∗, u1, . . . , up ∈ E.
Proof. It is enough to prove the formula on generators. By definition of µ, we have
p
µ
f
(v∗1 ∧ · · · ∧ v∗p) (u1, . . . , up) = µ(f (v∗1) ∧ · · · ∧ f (v∗p))(u1, . . . , up)
= det(f (v∗j)(ui))
= det(v∗j(f(ui)))
= µ(v∗1 ∧ · · · ∧ v∗p)(f(u1), . . . , f(up)),
as claimed.
23.15. EXTERIOR ALGEBRAS
643
The map
p f is often denoted f∗, although this is an ambiguous notation since p is
dropped. Proposition 23.21 gives us the behavior of f ∗ under the identification of
p E∗ and
Altp(E; K) via the isomorphism µ.
As in the case of symmetric powers, the map from En to
n(E) given by (u1, . . . , un) →
u1 ∧ · · · ∧ un yields a surjection, π : E⊗n → n(E). Now, this map has some section so there
is some injection, ι :
n(E) → E⊗n, with π ◦ ι = id. If our field, K, has characteristic 0,
then there is a special section having a natural definition involving an antisymmetrization
process.
Recall that we have a left action of the symmetric group, Sn, on E⊗n. The tensors,
z ∈ E⊗n, such that
σ · z = sgn(σ) z, for all σ ∈ Sn
are called antisymmetrized tensors. We define the map, ι : En → E⊗n, by
1
ι(u1, . . . , un) =
sgn(σ) u
n!
σ(1) ⊗ · · · ⊗ uσ(n).
σ∈Sn
As the right hand side is clearly an alternating map, we get a linear map, ι :
n(E) → E⊗n.
Clearly, ι( n(E)) is the set of antisymmetrized tensors in E⊗n. If we consider the map,
A = ι ◦ π : E⊗n −→ E⊗n, it is easy to check that A ◦ A = A. Therefore, A is a projection
and by linear algebra, we know that
n
E⊗n = A(E⊗n) ⊕ Ker A = ι( (A)) ⊕ Ker A.
It turns out that Ker A = E⊗n ∩ Ia = Ker π, where Ia is the two-sided ideal of T (E)
generated by all tensors of the form u ⊗ u ∈ E⊗2 (for example, see Knapp [62], Appendix
A). Therefore, ι is injective,
n
n
E⊗n = ι(
(E)) ⊕ E⊗n ∩ I = ι( (E)) ⊕ Ker π,
and the exterior tensor power,
n(E), is naturally embedded into E⊗n.
23.15
Exterior Algebras
As in the case of symmetric tensors, we can pack together all the exterior powers,
n(V ),
into an algebra,
m
(V ) =
(V ),
m≥0
called the exterior algebra (or Grassmann algebra) of V . We mimic the procedure used
for symmetric powers. If Ia is the two-sided ideal generated by all tensors of the form
u ⊗ u ∈ V ⊗2, we set
•
(V ) = T (V )/Ia.
644
CHAPTER 23. TENSOR ALGEBRAS
Then,
•(V ) automatically inherits a multiplication operation, called wedge product, and
since T (V ) is graded, that is,
T (V ) =
V ⊗m,
m≥0
we have
•
(V ) =
V ⊗m/(Ia ∩ V ⊗m).
m≥0
However, it is easy to check that
m
(V ) ∼
= V ⊗m/(Ia ∩ V ⊗m),
so
•
(V ) ∼
=
(V ).
When V has finite dimension, d, we actually have a finite coproduct
d
m
(V ) =
(V ),
m=0
and si