1. 18.757 Homework 4 solutions
1.1. Formal background. Suppose that (π, Vπ ) is a irreducible representation of
G over a field k, and that
(1a)
Dπ = HomG (Vπ , Vπ )
is the commuting algebra (endowed with the usual “reverse multiplication,” so that
Vπ is a right vector space over Dπ ). If W is any representation of G over k, the
multiplicity space and isotypic space are
(1b)
W π = HomG (Vπ , W ),
W (π) = largest sum of copies of Vπ in W .
π
Then W is a left vector space over Dπ (by the right action on Vπ ), and there is
an isomorphism
(1c)
Vπ ⊗Dπ W π → W (π),
v ⊗ A 7→ A(v).
Using these (easy) facts, it’s easy to make the following deductions.
(1) There is a one-to-one correspondence between G-invariant subspaces of
W (π) and (left) Dπ -subspaces of W π , S 7→ S π .
(2) There is a natural isomorphism
HomG (W (π), U (π)) ' HomDπ (W π , U π );
a map B on the left goes to the map on the right sending A ∈ W π to B ◦ A.
In class I discussed the corresponding definitions for quotients of W :
Wπ = HomG (W, Vπ ),
W [π] = lim (quotients isom to direct products of copies of Vπ ) .
←−
Then Wπ is a right vector space over Dπ (by the right action on Vπ ). The natural
map
W → HomDπ (Wπ , Vπ ),
w 7→ (T 7→ T (w))
(for T ∈ Wπ = HomG (W, Vπ )) gives rise to an isomorphism
W [π] → HomDπ (Wπ , Vπ ).
Notice that the right side is a direct product of copies of Vπ , indexed by a Dπ -basis
of Wπ .
In order to formulate a version of (1) above, we need
Definition 2. Suppose A and B are k-vector spaces. We define a topology on
Hom(A, B) by saying that a neighborhood of T1 ∈ Hom(A, B) is any subset U1
containing all linear transformations that agree with T1 on some finite-dimensional
subspace E ⊂ A. In particular, a neighborhood base at zero consists of the sets
VE = {T ∈ Hom(A, B) | T |E = 0},
with E a finite-dimensional subspace of A.
The main point in verifying that this defines a topology is that
VE1 ∩ · · · ∩ VEr = VE1 +···+Er ,
and E1 + · · · + Er is again a finite-dimensional subspace of A. Note also that if A
is finite-dimensional the topology is the discrete topology. You might like to check
that if A∗ = Hom(A, k) is topologized in this way, then the space of all continuous
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2
linear maps from A∗ to k (the latter having the discrete topology) is just A; that
is, that the (continuous) double dual of A is A.
Now the analogue of (1) above is
(10 ) There is a one-to-one correspondence between closed G-invariant subspaces
of W [π] ' HomDπ (Wπ , Vπ ) and (right) Dπ -subspaces of Wπ , S 7→ Sπ .
1.2. Homework problems for Thursday, March 14. For these problems, suppose Vπ is a finite-dimensional irreducible representation of G, and Dπ is as above.
Recall that you proved in the second problem set that Vπ∗ is an irreducible representation of G; you may assume that the commuting algebra is again Dπ , this time
acting on the left on Vπ∗ .
1. Prove: Vπ ⊗Dπ Vπ∗ is an irreducible representation of G × G.
Write GL × GR = G × G to distinguish the two factors. According to item
(1) above, any GL -invariant subspace of Vπ ⊗Dπ Vπ∗ corresponds to a Dπ -subspace
W ⊂ Vπ∗ . That is, any GL -invariant subspace S is of the form
S = Vπ ⊗Dπ W
(W ⊂ Vπ∗ Dπ -subspace.
This means in particular that W must be a k-subspace. Clearly such a subspace S
is GR -invariant if and only if W is a GR -invariant subspace of Vπ∗ . We saw in an
earlier homework that Vπ∗ is irreducible (as a representation over k, and a fortiori as
a representation over Dπ ); so the only candidates for W (corresponding to GL ×GR invariant subspaces) are 0 and Vπ∗ . That is, the only GL × GR -invariant subspaces
of Vπ ⊗Dπ Vπ∗ are 0 and the whole space. This is the irreducibility condition we
want.
2. Prove: HomDπ (Vπ , Vπ ) is an irreducible representation of G × G.
The proof is formally identical to that of the first problem. A point that requires a
little thought is how to formulate an analogue of (1) for quotients. For this problem
you could restrict attention to finite-dimensional quotients, in which case there is
an obvious version of (1). (I inserted above something fancier and more general.)
To change as few words as possible, you should make GL act on HomDπ (Vπ , Vπ ) by
acting on the range of maps, and GR by acting (with an inverse) on the domains.
3. Calculate the commuting algebra of one of these two irreducible
representations of G × G.
It’s easiest perhaps to talk about the first version, where the commuting algebra
is
Dπ⊗π∗ = HomGL ×GR (Vπ ⊗Dπ Vπ∗ , Vπ ⊗Dπ Vπ∗ ).
We first compute the larger algebra
E = HomGL (Vπ ⊗Dπ Vπ∗ , Vπ ⊗Dπ Vπ∗ ).
According to (2) above, this algebra may be naturally identified with
HomDπ (Vπ∗ , Vπ∗ ) ⊂ Homk (Vπ∗ , Vπ∗ ) = F.
Inside these algebras we are interested in the operators commuting with the action
of GR on Vπ∗ . To compute this, recall that for any finite-dimensional vector space
V , “transpose” is an algebra anti-isomorphism from Hom(V, V ) to Hom(V ∗ , V ∗ ).
It follows immediately that transpose is an algebra anti-isomorphism
HomGR (Vπ∗ , Vπ∗ ) ' HomGR (Vπ , Vπ ) =def Dπ .
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Recalling the strange definition of multiplication in Dπ , we see that the last equality
is also an algebra anti-isomorphism; so
HomGR (Vπ∗ , Vπ∗ ) ' Dπ
is an algebra isomorphism; the action on Vπ∗ is just the structure of left vector space
over Dπ .
The conclusion is that the Dπ -linear transformations of Vπ∗ commuting with the
action of GR are precisely the the vector space actions of the center of Dπ . That
is,
Dπ⊗π∗ ' Z(Dπ ) ⊃ k :
the commuting algebra we want is the center of the division algebra Dπ , which is
a field extension of k. It acts on the tensor product either by its right action on
the first factor, or by its left action on the second; these two actions are the same
because the tensor product is over Dπ .
4. Say as much as you can about the finite-dimensional irreducible
k-representations of a product group G × H, in terms of the irreducible
representations of G and of H.
You need the notion of a finite-dimensional representation Vπ of G over a fixed
division algebra E. This is a left vector space Vπ over E, endowed with an action
of G by E-linear transformations. One of the main subtleties is that scalar multiplication by an element of E need not be a linear transformation of Vπ , because it
need not commute with the action of E. The collection of linear transformations
HomE (Vπ , Vπ ) is a ring. We can (by choosing a basis) identify Vπ with the space E n
of row vectors, with scalar multiplication acting on the left. Then HomE (Vπ , Vπ )
consists of n × n matrices acting by right multiplication on these row vectors. The
action of constant matrices multiplies row vectors on the right rather than on the
left, and depends on the choice of basis; it is not the same as scalar multiplication on
Vπ . In particular, the ring HomE (Vπ , Vπ ) is not an algebra over E. The representation π is a homomorphism from G into the invertible elements of HomE (Vπ , Vπ ).
There is a version of Schur’s lemma, which says that if Vπ is irreducible, then
Dπ =def HomE,G (Vπ , Vπ )
is a division algebra. This time Dπ need not contain the base division algebra E,
because scalar multiplication by elements of E need not commute with E. If we
write K for the center of E (which is a field), then Dπ will contain K, and will be
division algebra over K. Here is what the Jacobson density theorem says.
Theorem 3. Suppose E is a division algebra with center K, and G is a group.
Write EG for the group algebra of G with coefficients in E (defined so that a
representation of G on a left E-vector space is the same thing as an EG-module).
If Vπ is an irreducible EG-module, then the commuting algebra Dπ is a division
algebra over K. The group representation defines an algebra homomorphism
EG → HomDπ (Vπ , Vπ ),
which has dense image. In particular, it is surjective if Vπ is finite-dimensional
over K.
Here is a description of representations of G × H.
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Proposition 4. Suppose G and H are groups and k is a field. Suppose (τ, Tτ ) is
a finite-dimensional irreducible representation of G × H over k.
(1) The restriction of τ to G is a (finite) direct sum of copies of a single irreducible k-representation ρ, Rρ of G. Write Dρ for the commuting algebra
of Rρ as in (??), so that
Tτ ' Rρ ⊗Dρ Tτρ ,
with
Tτρ = HomE,G (Rρ , Tτ )
a finite-dimensional (left) Dρ vector space (see (??)).
(2) The action of H on Tτ defines an irreducible Dρ -representation of H on
Tτρ .
Conversely, suppose that (ρ, Rρ ) is any finite-dimensional irreducible k-representation
of G, with commuting algebra Dρ , and that Mµ is any finite-dimensional irreducible
Dρ -representation of H. Then Rρ ⊗Dρ Mµ is a finite-dimensional irreducible krepresentation of G × H.
The proof is exactly parallel to the argument given for problem 1 above.
The proposition says that there is a one-to-one correspondence between finitedimensional irreducible k-representation of G×H and pairs ([(ρ, Vρ ), (µ, Mµ )], with
(1) ρ is a finite-dimensional irreducible k-representation of G, and
(2) µ a finite-dimensional irreducible Dρ -representation of H.
The representation of G × H is
Rρ ⊗Dρ Mµ ,
Of course this description is somewhat unsatisfactory because it is not symmetric
in G and H. One way to continue is to describe irreducible D-representations
(for any finite-dimensional division algebra D over k) in terms of irreducible krepresentations. Here is how that works.
Proposition 5. Suppose G is a group, k is a field, and E is a finite-dimensional
division algebra over k.
(1) Suppose (µ, Mµ ) is a finite-dimensional irreducible representation of G over
E. Then the restriction of µ to k (obtained by regarding Mµ as a k-vector
space) is a direct sum of copies of a single irreducible representation (σ, Sσ )
of G. Consequently
Mµ ' Sσ ⊗Dσ Mµσ ,
with Mµσ a finite-dimensional Dσ -module.
(2) The action of E on Mµ defines on Mµσ the structure of a finite-dimensional
E-module, commuting with the action of Dσ . In this way Mµσ becomes an
irreducible module for the tensor product algebra E ⊗k Dσ .
(3) Conversely, suppose M is any irreducible module for Dσ ⊗k E. Then Sσ ⊗Dσ
M is an irreducible representation of G over E.
This establishes a bijection between finite-dimensional irreducible representations of
G over E, and pairs ((σ, Sσ ), M ), with σ an irreducible representation of G over k,
an M an irreducible module for the tensor product algebra E ⊗k Dσ .
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Corollary 6. Suppose G and H are groups, and k is a field. There is a one-toone correspondence between irreducible representations of G × H over k and triples
((ρ, Rρ ), (σ, Sσ ), M ), with
(1) (ρ, Rρ ) is an irreducible representation of G;
(2) (σ, Sσ ) is an irreducible representation of H; and
(3) M is an irreducible module for the tensor product algebra Dρ ⊗k Dσ .
The representation of G × H is
[Rρ ⊗ Sσ ] ⊗Dρ ⊗Dσ M.
The tensor product algebra Dρ ⊗k Dσ is a direct sum of simple algebras Mm (E)
(with m a positive integer and E a finite-dimensional division algebra over k) in
characteristic zero. In any case such simple algebras are the only possible simple
quotients of the tensor product algebra. Each such quotient gives a simple module
Em.
The question of exactly how Dρ ⊗k Dσ decomposes is a question of number
theory, and I won’t say much more about it. Here are a few easy cases. Suppose
K is a degree n extension of k, with Galois group Sn . (That is, the smallest Galois
extension containing K has degree n!.) Adjoining to k two distinct roots of some
irreducible polynomial gives an extension E of degree n(n − 1). What happens is
K ⊗k K ' K ⊕ E;
the simple modules are therefore K and E. In general, if K1 and K2 are finite field
extensions of k, then K1 ⊗k K2 has as simple quotients all the field extensions E
of k generated by two subfields isomorphic to K1 and K2 .
Here is what happens for R:
C ⊗R C ' C ⊕ C
C ⊗R H ' M2 (C)
H ⊗R H ' M4 (R).
In the first case there are two simple modules, each isomorphic to C; the two actions
are
(z1 ⊗ z2 ) · x = z1 z2 x,
(z1 ⊗ z2 ) · x = z1 z 2 · y.
In the second case, there is single simple module, isomorphic to H ' C2 ; the action
is
(z ⊗ h) · x = zxh.
In the third case there is again a single simple module, isomorphic to H ' R4 ; the
action is
(h1 ⊗ h2 ) · x = h1 xh2 .