1. 18.757 Homework 3 solutions
1. Recall from the first problem set the space
H k (Cn )
of degree k harmonic polynomials in n variables, carrying a
representation of the orthogonal group O(n). Prove that the
restriction of this representation to O(n − 1) decomposes as a
direct sum
k
n
H (C )|O(n−1) '
k
M
H j (Cn−1 ).
j=0
(Possible hint: try to prove some statement about all polynomials first.)
The statement is not quite correct: for n = 1, one needs to define
(
C, j = 0
H j (C0 ) =
.
0, j > 0
Then (for n = 1) the statement in the problem is true only for j 6 1,
which are the cases when H j (C1 ) 6= 0.
We proceed with problems 1 and 2 together by induction on n; then,
for fixed n, by induction on k. For n = 1, the corrected assertion of
the problem is very simple, and I’ll skip it. So assume n > 2.
Writing S k (Cn ) for the space of homogeneous polynomials of degree
k in n variables, the statement the hint suggests is
(1)
k
n
S (C )|O(n−1) '
k
M
S j (Cn−1 )
(n > 1).
j=0
This is more or less obvious; the map from right to left is
(p0 , p1 , p2 , . . . , pk ) 7→ p0 + xn p1 + x2n p2 + · · · + xkn pk .
(Here pj is a polynomial homogeneous of degree j in the variables
x1 , . . . , xn−1 .) This isomorphism in fact respects the action not just of
O(n) but of the group GL(n−1) of invertible (n−1)×(n−1) matrices.
We also know from the first problem set that
(2)
S k (Cn ) ' H k (Cn ) ⊕ H k−2 (Cn ) ⊕ H k−4 (Cn ) ⊕ · · ·
1
(n > 1).
2
Plugging (2) into (1) gives
(3)
H k (Cn )|O(n−1)
k n−1
' H (C )
⊕ H k−1 (Cn−1 )
⊕ H k−2 (Cn−1 )
⊕H k−2 (Cn )|O(n−1) ⊕H k−4 (Cn )|O(n−1) · · ·
⊕H k−2 (Cn−1 )
⊕H k−4 (Cn−1 ) · · ·
⊕H k−3 (Cn−1 )
⊕H k−5 (Cn−1 ) · · ·
⊕H k−4 (Cn−1 )
⊕H k−6 (Cn−1 ) · · ·
..
.
(Here I have used (1) for n − 1, so I need n > 2.) By the inductive
hypothesis for Problem 2, all the representations appearing on the right
in this big formula are irreducible representations of O(n − 1). It’s an
easy consequence of Schur’s lemma that two direct sums of irreducible
representations are isomorphic if and only if each irreducible appears
with the same multiplicity in both direct sum decompositions. We may
therefore cancel terms that are known by induction to be equal. We
know by the inductive hypothesis on k that
H k−2 (Cn )|O(n−1) ' H k−2 (Cn−1 ) ⊕ H k−3 (Cn−1 ⊕ · · · .
That is, the terms in (3) in the “second column” on the right cancel
the second term on the left. The same argument applies to each later
column. After we cancel all these known equalities, we conclude that
the first term on the left is isomorphic to the sum of the first column
on the right, which is what we wanted to show.
2. Prove that H k (Cn ) is an irreducible representation of
O(n). (Hint: the Frobenius reciprocity result proved in class
can be helpful.)
This is wrong: the correct statement is that H k (Cn ) is irreducible
or zero; and that zero occurs only if n = 0 and k > 0, or n = 1
and k > 2. In order to use this result to help with Problem 1, we
also need to prove the additional statement that the nonzero H k are
non-isomorphic representations of O(n).
Here is a proof. If n = 0, then the statements are obvious: the group
O(0) is trivial, the representation H 0 = S 0 is trivial, and all the higher
S k are zero. (For n = 0 the results of the first problem set about
functions on the sphere are not available, but these representationtheoretic statements nevertheless make sense.)
So suppose n > 1. We will show first that H k (Cn ) is irreducible or
zero. If n = 1, H k (C1 ) has dimension 1 or zero, so the statement is
clear. So assume n > 2. In order for Frobenius reciprocity to help,
we need to make use of the fact that O(n) acts transitively on S n−1
(by Witt’s theorem, for example), and that the stabilizer of the last
3
coordinate vector en ∈ S n−1 is O(n−1). From these two facts it follows
that
S n−1 ' O(n)/O(n − 1),
and therefore (Frobenius reciprocity) for any representation (π, Vπ ) of
O(n),
(4)
HomO(n) (Vπ , {functions on S n−1 }) ' HomO(n−1) (Vπ |O(n−1) , C).
We know by induction that the Problem 1 decomposition of H k (Cn )
is into inequivalent pieces which are irreducible or zero. If (π, Vπ ) is
any subrepresentation of H k (Cn ), it follows that
M
(5)
Vπ |O(n−1) '
H j (Cn−1 ),
j∈J
with J ⊂ {0, 1, . . . , k}. Schur’s lemma says that for an irreducible
representation (τ, Eτ ) of H, we have
(
C, τ ' trivial
(6)
HomH (Eτ , C) =
.
0, τ 6' trivial
Plugging (6) into (5), we conclude that
(
C, 0 ∈ J
HomO(n−1) (Vπ , C) =
.
0, 0 ∈
/ J.
Plugging into (4) gives
(
C, 0 ∈ J
HomO(n) (Vπ , {functions on S n−1 }) '
.
0, 0 ∈
/ J.
We got this conclusion for any subrepresentation of H k . But obviously
any nonzero subrepresentation of H k is contained in functions on S n−1 ,
so this space of homomorphisms has dimension at least one. We have
therefore shown that
any O(n)-subrepresentation of H k (Cn ) must contain the
one-dimensional subspace of O(n − 1)-invariant vectors.
It follows in turn that
H k (Cn ) has a unique irr O(n) subrep, which is
generated by the one-diml subspace of O(n − 1)-invt vectors.
When I assigned the problem, I thought that one could use the complete
reducibility of H k under O(n − 1) (known by induction) to conclude
the complete reducibility under O(n). But I do not see how to do that,
unfortunately. All I know how to do is use
4
Proposition 1.1. Any finite-dimensional complex continuous representation of a compact group G is a direct sum of irreducible representations.
We will prove this in class in a few days. Combined with the statement above about H k , it follows that H k is irreducible.
Finally, we need (for the argument I gave for Problem 1) to show
that the nonzero H k are inequivalent. For n = 1 this is easy to check:
H 0 is trivial, and H 1 is not. For n = 2, one can look at the restriction
to SO(2) ⊂ O(2): by the description of H k in the first problem set, it’s
clear that SO(2) acts on H k by the sum of the characters ξk and ξ−k ,
where
cos θ sin θ
ξm (
= eimθ .
− sin θ cos θ
Since these one-dimensional representations ξm are all different, it follows that the H k are all different.
For n > 2, if k > k 0 , then H k (Cn ) contains the irreducible H k (Cn−1 )
0
of O(n − 1), and H k (Cn ) does not. So they are inequivalent.
3. Give an example of a finite group G, a field k, and an
irreducible representation Vπ over k, with the property that
Dπ = k, and the composite of the maps
∼
∼
C(G, k)(π) ,→ C(G, k) Endk (Vπ ) −→ Vπ ⊗k Vπ∗ −→ C(G, k)(π)
is zero. Here the second map is the operator-valued Fourier
transform
X
f 7→
f (x)π(x),
x∈G
and the last is the inverse Fourier transform discussed in class.
Take G = Z/2Z and k any field of characteristic 2. There is only one
irreducible representation π, the trivial representation. For any group
and any field, C(G, k)(trivial) consists of the constant functions. But
if f is a constant function in our case,
X
f (x)π(x) = f (1) · |G| · π(1) = 2f (1) = 0.
x∈G