Fast Fourier Analysis for Abelian Group Extensions

45 Pages·2007·0.3 MB·English

Fast Fourier Analysis for Abelian Group Extensions
(cid:3)
Daniel Rockmore
Harvard University
Department of Mathematics
July 11, 1995
Abstract
Let G be a (cid:12)nite group and f any complex-valued function de(cid:12)ned on
G and (cid:26) an irreducible complex matrix representation of G. The Fourier
P
transform of f at (cid:26) is de(cid:12)ned to be the matrix s2Gf(s)(cid:26)(s). The Fourier
transformsoff atalltheirreducible representationsofGdeterminef viathe
P
1 b (cid:0)1
Fourier inversion formula f(s) = jGj (cid:26)d(cid:26)trace(f((cid:26))(cid:26)(s )): Direct compu-
2
tation of all Fourier transformsof f involves on the order of jGj operations
as does direct computation of Fourier inversion. Here fast algorithms are
obtained forbothoperationsin thecase in which Gcontains somenontrivial
normalsubgroupK suchthatG=K isabelian. Consequently, fastalgorithms
for computing convolutions on G in this situation are also determined. Un-
der the simpli(cid:12)ying assumption of exponent for matrix multiplication equal
to 2 (it is 2.38 as of this writing) it is shown that the number of operations
jGj jGj
needed to compute all Fourier transforms on G is O( T(K)+jGjlog( ))
jKj jKj
where T(K) is the number of operations needed to compute Fourier trans-
forms on K. An analogous result is obtained for Fourier inversion and more
careful estimates made for arbitrary exponents of matrix multiplication. In
particular, in the case in which G is metabelian (an abelian extension of an
abelian subgroup) the assumptions hold and the (cid:12)rst large class of noncom-
mutativegroupsisobtainedforwhichFourierinversionandthecomputation
of all Fourier transforms may be performed in O(jGjlog(jGj)) operations.
(cid:3)
Supported by IBM and NSF Graduate Fellowships.
0
1. Introduction
Let G be a (cid:12)nite group, f a complex-valued function on G and (cid:26) any
complex irreducible matrix representation of G. Then the Fourier trans-
form of f with respect to (cid:26) is the matrix
X
b
f((cid:26))= f(s)(cid:26)(s):
s2G
The Fourier transformsat all the irreducible representations of G deter-
mine f via the Fourier inversion formula
1 X b (cid:0)1
f(s)= d(cid:26)trace(f((cid:26))(cid:26)(s )
jGj
(cid:26)
where the sum is over all irreducible representations (cid:26) of G.
LetT(G)denotethenumberofoperationsneeded tocomputeall Fourier
transforms on G and I(G) the number needed to perform Fourier inversion.
2
Direct computation of all Fourier transformswould yield T(G)=jGj . Sim-
2
ilarly, naive computation of Fourier inversion gives I(G)=jGj . For groups
of large order this cost is prohibitive. Fast algorithms must be derived.
Motivated partly by their manifold applications, much is known for
the case in which G is abelian. A large class of algorithms, collectively
known as the \Fast Fourier Transform" (\FFT") have been developed. It
is now a well known result that for abelian groups both T(G) and I(G) are
O(jGjlog(jGj)). Over the years these fast algorithms have yielded a diver-
sity of important applications. Aho, Hopcroft, and Ullman give a thorough
treatment of the basic algorithm and its uses ([AHU], Chapters 7 and 8).
The problem for general (cid:12)nite groups has only recently begun to attract
attention. In the most general setting, in joint work with Diaconis ([DR]),
prompted by the study of spectral analysis for (cid:12)nite groups ([D1], [D2]),
an algorithm for e(cid:14)cient computation of Fourier transforms for arbitrary
groups has already been presented. There it is shown that great savings
may be achieved by iterating a simple recurrence which allows the Fourier
transforms of f to be written in terms of Fourier transforms on a subgroup
of G. This is in fact the idea at the heart of many of the fast algorithms in
the abelian case.
The e(cid:14)cacyof themethods in [DR]depends stronglyon the exponent of
matrixmultiplication andtheexistence ofrepresentationsofGwhich\split"
when restricted down agiven towerofsubgroups of G. In particular, forthe
1
symmetric group Sn this situation exists and practical implementations for
analyzing ranked data are possible and useful ([R1]).
Fourier analysis is also useful in the study of random walks on groups.
([D2], [DS]). A probability measure de(cid:12)ned on the group gives rise to a
random walk on the group. Explicit knowledge of the matrices for the
Fourier transforms determined by the measure are useful in studying the
rate at which the walk becomes random.
A problem left open in much of the work on fast algorithms for Fourier
analysis is the e(cid:14)cient computation of Fourier inversion. In the case of
b
abelian groups the group structure on the dual G allows the techniques for
computing Fourier transforms to be applied. In general there is no such
structure and new methods must be developed.
In this paper the case of solvable groups is considered, and more gener-
ally, thecaseofabelian groupextensions. Here,modi(cid:12)cations ofthe ideas in
[DR] combined with the use of induced and projective representations yield
fastalgorithmsforbothFourierinversionandcomputingFouriertransforms.
One main result is
Theorem 1 Let G be a (cid:12)nite group containing K as a normal subgroup
with quotient G=K an abelian group. Let f(cid:17)igi2I be a complete set of rep-
resentatives for the orbits of the irreducible representations of K under the
action of G. Denote the orbit of (cid:17) as (cid:1)((cid:17)). LetH(cid:17)i be a subgroup of G (con-
taining K) such that (cid:17)i extends to a representation of H(cid:17)i but no further.
Then T(G) is bounded by
jGj X jH(cid:17)ij jGj 2 (cid:11) jGj (cid:11) 2 jH(cid:17)i j
jKj(cid:1)T(K)+i2I( jHj (jH(cid:17)i j) (cid:1)d(cid:17)i+O(jKj(cid:1)j (cid:1)((cid:17)i)j(cid:1)(d(cid:17)i+d(cid:17)ilog( jKj )))
where the O-notation indicates a universal constant determined by the by
the FFT on cyclic groups. Furthermore, I(G) is bounded by the same sum
but with I(K) replacing T(K).
InTheorem1(cid:11)denotestheexponentofmatrixmultiplication, (currently
2.38,see[W]).Itisuseful topointoutthatunderthesimplifying assumption
of (cid:11) = 2 the bounds becomes
jGj jGj
T(G)= (cid:1)T(K)+O(jGj(cid:1)log( ))
jKj jKj
and
jGj jGj
I(G)= (cid:1)I(K)+O(jGj(cid:1)log( ))
jKj jKj
2
with the O-notation as in Theorem 1.
In brief, the argument for Theorem 1 relies on understanding how the
irreducible representations of G can be constructed, given the representa-
tions of some normal subgroup K such that G=K is abelian. G acts on the
representations of K, partitioning them into orbits. The representations in
any single orbit are then extended maximally using the theory of projective
representations. When they can be extended no further, the representation
is induced to G and necessarily gives an irreducible representation of G.
Careful rewriting of the Fourier transform and analysis of the structure of
the resulting matrices gives the above result.
Iterating these ideas gives e(cid:14)cient methods for treating any solvable
group (asequence of abelian extensions). In thesimplest nontrivial instance
of this case, the so-called metabelian groups, (abelian extensions of abelian
groups) this paper gives the (cid:12)rst example of a class of non-abelian groups
such that
T(G)= I(G)= O(jGjlog(jGj)):
Group convolutions may be computed using Fourier transforms(see sec-
tion 2C). Both naive methods using Fourier transforms and direct compu-
2
tation require on the order of jGj operations. To obtain an advantage by
using the method of Fourier transforms fast algorithms for both computing
Fourier transforms and performing Fourier inversion are necessary. Thus as
an immediate consequence of Theorem 1 fast group convolution algorithms
for abelian extensions are determined (Theorem 7).
Previous fundamental work was done by Beth in the (cid:12)rst serious treat-
ment of noncommutative Fourier analysis ([B]). Beth sketches ideas for
achieving savings in the case in which G contains some nontrivial normal
subgroup with more careful analysis being given to the case of normal sub-
groups of prime index. Also, novel applications to the theory of coding,
VLSI design and vision are given.
Independently, Beth has also made a strong start on fast algorithms for
abelian extensionsbyderiving somespeedup in thecaseofprime extensions.
In [B] a result like Theorem 1 given here (specialized to this situation) is
obtained.
Clausen also derives speedups for the general computation of Fourier
transforms as well as in the speci(cid:12)c case of Sn ([C1], [C2]). This work is
doneinthecontextofVLSI designandthestudyofWedderburntransforms.
Also, it has recently been brought to my attention that simultaneous inde-
pendent workby Clausen studies the case of Fourier analysis for metabelian
3
groups. Again, in the setting of VLSI design, he obtains the jGj log(jGj)
result ([C3]).
Collectively, this work makes a good start towards obtaining general
results on more e(cid:14)cient computation of Fourier inversion. Work on this
problem for general (cid:12)nite groups is in progress and will appear presently
([R3]).
The organization of this paper is as follows: Section 2 provides neces-
sary group theoretic background, in particular giving a highly constructive
treatment of induced representations. Section 3 explains the basic idea of
the algorithm, providing a \road map" for the rigorous derivations of the
main results in section 4. The methods of section 4 are illustrated with two
examples in section 5. There the algorithm is applied to metabelian groups
2
G such that G=K is of order p or p for some prime p. In the former case
the example of the Heisenberg group is carried along in detail as a concrete
illustration. Theideasinsection4suggestalgorithmsforanygroupin which
the irreducible representations are given as induced representations. As a
start, in section 6, these ideas are explored brie(cid:13)y in the case of semidirect
products by abelian subgroups.
2. Background
A. Representation Theory
Here the most basic concepts of representation theory are presented
brie(cid:13)y. Proofs of all unproved assertions may be found in Serre ([S]).
Let G be a (cid:12)nite group. Recall that a representation (cid:26) of G is a map
assigining matrices to group elements such that (cid:26)(st) = (cid:26)(s)(cid:26)(t) for all s
and t in G. Thus, (cid:26) is a homomorphism from G to GL(V) with V a vector
space of dimension d(cid:26), the dimension or degree of (cid:26). Two representations
are equivalent or isomorphic if they di(cid:11)er only by a change of basis. The
representation is irreducible if and only if for any subspace W (cid:18) V; if
(cid:26)(s)W (cid:18) W for all s 2 G then either W = f0g or W = V. A basic fact
due to Schur is
Theorem (Schur's lemma, [S], Proposition 4) Let (cid:26) be an irreducible
representation of G and let A be an invertible matrix such that
A(cid:1)(cid:26)(s)= (cid:26)(s)(cid:1)A
for all s 2 G. Then A is a scalar multiple of the identity.
4
Thenumberofinequivalent irreducible representationsofGis(cid:12)nite with
the degrees satisfying
X
2
d(cid:26) =jGj
(cid:26)
where the sum is over all irreducible representations of G.
Any representation (cid:26) of G may be decomposed as the direct sum of
irreducible representations. Thus, suppose (cid:26) has the decomposition
(cid:26)= (cid:26)1(cid:8):::(cid:8)(cid:26)m (1:1)
wherethe(cid:26)i areirreducible (thoughnotnecessarily distinct) representations
of G. In the language of matrices, this says that with respect to some
basis, (cid:26)(s) is block diagonal with the representations (cid:26)i(s) appearing on
the diagonal. If exactly j of the (cid:26)i are equivalent, then we say that the
irreducible (cid:26)i appears in (cid:26) with multiplicity j.
To determinine the decomposition of a representation the notion of the
character ofarepresentationis important. If (cid:26)is arepresentationofGthen
the character of (cid:26), written (cid:31)(cid:26) is de(cid:12)ned as
(cid:31)(cid:26)(s) = trace((cid:26)(s))
for all s 2 G. As the trace is invariant under conjugation, the character is
an invariant for any class of equivalent representations. If a representation
is one-dimensional then it is equal to its character.
Let L(G) denote the space of complex-valued functions on G. There is
a natural inner product on L(G) given by
1 X
hf1;f2i= f1(s)f2(s):
jGj
s2G
The irreducible characters are orthonormal with respect to this inner prod-
uct. In terms of characters equation (1.1) becomes
(cid:31)(cid:26) =(cid:31)(cid:26)1 +:::+(cid:31)(cid:26)m:
Orthonormality implies the following useful result:
Theorem ([S], Theorem 3) Let (cid:26) be an irreducible representation and (cid:17) be
any representation of G then the multiplicity of (cid:26) in (cid:17) is equal to h(cid:31)(cid:26);(cid:31)(cid:17)i.
In particular, (cid:17) is irreducible if and only if h(cid:31)(cid:17);(cid:31)(cid:17)i= 1.
5
If (cid:26) and (cid:17) are representations of G we will sometimes write h(cid:26);(cid:17)i for
h(cid:31)(cid:26);(cid:31)(cid:17)i. An easy consequence of the orthogonality is that distinct irre-
ducible representations have distinct characters.
B. Constructing Representations
Let X = fx1;:::;xng be some (cid:12)nite set on which G acts. That is for
all s;t 2 G and x 2 X, sx is an element of X such that (st)x = s(tx):
Associated to this action is the permutation representation of G on X.
To de(cid:12)ne this, let V(X) be the complex vector space generated by X,
V(X)= Cx1 (cid:8):::(cid:8)Cxn:
G acts on V(X) in the obvious way. An element s of G sends the basis
vector exi to esxi. Thus, the corresponding matrix assigned to s, (cid:26)X(s), has
the form,
(cid:26)
1 if sxj = xi
((cid:26)X(s))i;j = :
0 otherwise
The most natural (cid:12)nite set on which G acts is itself, by left multi-
plication. Here the associated permutation representation is called the
regular representation of G. The associated matrices with respect to the
standard basis fes js 2 Gg have the following properties:
Proposition 1 Let (cid:26)G be the regular representation of G with respect to the
standard basis of G. Then
(i) The matrices f(cid:26)G(s) j s 2 Gg have a unique nonzero entry in each
row and column. This entry is equal to 1.
(ii)If s and t are distinct elementsof G then all nonzeroentries of (cid:26)G(s)
and (cid:26)G(t) occur in distinct positions. (Another way to say this is that the
P
matrix s2G(cid:26)G(s) is the jGj by jGj matrix of all ones.)
0
Proof: (i) is clear from the de(cid:12)nition. As for (ii), note that s(cid:1)t = s (cid:1)t if and
0
only if s =s,
(cid:127)
Let H be a subgroup of G. Representations of H and G are related
by the dual constructions of induction and restriction. A very explicit
andconcreteexplanantion ofinduced representationsfollowing thebeautiful
exposition of Coleman ([Co]) can be given.
6
Let (cid:17) be a matrix representation of H. Then (cid:17) gives rise to a represen-
tation of G called the induced representation of (cid:17) to G. This is denoted
((cid:17) " G). To describe this, let k = [G:H], the index of H in G, and (cid:12)x
a set of right coset representatives fs1;:::;skg, with s1 the identity, for the
coset space G=H. Let s2 G, then ((cid:17) "G)(s) will be a k by k block matrix,
with blocks of size d(cid:17) by d(cid:17). Let (((cid:17) "G)(s))[i;j] denote the i;j block of the
matrix ((cid:17) "G)(s). Then
(cid:0)1
(((cid:17) "G)(s))[i;j] =(cid:17)~(si ssj)
where for all t 2 G,
(cid:26)
(cid:17)(t) if t 2 H;
(cid:17)~(t)=
0d(cid:17)(cid:2)d(cid:17) otherwise.
It is readily checked that this de(cid:12)nes a representation of G([Co],section
4). Also it is straightforward to show that induction is transitive ([Co],
Theorem 4). That is, if
G (cid:19) H (cid:19) K;
and (cid:17) is a representation of K then,
(((cid:17) " H)" G)= ((cid:17) " G):
The block structure of these matrices will prove to be of great impor-
tance. Several key properties follow immediately from the de(cid:12)nition.
Proposition 2 (i) ((cid:17) " G)(s) is a k by k block matrix with blocks of size
d(cid:17) by d(cid:17) with at most one nonzero block in each row and column. Thus, the
dimension of ((cid:17) " G) is [G:H]d(cid:17).
(ii) Suppose H is normal in G. If s and t are elements of G that lie in
the same coset of G by H then ((cid:17) " G)(s) and ((cid:17) " G)(t) have the same
structure (ie. nonzero blocks in the same position).
(iii)IfH isnormal in G thenthe matricesf((cid:17) "G)(si) j1(cid:20) i(cid:20) kg have
the same \structure" as the matrices for the regular representation on the
group G=H. (That is, there are nonzero blocks in the induced representation
in the same positions as the nonzero entries in the regular representation of
G=H.) In particular, the matrices for the elements of H are block diagonal
and elements not in H will have no blocks on the diagonal.
7
Proof: (i) is clear from the de(cid:12)nition. To show (ii) and (iii) suppose now
that H is a normal subgroup of G. s and t are in the same coset of G=H
(cid:0)1 (cid:0)1
if and only if s t 2 H. Suppose si ssj 2 H. Because H is normal in G
(cid:0)1 (cid:0)1
sj s tsj 2 H as well. Consquently, the product,
(cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1
si ssj (cid:1)sj s tsj = si tsj
is in H. This proves (ii).
For (iii), note that
(cid:0)1
si ssj 2 H () ssj 2 siH
which is if and only if
ssj = si
where the bar denotes the image in G=H. This proves (iii).
(cid:127)
Dual to the operation of induction is the process of restriction. That
is, given a subgroup H of G and a representation (cid:26) of G we obtain a rep-
resentation of H by restricting (cid:26) to H. This is denoted ((cid:26) # H). Clearly,
this too is a transitive operation. Induction and restriction are related by
Frobenius reciprocity.
Theorem (Frobenius reciprocity, [Co], Theorem 6) Let (cid:17) be a represen-
tation of H and (cid:26) a representation of G. Then,
h((cid:17) "G);(cid:26)i= h(cid:17);((cid:26)# H)i:
When restricted to subgroups, irreducible representations need not re-
main irreducible. Frobenius reciprocity is a useful tool in determining the
structure of a restricted representation. In general this can be a very di(cid:14)-
cult task. In the case of normal subgroups, however,much can be said. The
study of the structure of restrictions of representations tonormal subgroups
is often known as Cli(cid:11)ord theory or Mackey's theory of little groups.
Let H be a normal subgroup of G and (cid:17) be a representation of H. Then
given any s 2 G a (possibly) new representation of H may be de(cid:12)ned, a
(s)
conjugate representation of (cid:17), (cid:17) by
(s) (cid:0)1
(cid:17) (t) = (cid:17)(s ts):
8
Note that the conjugate of an irreducible representation is still irreducible.
Consequently,thereisanactionofGonthesetofinequivalent irreducible
representations of H. For any irreducible representation (cid:17) of H let (cid:1)((cid:17))
denote its orbit under this action. These are simply all the inequivalent
conjugates of (cid:17). Abusing notation, (cid:1)((cid:17)) will sometimes denote the direct
sum of these representations as well. (cid:1)((cid:17)) is often called the star of (cid:17). Let
StabG((cid:17)) be the isotropy subgoup of (cid:17) under the action of G. Thus,
(s)
StabG((cid:17))= fs 2 Gj (cid:17) (cid:24) (cid:17)g
where \(cid:24)" denotes equivalence of representations. In particular
Stab((cid:17))(cid:19) H.
The language of conjugate representations and restrictions gives a nice
way to explicitly restate part (iii) of Proposition 2.
Theorem 2 (Cli(cid:11)ord Theory) Let H be a normal subgroup of G.
(i) ([Co], Theorem 10) Let (cid:17) be a representation of H. Then
(((cid:17) " G)#H)= [Stab((cid:17)):H](cid:1)(cid:1)((cid:17)):
(ii) ([Co], Theorem 14) Let (cid:26) an irreducible representation of G. Then
d(cid:26)
((cid:26)#H)= (cid:1)(cid:1)((cid:17))
[G:Stab((cid:17))]d(cid:17)
where (cid:17) is any irreducible representation of H which occurs in ((cid:26)#H).
Related to restriction and induction is the notion of extension. If (cid:26) is
an irreducible representation of G such that
((cid:26)#H)= (cid:17)
then (cid:26) is said to extend (cid:17). The most natural setting in which extensions
arise is in the theory of projective representations. A thorough discus-
sion of this is given in the (cid:12)rst edition of Curtis and Reiner ([CR], section
51).
C. Fourier Transforms
Let G be a (cid:12)nite group, (cid:26) a representation of G and f 2 L(G). Then
b
the Fourier transform of f at (cid:26), denoted as f((cid:26)), is the matrix given by
X
b
f((cid:26))= f(s)(cid:26)(s):
s2G
9