Eftlcient Computation of Fourier Inversion for

Finite Groups

DANIEL N. ROCKMORE

Haruurd Unil’ersiV, Canrbricige, AlassachLtsetts

Abstract. Let G be a finite group and let f’ be a complex-valued function on G. If p is a matrixrepresen~ation of G (of dimension LIP)then the Fourier trarzsfiirnz o~~ at p k defined to be thematrix .f( p) = X, c ~f(s)p(s). Let % be a complete set of inequivalent irreducible matrixrepresentations of G. The Foaner trarzsfomr of f (with respect to %’) is defined as the set ofmatrices {jl p)},,~ ~. Recove~ of ~ from its Fourier transform may be accomplished via theFourier inlerscon formula,

f(s) = ~ ~ dPtrace(f( P) P(S)-’).pe9

Given the Fourier transform of f, direct computation of ~ by Fourier inversion requires on theorder of IGI2 operations. In this paper, using the techniques of induced representations, generalresults for more efficient computation of Fourier inversion are given. In combination with earlierresults for fast computation of the Fourier transform [Diaconis and Rockmore, 1990] fastalgorithms for computing group convolutions (or equivalently, multiplication in the group algebra)also are obtained. In a particular case of interest, when G is the symmetric group Sn, the methodsare readily applied and careful analysis reduces (n! )2 to (n( n!)) a“z where a k the exponent ofmatrix multiplication (2.38 as of this writing). A similar bound is achieved for computingconvolutions on S.. To illustrate the techniques, the explicit constructions are carried out for S3.These ideas may be implemented in a straightforward manner on a computer. A practicalalgorithm is given for S,,.

Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]:Numerical Algorithms and Problems—computation of transforms; computations on matrices; G. 1.0[Numerical Analysis]: General —rzarnerical algorithm; G.2.1 [Discrete Mathematics]: Combina-tories—pennatat~ons and cornbinattom; G.4 [Mathematical Software]: algorithm analysis; J.2[Physical Sciences and Engineering]: rnafhernatics and statistics

General Terms: Algorithms, Theory

Additional Key Words and Phrases: Fast Fourier Transform, finite groups, Fourier inversion,group convolution, group representations, symmetric group

A preliminary version of much of this work first appeared in the author’s doctoral dissertation,“Fast Fourier Analysis for Finite Groups”. Harvard University, Department of Mathematics,May, 1989.

The author was supported by IBM and NSF Graduate Fellowships.

Current address: Depart ment of Mathematics and Computer Science, Dartmouth College,Hanover, NH 03755.

Permission to copy without fee all or part of this material is granted provided that the copies arenot made or distributed for direct commercial advantage, the ACM copyright notice and the titleof the publication and its date appear, and notice is given that copying is by permission of theAssociation for Computing Machinery To copy otherwise, or to republish, requires a fee and/orspecific permission.@ 1994 ACM 0004-5411/94/0100-0031 $03.50

Journal of the Aw>c,.tlon for Computmg Machme~, VOI 41, No 1,January 1994,PP 31-66

32 D. N. ROCKMORE

1. Introduction

Let G be a finite group and ~ any complex-valued function on G. If p is any

matrix representation of G then the Fourier transform off at p is the matrix

The dual of G, denoted d, is the set of equivalence classes ofa irreducible

representations of G. Let W be any set of representatives for G. Then the

F~urier trarzsfonn of f (with respect to ~) is the collection of matrices

{f( /J)}p= q.The function f may be recovered from its Fourier transform via the Fourier

irmersion formula

f(s) = & ~=d,trace(~( P) P(S-l)),pe

(1.1)

where p has dimension dP.

Noncommutative Fourier transforms have appeared in applied problems

such as spectral analysis of permutation data [Diaconis, 1989]. They also allow

computation of convolutions as in the abelian case. Beth [1984] and Diaconis

[1988] discuss applications.In a previous paper, the problem of efficiently computing the Fourier

transform of f was discussed [Diaconis and Rockmore, 1990]. The present

paper is a complement to this earlier work. It solves the problem of efficient

computation of the inverse transform (Fourier inversion) for arbitrary finite

groups. In the abelian case, the dual is again an abelian group, so no new ideas

are required.

To state the problem more rigo~ously, assume that for all irreducible

representations p G W, the matrices f( p) are given (i.e., stored in memory) as

are the matrices p(s) for every s = G. In this paper the computational

problem of efficiently recovering the values f(s) for every s = G from this

initial data is investigated.

In a standard model of computation, a single “operation” is often defined to

be a complex multiplication followed by a complex addition. For example,

direct computation of the standard inner product of two complex d-tuples

requires d operations in this model.

With respect to this computational model, direct computation of (1.1) for any

fAked s = G takes at most IGI operations. (Since only the trace of the matrices

~( P) P(S-l ) is needed, only the diagonal elements of these matrices must becomputed. To compute any single entry of a product of two d by d matrices at

most d operations are needed. Thus, computation of the trace requires d2

operations. Consequently, calculating (1.1) directly requires at most XPd~ = IGI

operations.) Thus, to compute (1.1) for every s would require IG12 operations.

To make this feasible for large groups, fast algorithms must be derived.

For abelian groups such as the integers mod n or the group of binary

n -tuples, fast algorithms for Fourier inversion already exist. These all take

advantage of the fact that in this case the irreducible representations of G

form a group d (the dual group) isomorphic to G. Fourier inversion then

amounts to computing Fourier transforms on G. Efficient algorithms exist for

Efficient Computation of Fourier Iru?ersion for Finite Groups 33

this task, permitting computation of the Fourier transform (and consequently,

performing Fourier inversion) in 0(1 Gllog(lG\)) operations. These ideas havebeen a source for many important applications. Aho et al. [1976] and Elliot and

Rao [1982] contain thorough treatments of the basic algorithms and their uses

as well as pointers to the wealth of relevant literature.

For non-abelian groups, however, relatively little is known. Here ~ is no

longer a group, necessitating the introduction of new ideas. In particular,

induced representations provide the key to developing the general algorithm”.

In brief, let H s G (throughout, for a given group G, let H s G denote that

H is a subgroup of G) and suppose that {sl ( = 1), Sz,..., s~} is a complete set

of left coset representatives of H in G. Then f is completely determined by

the k functions on H,

{f, >..., fA}

defined by

f(t) =f(s,t)

for every t E AH.The idea is to find a way to recover efficiently the “restricted

transforms” ~(q) for a complete setA of irreducible representations q of H

from the original set of matrices {f( p)}. This will reduce the problem to

performing Fourier inversion for the k functions ~, on H. This is made

possible using the techniques of induced representations.

Specifically, if H s G, then representations of G may be constructed from

representations of H by a process known as induction. If q is a representation

of H, then q gives rise to a representation of G, denoted (q ~ G) (read q

induced to G) of degree (d~ . /G\/1 H l). In general, (q ~ G) is reducible. Let

be the direct sum decomposition of (~ ~ G) into irreducible representations of

G. Thus, there is a change of basis so that the matrices (q ~ G)(s) are block

diagonal for every s e G with blocks ~given by the matrices p,(s). Conse-

quently, with respect to such a basis f(q ? G) is block diagonal with blocks

f( p,).The main fact which is now employed is that in the “n~tural” basis for the

induced representation (q ~ G) the Fourier transform f(q ~ G) has a more

~seful form. Specifically, with respect to this other basis the Fourier transform

f(q T G) my be viewed as a k X k block matrix in which the first column isgiven by the blocks L(q).

Now the “algori~hm” is clear. Given the original Fourier transform, the block

diagonal form of f (q ~ G) may be constructed. An ~ppropriate change of basis

allows the recovery of the restricted transforms {f,(q)}, If this is done for a

complete set of irreducible representations q of H, then the problem has been

reduced to performing Fourier inversion for the functions {f,}. This may of

course now be iterated through subgroups of H. This reduction can yield great

savings.The organization of this paper is as follows: Section 2 contains some of the

necessary group theoretic background, in particular the construction of in-

duced representations is reviewed in a computationally useful way. Section 3

presents the main idea as summarized above as well as some of its immediate

34 D. N. ROCKMORE

consequences. This includes a discussion of fast algorithms for computing

group convolutions which arise as a result of having both fast Fourier inversion

and fast Fourier transform algorithms.As explained, the Fourier inversion algorithm requires the construction of

certain change of basis matrices for every induced representation (q ~ G).

These are readily obtained by a neat application of Frobenius reciprocity. This

is detailed in Section 4. Section 5 applies the techniques of the previous

sections to the symmetric group S,l. There, more careful analysis of the running

time is carried out, using bounds on the maximum degree of the irreducible

representations of S.. Combined with the careful bounds given earlier for

computing Fourier transforms on S. [Diaconis and Rockmore, 1990] explicit

bounds for computing convolutions on S. are given as well. As illustration of

the techniques, Section 6 works out the explicit constructions for performing

Fourier inversion on S~. In closing, in Section 7, a practical algorithm is given

and ideas for implementation on a computer are presented.

A first step in developing more general methods for computing the inverse

transform has already been presented by the author in studying fast algorithms

for Fourier analysis for abelian group extensions [Rockmore, 1990].

A group G is said to be an abelian extension of the group K if K is a normal

subgroup of G and the quotient group G/K is abelian. In [Rockmore, 1990], it

is shown that by using the special structure of an induced representation from

a normal subgroup with abelian quotient, savings can be obtained. Thus, for

the class of solvable groups (groups that may be written as a succession of

abelian extensions) fast algorithms for inversion are obtained.

In particular, specialization of these methods to the “metabelian groups”-

groups containing abelian normal subgroups such that the quotient is abelian

—yields a class of non-abelian groups such that both Fourier inversion and

computing all Fourier transforms may be done in 0(\ G] log( IG l)) operations.

These results for metabelian groups have also been obtained by Clausen

[198%1, working in the context of VLSI design.Fundamental work on this problem has been done by Beth [1984] in the first

serious treatment of computational aspects of noncommutative Fourier analy-

sis. 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 prime extensions. Also, novel applications to the theory of coding, VLSI

design and vision are given.

Independently and with a different approach, Clausen has also obtained

results for fast computation of Fourier inversion similar to some presented

here [1989b]. In work that appeared after submission of this manuscript,

Clausen derives a recurrence ([1989b], see proof of Theorem 1.5) for comput-ing Fourier inversion, much hke that given here m Theorem 1.

Clausen works directly in the group algebra. To explain his ideas in brief,

recall the natural correspondence between a function ~ E L(G) and an ele-

ment Z,. ~ f’(s)s in the group algebra C[G]. The matrix coefficients of anycomplete set of irreducible representations ~ of G provide a basis for C[G].

The Wedderburn transfoim (with respect to W) Z> is the change of basis from

the basis of delta function: of C[G] to the basis of matrix coefficients in %.

TheA Fourier transforms f( p) for p = ~ express f in the latter basis. Let@Pf( p) denote the group algebra element (with respect to the latter basis). In

Efficient Computation of Fourier Iru)ersion for Finite Groups

this setting, computing Fourier inversion amounts to computing

35

‘iZZj,’(OP~( p)) = & ,2,( ,~yd,trace(~( p)p(,s)))s-l.

Let H < G with t!%” a set of representatives for H. Assume that the

representations p = % are such that when restricted to H, they become block

diagonal with the representations of ~’ as blocks. Slightly modifying Clausen’s

terminology, say that J% is H-adapted.

Under the assumption that @ is H-adapted, by looking at the group algebra

element as a whole, Clausen is able to decompose the computation, recogniz-

ing it as a sum of inverse transforms of elements of the subgroup algebra

C[H],

2zj9’(@pf(p)) = ~ 7q?(@J(q))s,-’.j<[G:H]

Here {s,} is a set of coset representatives for G/H. The recurrences obtained

for the general case in both approaches are comparable (see Remark 1

following the proof of Theorem 1).

For any specific class of groups, it maybe possible to take advantage of extra

structure in order to reduce the complexity of performing Fourier inversion.

For example, this is the case with metabelian groups [Clausen, 1989a; Rock-

more, 1990] and as Clausen shows, also true for the symmetric groups (see the

Remark following the proof of Theorem 5).1

It should also be pointed out that advances are also being made in the

setting of compact groups. Recent work of Driscoll and Healy [1989] gives the

first efficient algor~thm-s for recovering bandwidth-limited

sphere from their spherical transforms.1

2. Induced Representations

Let G be a finite group. Recall that a representation p of G

matrices to group elements such that p(st ) = p(s) p(t ) for

functions- on the

is a map assigning

all s and t in G.

Thus, p is a ho-momorphism from G to GL(V) with V a vector space of

dimension dP, the dimension or degree of p. Choice of basis for V realizes p as

a matrix representation of dimension dP. Two representations are equivalent or

isomophic if they differ only by a change of basis. The representation is

irreducible if and only if for any subspace W G V, if p(s)W c W for all s = G,

then either W = {O} or W = V.

The number of inequivalent irreducible representations of G is finite with

the degrees satisfying

(2.1)

P

where the sum is over all irreducible representations of G.

Any representation p of G in a complex vector space V maybe decomposedas the direct sum of irreducible representations. That is, there exist subspaces

1See Note Added in Proof.

36 D. N. ROCKMORE

VI,.. ., V, such that

v=vlfB””” @v/

and p(s) restricted to ~ is an irreducible representation of G for each i. Thus,

in a suitable basis p(s) can be written as the block diagonal matrix

[ ““ )7p~(s)

p,(s)

where p, is an irreducible representation of G equivalent to p restricted to ~.

Note that these irreducible representations need not be distinct. If exactly j of

the pl are equivalent to p,, then the irreducible representation p, is said to

appear in p with multiplicip j. This decomposition is denoted as

p=mlp~+... +m~pj,

where the p: are the inequivalent irreducible representations of G, which

appear in p, each with multiplicity m,.

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 and concrete

explanation of induced representations following the beautiful exposition of

Coleman [1966] can be given.

Let q be a matrix representation of H. Then T gives rise to a representation

of G called the induced representation of q to G. This is denoted (q ~ G). To

describe this, let k = [G:HI = lG1/lHl, the index of H in G, and fix a set of

left coset representatives {sl, ..., s~}, with sl the identity, for the coset space

G/H. Let s e G, then (q T G)(s) will be a k x k block matrix, with blocks of

size d~ x dq. Let ((q ! G)(s)),,,,] denote the i, j block of the matrix (q ~ G)(s).

Then

where for all t e G,

It is readily checked that this construction defines a representation of G

[Coleman, 1966, Sect. 4]. Also it is straightforward to show that induction is

transitive [Coleman, 1966, Theorem 4.]. That is, if

and q is a representation of K then,

The block structure of these matrices will prove to be of great importance.

The key properties that follow from the definition are collected in the follow-

ing proposition.

PROPOSITION 1. For alls = G, (q ? G)(s) is a k x k block matni with blocks

of size dq by dq. There is exactly one nonzero block in each row and column. Thedimension of (q T G) is (lG1/lHl) “ dq.

Efficient Computation of Fourier Inversion for Finite Groups 37

Dual to the operation of induction is the process of restriction. That is, given

H < G and a representation p of G a representation of H is obtained by

restricting p to H. This is denoted ( p ~ H). Clearly, this too is a transitive

operation. The sense in which induction and restriction are dual will play a key

role in the algorithm. This is explained in Section 4.

3. Main Idea and General Results

The main result of this section (and paper) is Theorem 1 below, which derives a

recurrence for performing Fourier inversion by “reducing” the problem to

performing Fourier inversion for functions defined on a subgroup.

To state the problem precisely, assu~e that for a complete set of irreducible

representations p of G, all matrices f( p) are given (i.e., stored in memory) as

are the matrices p(s) for every s = G. How may the function f be recovered

quickly via Fourier inversion?

Throughout, if G is a group, then let 1(G) denote the number of operations

needed to recover a function f = L(G) given the initial data of the Fourier

transform of f and the matrices {p(s)}, ~ ~. As discussed in the introduction,

direct computation of f(s) for all s = G using the Fourier inversion formula

requires on the order of IGl 2 operations. Use of induced representations gives

the following main recurrence from which all savings are derived.

THEOREM 1. Let G be any finite group and H < G. Then

(3.1)

where the sum is over all irreducible representations q of H and a is the exponent

of matrix multiplication.

Before giving the proof of this result it is useful to prove some immediate

consequences of Theorem 1. To begin, rewrite (3.1),

where d~QX(H) denotes the maximum degree of an irreducible representation

of the group H. Note that the last line follows from (2.1). Further simplifica-

tion is achieved by the estimate dn., (H) s IH11’2 (which also follows from

(2.1)). The consequent (coarse) estimate of the sum Zqdy by

will allow some insight into the bound given by (3.1). (It would be useful to

derive a better estimate of the sum Zqd: and dn~X(H) to obtain a better

estimate of I(G).)

38 D. N. ROCKMORE

COROLLARY 1. Let all notation be as in Theorem 1, then

IGI IGI a

(-)l(G) s IHI1(H) + 2 IH[lHla/2. (3.3)

Assuming that Fourier inversion on H is then done directly, so that I(H) =

/H12, (3.3) would be minimized for a subgroup of size

@(lGl(2a–2)/(a+2)

).

Assuming that a subgroup of such size exists, this would then yield a bound of

I(G) < (XlG13a’(a+2)).

In most applications a = 3 and thus assuming that a subgroup of size

@(lG14’5) can be found,

I(G) < O(lG19i5). (3.4)

At the other extreme consider the simplifying assumption of a = 2.

COROLLARY 2. With Ihe notation as in Theorem 1, assuming exponent of

matrix multiplication equal to 2

I(G) I(H)

IGI < [HI

+ 21GI

IHI “

PROOF. Using (2. 1), the recurrence (3. 1) becomes

Again, if inversion on H is performed directly then I(H) = IH Iz and

I(G) IGI

IGI “H’+21HI’

This is minimized (assuming the choice is possible) by choosing H such that

IHI = @(m). This gives (21GI)’1’ as an upper bound for 1(G).The recurrence (3. 1) may be iterated through any tower of subgroups

G= Go> G,>... >Gm>G,n+l= {l} (3.5)

to obtain even greater savings. Here the last step would be computed directly.

The following proposition shows that using as “refined” a tower as possible

leads to the fastest algorithm.

PROPOSITION 2. Let G be a finite group. Assuming exponent of matrixmultiplication equal to 2, the tower of subgroups minimizing 1(G) ( using the

39Eficient Computation of Fourier Iruersion for Finite Groups

algorithm indicated by Theorem 1) is such that

m–l

E [G,:G,+,I + )%1~=()

is minimal.

PROOF. To simplifj notation, let k, = [G, : G, + ~]. It will be argued that

using the tower in (3.5), 1(G) is given by

(

m– 1

)1(G) < 21GI . ~ k,+ ;IGJ . (3.6)

~=o

This is done by induction on m. From the basic recurrence

1(G) < koI(Gl) + 2kOlG1.

Thus, iterating yields

1(G) < ko(klI(Gz) + 2kllGl)

IGI—. —I(G2) + 21Glkl -t

IGZI

+ 2ko\G\

21Glko.

Hence, for any 1,

IGII(G) < —I(Gl_l) + 21Gl(kl + . . . +kl_z)

lG1_,l

IGI—– —I(G1) + 21Gl(k,l + . . . +kl_l)

IG,I

Taking 1 = m and using KG~) = lGmZ12, proves (3.6) and hence, the

theorem. ❑

Remark 1. In any particular example, choice of the tower of subgroups will

depend highly on the availability of explicit representations for the various

subgroups chosen, as well as potentially specially tailored algorithms for

specific groups.

The problem of generating irreducible representations recently has been

studied by Babai and R6nyai [1990]. They obtain polynomial time algorithms

for computing all irreducible representations for a finite group from its

multiplication table.

Remark 2. If G is cyclic of prime order p, then no nontrivial subgroups

exist. In this case, G is abelian so that Fourier inversion amounts to computing

the Fourier transform on the dual group. For this, algorithms such as the

chirp-z and primitive root transforms perform inversion in O(p log p) opera-tions. Diaconis [19S0] briefly describes these techniques and gives references to

the literature.

The existence of fast algorithms for both computation of Fourier transforms

and performing Fourier inversion yields more efficient methods of computing

40 D. N. ROCKMORE

group convolutions. Recall that if f, g = L(G), then the convolution off with

g, written f * g is in L(G) and defined by

f* g(s) = x f(st-’)g(t)t~G

for any s = G. For a given s, the above computation requires IGI operations, so

that IGI 2 operations are required to compute f * g directly.

If p is a representation of G, then the computation of the Fourier transform

at p converts convolution into matrix multiplication by

Consequently, a straightforward ~lgorithm for computing f * g is to first

compute all Fourier transforms f( p) and P( p), multiply them together to

obtain the Fourier transforms f * g( p) and then perform Fourier inversion to

recover f * g. This algorithm gives the following proposition:

PROPOSITION 3. Let T(G) denote the number of operations needed to compute

the Fourier transform over G and I(G) the number of operations needed to

perJorm Fourier inversion over G. Then the number of operations needed to

compute the convolution of two functions defined on G is at most

2T(G) + I(G) + ~d~,

P

where a is the exponent of matrix multiplication and the sum is ouer all irreducible

representations of G.

In [Diaconis and Rockmore, 1990], the following bound on T(G) is proved.

THEOREM ([DIACONIS AND ROCKMORE, 1990], PROPOSITION 1). Let G be a

groLlp and H a subgroup. Then if bases are chosen for the matrix representations of

G such that certain “splitting properties” hold, then

IGI IGIT(G) < —

IHI (- i1 Ed:,

‘(H) + IHI - ,(3.7)

where the sum is ol~er all irreducible representations of G and w is the exponent of

matrix multiplication.

The “splitting properties” mentioned here mean that the matrix representa-

tions when restricted to H become block diagonal with irreducible representa-

tions of H as the blocks. This assumption is not crucial and given any

representation, change of basis so that the representations have this splitting

property is easily computed [Dixon, 1970] and may be performed as part of aprecomputation so as not to affect the running time.

Thus, by applying (3.7) in combination with Theorem 1 to Proposition 2, a

bound on the number of operations to perform convolution on G may be

given.

Eficient Computation of Fourier Inversion for Finite Groups 41

THEOREM 2. Let G be a finite group with f and gin L(G) and let H < G be a

subgroup. Then the convolution f * g can be computed in at most

IGI Id

()~(2T(H) + I(H)) + ‘2E -1 ~d:

P

IGI m

()— . ~d~

‘21HI ~

operations where a is the exponent of matrix multiplication and the sum ouer p is

ouer all irreducible representations of G, while the sum over q is ouer all irreducible

representations of H.

Consequently, applying Theorem 2 and inequality (3.2) shows

COROLLARY 3. Let all notation be as in Theorem 2, then the convolution of

two functions on G may be computed in at most

IGI IGI a

()~[2T(H) + I(H) + 21Gl”/2] + 2 ~ lHl”/2

operations,

Again, for a = 2 things simplify greatly.

COROLLARY 4. With notation as in Theorem 2, assuming exponent of matrix

multiplication equal to 2, the convolution of two jimctions on G may be computed

in at most

IGI~(2T(H) + I(H) + 41GI) (3.8)

Once more, assuming that Fourier analysis on 11 is performed directly, (3.8)

becomes

IGI~(31H12 + 41GI),

which is minimized (assuming the choice possible) for H such that IH \

= ~m. This gives 4tilG13’2 as an upper bound on the number of

operations needed to compute convolutions.

Computation of inversion and the transform may be done independently so

that the number of operations need to compute convolution in this fashion

would be minimized by choices of potentially different subgroups which mini-

mized each computation independently.

By applying (3.2) to (3.7), a bound of

IGI—(T(H) + lGl”/2)

‘(G) s IHI

is obtained. This is minimized by choice of a subgroup H (if possible) of size

lGl”/4. For such a choice, T(G) < 21Gl(”+4)/4, so if a = 3, T(G) < 21G17/4.

Thus, assuming the existence of subgroups of size on the order of IGI 3’4 (for

42 D. N. ROCKMORE

computing transforms) and IGI 4’5 (for computing inversion, see (3.4)) a coarse

bound of

O(IG17’4 + IGIY’5)

for computing convolution is obtained.

These bounds for computing convolutions also apply to performing computa-

tions in the complex group algebra on G, denoted C[G]. There is a natural

correspondence between elements in L(G) and elements in ~[ G]. Convolution

of functions in L(G) becomes multiplication of elements in ~[ G]. Thus, the

algorithm given above presents savings for performing these calculations as

well.

It is also worth making a remark here about assumptions on the exponent

for matrix multiplication, a. In practice and in almost all implementations

a = 3. There is active research in pursuing what is thought to be the theoreti-

cal limit of a = 2. Currently, the best theoretical result is a = 2.38 [Coppers-

mith and Winograd, 1987]. Results similar to those derived above could be

obtained under assumptions like a = 2 + e or assuming that d2 log d opera-

tions are needed to multiply d X d matrices.

Having worked out some of the consequences of Theorem 1, it is now time

to provide its proof.

PROOF OF THEOREM 1. Let q be an irreducible representation of H and let

~ G L(G). Consider the matrix

which may be rewritten as

where m = [G:H] and {Sl(= 1), S2, . . . . s,.} are left coset representatives for

G/H, and fi ● L(H) is defined by

f(t) =f(s[t)

for all t G H.

By Proposition 1, for any s ~ G the j, k block of the matr~ ‘ “‘ ‘

(denoted (q ~ G)(s),, ~) is given by

(TIT G)(s)J, /c =

{

‘r)(s,- ‘ ss~) if ,S–l SSL G H:

o otherwise,

so that there is exactly one nonzero block in each row and column.

Consider column 1 of the matrix (q ~ G)(s, ). The nonzero entry

row j for the unique j such that

S,-ls, “ 1 G H,

Thus. it will be in row i, and the entry will be the dq X d~ identity matrix,

denoted Id .7

Eficient Computation of Fourier Inuersion for Finite Groups

Thus, (q ~ G)(s, ) will be of the form

o * ““” *. . .. .

0 i ““”” iId o ““” o

0“ * ““” *. . .. . . .

0 i ““: i

~(row i),

43

where * denotes an unspecified block and Id is @ row i.

Now consider the form of row 1 in the mat~ix f,((q ~ G) $ H). For any t e H

l“tsk~Htik=l.

Thus,

(q ~ G)(t) =

so that

q(t) o ““” o

0 * “’” *. .. . .

0 i ““”” i

f(q) o ““” o

0 * ‘.. *. .

Consequently, an explicit rewriting of ~(q ? G) gives

o * ““” *. . .. . . .

0 : ““”” iId o . . . 0

0“ * ““” *. ...” ““.

o : ... io*.,.

o*

o*

o*

. . . *

. .. .

. . . *

. . . *

. . . *

. .. . . *

. . . 0

. . . *

. .. . . *

44 D. N. ROCKMORE

——

m) * ““” *. .

. .

l%?) * ... *

1:”’. .. .

Z(T)) * ““” *

(3.9)

The problem now is that the above matrix (3.9) cannot be written down

directly- from the given Fourier transforms {~( p)}P. However, suppose that the

decomposition of (q t G) into irreducible representations is

(q~G) =pl + ““” +p,

(where the o, need not be distinct). Then, with respect to another basis, the,. . .matrix jlq t G) has the form

/f(p, ) o ““” o

0 f(p,) ““” o

0 0 ““” f(pr)\

(3.10)

As the matrices ~( p) for all irreducible representations p of G are given,

the above matrix (3. 10) may be constructed ‘directly. The matrices (3.9) and

(3.10) are related by a change of basis. Thus, there exists an invertible matrixAv which depends ~nly on ~, H, and q such that

‘f(p,) o ““” o\

o f(p,) ““” oAT “ ... . .A~l

\o 0 ““” f(pr)l

—

E(q) * ““” *. .

. .

(3.11)

The above matrices are of dimension (m “ dv ). Consequently, performing

conjugation for each irreducible representation q of H requires at most

this

operations. So, assuming that for each irreducible representation ~q of H the

matrix A7 can be constructed, the restricted Fourier transforms {~,(q )}1~, may

be recovered. In Section 4, this change of basis construction will be explained.

Eflicient Computation of Fourier Iru)ersion for Finite Groups 45

Since f is equal to the sum of the restricted functions ~, the problem has been

reduced to that of inverting these functions on H. This requires an additional

(lG1/lHl) “ 1(H) operations. ❑

Remark 3. As one of the referees points out, it is possible to take advantage

of the fact that only the first column of blocks in the matrix multiplication

(3.11) are of interest. To recover this first column consisting of lG1/lHldq X d blocks only 2(\ G\/lHl)2(d;) operations are needed (with an additional

2((lG1/~H I)dm)z additions). This observations simplifies the recurrence (3.1) to

(3.12)

with an additional 2(1GI 2/1 H 1) additions. Setting a = 3 essentially gives the

recurrence obtained by Clausen ([1989b], p. 62).

Remark 4. It is instructive to point out here that the algorithm described in

the proof of Theorem 1 is essentially the “running backwards” of the algorithm

given in [Diaconis and Rockmore, 1990] for fast computation of the Fourier

transform.

In [Diaconis and Rockmore, 1990], it is shown that to compute the Fourier

transform of some given function f = L(G) one considers some fixed tower of

subgroups

{l}= GO< G1<O.. <GH. I< G,I=G (*)

in G. The Fourier transform at any given irreducible representation of G is

then built up by successively computing transforms from GO up to G. (= G).

The Fourier transforms for any G~ are built directly from those of G~ _ ~ by use

of restricted representations “glued together” with the irreducible representa-

tions of some fixed set of coset representatives for G~/G~ _ ~.

In order to recover a function f e L(G) frorh its Fourier transform, one may

again consider the tower ( *). This time one moves from G,, ( = G) down to Go,

using induced representations (the actual “dual” of restricted representations)

to move from Gk to G~ _ ~. Furthermore, as will be shown in Section 4, the

change of basis matrices necessary to obtain the reduction from Gk to Gk. 1

are built out of the irreducible representations of the coset representatives for

Gk/Gk_l.

Remark 5. In estimating running times, it is assumed that all necessary

representation theoretic data (i.e., the actual representing matrices, dimen-

sions, etc.) are available for “free”. As the change of basis matrices An (in

(3.1 1)) depend only on this data for G and some fixed subgroup of G,

computation of A7 may be done independently of Fourier inversion. Thus, the

calculation of A ~ may be viewed as part of a “start-up cost” and does not

figure in the determination of the running time for the algorithm. The next

section shows how Aq is computed.

4. Construction of the Change of Basis Matrices

Let G be a group and H a subgroup of G and q an irreducible representation

of H. The key observation in the proof of Theorem 1 is that the Fouriertransform at the induced representation (q ? G) in its “natural” basis contains

46 D. N. ROCKMORE

as blocks the restricted transforms at the representation q. How~ver, the initial

data only allows one to build the block diagonal form of ~(q ? G) (3.10)

directly, so that to obtain the “natural” form of this transform (3.9) a change of

basis is required. As is remarked at the end of the last section, this change of

basis depends only on the groups and representations of interest and not on

the function so that such constructions can be viewed as part of a start-up cost

and do not affect the speed of execution of the algorithm. In this section, the

construction of these change of basis matrices is explained.

A nice application of one form of Frobenius reciprocity allows the construc-

tion to be done neatly. A comprehensible statement of this property, which

relates induced and restricted representations, requires a more detailed under-

standing of several basic ideas from representation theory.

Let G be a group and p a representation of G in GL( V) for some complex

vector space V of degree n. In this manner, V becomes a G-space. That is to

say, there is an action of G on V given by

sL1d5p(s)L)such that

(st)u = s(w),

where s and t are any elements of G and v = V.

Let H be a subgroup of G. Then V is also an H-space under restriction of

the action of G to H. Considered as an H-space, it is denoted Res( V). Thus,

Res(V) and V are the same vector spaces, but considered with different group

act ions.

In Section 2, induced representations were introduced. It is now useful to

reintroduce them in a coordinate-free fashion.

Let q be a representation of H in GL(W’) for some complex vector space W’.

Let {sl (= 1),..., Sn} be a complete set of coset representatives for G/H.

Consider now the new vector space C[G] ~c[~l W where CIGI and CIH1denote the complex group algebras of G and H respectively. Any element of

CIGI 18c[~1 W can be written as a sum of vectors of the form s, @ w where wis an element in W. C[G] ~c[~l W has a natural structure as a G-space. Ifs = G is such that

S“sl=sk”t

for some t = H, then

S(S1 8 w) = Sk @ tw.

Considered with this structure, C[G] @c[HI W is denoted as ~nd(W).

The G-space lrzd( W) is nothing but the abstract definition of the induced

representation described here in Section 2. To actually get the matrix represen-

tation described, there a choice of basis must be made. The “obvious” choice is

the right one.

THEOREM ([CURTIS AND REINER, 1988], SECTION 12D). Let G be a HOLlp and

H a subgroup of G. Let q be a matrix representation of degree r of H in GL( W)

with respect to the basis {e,, . . . . e,}. Let {s I,..., s~J be a complete set of cosetrepresentatives for G/H. Then the rnatrti representation of G in GL( Ind( W )) with

47Eficient Computation of Fourier Inlersion for Finite Groups

respect to the ordered basis

{sl~el,..., sl~e,, s~~el, ~,s~..., s~ Be,, }

is exactly that described by Proposition 1.

Let P, be a representation of G in ~ for i = 1,2. Then the set of linearmaps A from V, to Vz that commute with the respective G-actions is denoted

as Horn~(Vl, Vz). That is

HornG(vl, v2) = {A c Honz(vl, v2) I p2(s)A =Apl(s) for all s = G}.

After choice of bases for VI and Vz, Horn ~(Vl, Vz) becomes the set of

matrices that intertwine the matrix representations PI and pz.

Returning to the problem at hand, recall eq. (3.11):

h,) o

0 f( p2)+ . . .

0 0

—

fh) *

f(rl) *

. . . 0

. . . 0

““” lb,. . . *

. .. . . *

. .

. . . *

(3.11)

The object is to find this matrix A. giving the change of basis from the“standar&’ basis for the induced repre~e~tati& (the rig~thand side of (3.11))

to the “diagonal” basis (the matrices being conjugated on the lefthand side of

(3.11)). That is, let q be an irreducible representation of H in W. Let {e,} bethe basis of W’ that allows q to be realized as the matrix representation of H

assumed in (3.9). Let p be the representation of G equivalent to (q ~ G) given

with respect to a basis {fi} of a vector space V that gives rise to the block

diagonal matrices on the lefthand side of (3.11). Then the problem is to find

the matrix Aq G Horn~(Ind( W’), V) intertwining these representations. In this

setting use of Frobenius reciprocity reduces this to a potentially easier problem

of finding an element in Horn~(W’, Res(V)) by relating Horn~( W, Res( V)) and

Horn~(Irzd(W), V). The following is a pertinent restatement of Frobenius

reciprocity.

THEOREM ([CURTIS AND REINER, 1988], THEOREM 43.14). Let H < G be a

subgroup and j& coset representatil’es {sl ( = 1),. ... s~} for G/H. Let q be a

representation of H in a uector space W and p a representation of G in a vectorspace V. Then there is a canonical isomo~hism (thus, written as an equality)

between

Horn~(W, Res(V)) = Hom~(Ind(W), V),

48 D. N. ROCKMORE

where Res( V) is V considered as an H-space under the action of p restricted to H

and Ind(W) is the G-space CG ~c~ W.

The isomorphism mentioned here is the “natural” one. If @ ~

HomJInd( W), V), then an element O = Horn~( W, Re.s(V)) is obtained by

defining

+(w) = @(l @ w).

Similarly, given @ E Horn~( W, Res(V)), define @ E Hom~(lnd( W), V) by

0(s, 8 w) = s, “ ~(w)

and extending linearly. (Note that @(w) E Res(V). Since Res( V) = V as vector

spaces, the action of s, on @(w) is defined. So the righthand side above makes

sense.)

Suppose that in addition, V decomposes as a direct sum of G-invariant

subspaces

v= qe, q.

Let @ c Horn~(W, Res(V)). Then, for any w ● W,

@(w) = ~@(w)L1s1

where O(W), E ~ and in fact it is easy to check that the map

@l:w+y

defined by

4,(W) = +(w)

is an element of Hom~ ( W, ~). Thus, in this case

Hom~(W, Res(V)) = Q,c IHom~(W, Res(~)). (*)

Now specialize to the case at hand. Let p be the representation of G

equivalent to (q ? G) such that

(1) The matrices { p(s)},. ~ are block diagonal of the form

where

(2) When{ P,} are the irreducible representations of G occurring in (q T G).restricted to H, the representations p, are block diagonal with

distinct blocks for every occurrence of q. Furthermore, the blocks corre-

sponding to v are equal (as matrices) to the matrix representation of q.

It is worth explaining condition (2) in slightly more detail. Let q be a fixed

irreducible matrix representation of H and suppose that p is an irreducible

representation of G such that ( p J H) contains ~ with multiplicity m > 0.

Efficient Computation of Fourier Inuersion for Finite Groups 49

Condition (2) requires that for every t G H the matrix p(t) is of the form

‘qI(t) o ““”

o ‘q2(t) ““”

\o 0 . . .

where either T, = q or q, is a representation

(Thus, there exist {ii,..., im} such that q,(t)

0

‘1.,q~(t)

of H containing no copies of q.

=q(t)for j=l,..., mand for

i~{il,. ... im} q, is a representation of II’containing no copies of q.)In fact, condition (2) is a slightly relaxed version of the conditions under

which efficient algorithms for the computation of Fourier transforms are

derived in [Diaconis and Rockmore, 1990]. That is, condition (2) requires that

the representations p, become block diagonal with respect to H in such a way

that all occurrences of q appear explicitly. In [Diaconis and Rockmore, 1990],

it is assumed that the irreducible representations of G decomposed with

respect to H so that every irreducible representation occurred. Change of basis

to such bases may always be determined [Dixon, 1970]. As will now be shown,

assumptions (1) and (2) permit the direct determination of the matrices AT.

Let V = @~=~~ be the assumed direct sum decomposition of V into

irreducible G-spaces, with ~ of dimension d, and let W be of dimension d. ( * )

implies, that any element of HornJ Irzd( W), V) can be described by consider-

ing separately its effect as an element of HonzJInd( W ), ~) for each i. By

Frobenius reciprocity, it is enough to understand the associated element of

Hom~( W, Res(~)).

Condition (1) above says that the representation p is given with respect to a

basis {fi} such that {~}~~ ~ span the G-invariant subspace ~ and that theassociated representation of G restricted to ~ with respect to this basis is p,.

Let v be the matrix representation of H in W given with respect to the basis

{e,,..., e~]. Then, condition (2) says there exists a direct sum decomposition of

Res(~)

Res(~) = W; EBo“” 69 W;,,

where the subspaces lift are H-invariant and either:

(i) The representation of ~ restricted to ~’ is equal to q

(ii) The representation of H restricted to ~’ contains no copies of q.

Furthermore, the block diagonal structure of the matrices again implies that

the basis {~Jz} of ~ may be partitioned into subsets, each making up the basis

of ~’. Furthermore, the block diagonal structure implies that these subsets

index consecutive columns of the representing matrices p,(s).

As before, since the subspaces ~’ are H-invariant, a similar argument shows

that

Horn~(W, Res(~)) = ~“~1 Hom~[W, ~’).

Now recall the following lemma of Schur:

LEMMA ([CURTIS AND REINER, 1988], THEOREM 27.3). Let ql and qz be

representations of a jinite group G such that they have no irreducible components

50 D. N. ROCKMORE

in common. Then the only matrix intertwining these representations is the matrix of

all 0’s.

Thus, for the ~’ with no copies of q,

However, for the other ~’ which give representations of H equal to q by

restriction, condition (2) says that there is a subset of the basis vectors of ~,

with { f~,, ~)} forming a basis for ~’ such that the map sending

for each k extends linearly to an element of Hom~( W, ~’ ). (Note that

implicitly, (j, k) denotes here a map that takes values in the subscript set.) That

is, f;, k, is the basis vector of ~’ which “acts like” e~ with respect to the

restricted action of p,.

Let ~(q) denote the set of indices j such that j = ~(q) if and only if the

representation of H on ~’ is equivalent to q (i. e., W).

Consequently, there is an associated element of Hom~(W, Res( J()) sending

Using reciprocity an element of Hom~(Ind(W), ~) is obtained by the map

sending

Note that since the vectors ~j ~, are among the basis with respect to which the

matrices p,(s) are given, any single product

PJ(sq)f(\.k)is simply the (j, k) column of pl(s~ ). Thus, the sum (4.1) is just a sum across

columns of p,( s~).

Now, by summing over i an element of HonzG(Imf( W), V) is obtained, given

by the map which sends

By the G-invariance of the vector spaces ~ and condition (l), (4.2) is simply

the column vector given by adjoining the columns in (4.1) for i from 1 to r.

To summarize, what has been shown above is how any particular column of

the change of basis matrix AT is to be computed. This requires determining

the image of basis vectors Sy @ ek in terms of the basis vectors fi. The

G-invariance of the subspaces ~ G V and condition (1) make this the “con-

junction” of the images in any particular ~. By Frobenius reciprocity, this is

reduced to finding the image of e~ in ~. Condition (2) shows that the image of

ek is a simple sum of basis vectors for ~ and hence, explicitly a column vector

Efficient Computation of Fourier Im?ersion for Finite Groups 51

of 1’s and 0’s. Thus, p,(s~) is the sum over the appropriate columns of p,(s~).

Adjoining these column vectors as i goes from 1 to r then gives the image of

s~ @ e~.

Note that in the case in which q appears exactly once in each ~ the change

of basis matrix A ~ is given by columns which are just conjunctions of certain

columns of the matrices p,(s~). Consequently, essentially no additional compu-

tation need be performed for its construction. The properties of the above

construction are collected in the following theorem:

THEOREM 3. Let G be a group and H < G with {sl, . . . . sk) a complete set of

left coset representatives for G/H. Let { p,}, ● ~ and {qJ}J~ ~ be complete sets of

irreducible matrix representations for G and H, respectively, related in the following

manner:

If (p, J H) is equiltdent to Z;, = ~q,m, then

q,(t) o ““” o

0 q,,(t) ““” op,(t) = . . . .

\o 0 . . . rl,Jt)

for any t G H.

Let q be an irreducible representation of H and let

(T I?G) ‘p,, o “. ~Pl,

be the decomposition of (q ~ G) into irreducible. Then the change of basis matrix

A ~ such that (3.11) holds may be constructed directly from the entries of the

matrices { p,,(s~ )} for 1 < j < r. With the notation as in (4.1) the column indexed

by the elements~ 8 e~ is gil)en by the transpose of the row oector

(4.3)

where the superscript t here denotes transpose.

It is instructive to examine the complexity of the change of basis calculation

detailed above. Let m = lG1/lHl. Then, in (4.3) above, as has been previously

noted each of the column vectors

is just a sum over columns of ~l~s~). Assuming that additions take negligible

time, then so does the construction of Aq.

The above discussion uses very strongly the assumptions (1) and (2) in the

change of basis construction. Under no assumptions at all, the change of basis

can still be calculated. It can be shown [Dixon, 1970] that in general, on the

order of at most

z,4

52 D. N. ROCKMORE

operations will be needed to compute AT. However, h is worth pointing out

that in most applications in which one wants to perform Fourier inversion (e.g.,

computing convolutions) Fourier transforms are to be computed as well. To

obtain savings computation of the transforms should be with representations

given with respect to bases that decompose completely with respect to some

fixed subgroup, so that they satisfy the conditions of Theorem 3.

5. Fourier Inversion for the Symmetric Group

The ideas presented in sections three and four of this paper lend themselves

naturally to performing Fourier inversion for the symmetric group S.. In order

to explain this, it is necessa~ to give a very brief introduction to the represen-

tation theory of S.. All this and much more may be found in James and

Kerber’s encyclopedic work [1981] or in James [1978].

Let S. denote the symmetric group on n letters. It is a fundamental fact that

the irreducible representations of S,, are indexed by partitions of n. To fix

some terminology, let A = {Al, . . . . A,} be a partition of n. This is often

denoted as A R n. Thus

A,> A,+ I>O,

where the integers {A,}:. , are called the parts of A and A is said to have length

r. If A is a partition of n then the corresponding irreducible representation of

S. is denoted as pA and is of degree dA. For example for S~, the table below

lists the partitions of 5 and the dimensions of the associated irreducible

representations.

A (5) (4,1) (3,2) (3,1,1) (2,2,1) (2,1,1,1) (1>1,1 ,1,1)

dA 14 5 6 5 4 1“

The asymptotic of the degrees of the irreducible representations of S. have

been studied extensively, and in particular, the asymptotic of the maximum

degree are well-understood. Bounds for the maximum degree are useful in

estimating the running time of the algorithm for S.. Vershik and Kerov [1985]

prove:

THhOREM ([KEROV

C,), c1 > 0 SUCh that

AND VERSHIK, 19S5] THEOREM 2). There exist constants

(5.1)

Presently, it is known that one can take co = 0.1156 and c1 = 1.238.

Associated to any partition of n is its Young diagram (after the Englishmathematician Alfred Young). For A as above, this will be a left-justified

arrangement of empty boxes (nodes) with A, boxes in row i. If the Young

diagram corresponds to the partition A, then it is said to be of s)zape A. The

Young diagram of shape (3,2,2, 1) is shown below.

Ejj5cient Computation of Fourier Inversion for Finite Groups 53

Recall that S._ ~ sits inside S. as the subgroup of permutations fixing the

symbol n. The Branching Theorem explains how a representation from S,, splits

when restricted to S,, _ ~.

THEOREM [BRANCHING THEOREM ([JAMES 19781, THEOREM 9.2)1. Let A be a

partition of n and ph the associated irreducible representation. Then

(pAJsn_~) = q P*,

where h’ run ouer all partitions of n – 1 with diagrams obtainable from that of h

by remouing a single block.

In the context of induced representations the branching theorem takes the

form of

COROLLARY. Let h’ be a partition of n – 1 and ph, the associated irreducible

representation. Then

(P, T~,l) = q P,4

where A runs over all partitions of n with diagrams obtainable from that of h’ by

adding a single block.

For example,

(P(3,2,z,I)~~7) = ~(2,2,2,1) o p(3,2.1 ,1) o p(3,221

and

( P(3,2,Z)~ ‘S) = P(3.2,Z,I) @ p(3,3,2) o P(d,z.z)”

The representation theory of S. essentially is completely determined by the

combinatorial properties of Young diagrams, and their generalizations, the

so-called Young tableaux. Given a Young diagram for a partition of n, any

filling of the boxes without repetition from the numbers {1,..., rz} is a Young

tableau of size n. If the entries are increasing from left to right and top to

bottom then the tableau is said to be standard. Below are the standard tableaux

of shape (3, 2).

135 125 134 124 123

14 34 25 35 45

There is a natural total order on the standard tableaux of a fixed shape, the

last-letter order. It is defined as follows: Consider two standard tableaux T and

T‘ of size n and of the same shape. If n is in a row higher up in T than in T‘,then declare T < T‘. If n is in the same row in both (necessarily at the end of

this row), then delete n from both (leaving a standard tableau of size n – 1)

and now consider the position of the entry n – 1,etc. The five tableaux above

are shown in increasing order from left to right.

In particular, there are two types of irreducible representations of S,, which

can be defined directly from the standard tableaux, Young’s seminormal form

and Young’s orthogonal form. For the purposes of this paper, these will be the

only irreducible representations of interest.

The matrix entries of these representations are defined in terms of the axial

distance. For this, consider a tableau T (standard or not) of shape A For

a={l,2, ..., n}, let rJ a) be its row position and c~(a) its column position.

54 D. N. ROCKMORE

Given two entries a and b (1 s a, b < n), define the axial distance from a to b

as

o!~(a, b) = (c~(a) – c~(b)) + (r~(b) – rr(a)).

This is the number of moves it takes to go from a to b in T counting moves to

the left or down as positive, and moves to the right or up as negative. For

example, if

T= 123

45’

then

d7(3,4) = 3 and d~(5,1) = O.

Note that this is a signed distance and

dr(a, b) = –d~(b, a).

Let d’(a, b) denote the axial distance for the ith standard Young tableau in

the last letter order. Dependence on the shape A is suppressed in this notation.

Fix a shape A. Given a pairwise adjacent transposition (t,t + 1) = S., define

a matrix a ( t,t + 1) = (( mt, (t, t + l))) with rows and columns indexed by the

standard Young tableaux < of shape A in the last letter order, by the following

rule:

(a)

{

+1 if tandt+l are(rL, =

–1 if tandt+l are

(b) if (t,t + 1)~ = ~ for i <j, then let

LrLL u,, –d’(f,t + 1)-1

CT,[ UJ1—1

(Here (t, t + 1)~ is the tableau obtainedt + 1.)

(c) u,, = O, otherwise.

in the same row of T,;

in the same column of T,.

1 – Li’(t,t + 1)-2

Li’(t,t + l)-’

form ~ by interchanging t and

Similarly, define a matrix dt, t = 1)by changing (b) above to

(b) if (t, t + 1)~ = ~ for i <j, then let

tiL[ Wt, –LiP(t,t + 1)-’ J1 –LT(t, t+ 1)-2

‘]1 ‘JJ J1 –d’(t, t+ 1)-2 ~’(t,t+ 1)-I ‘

Chapter 3 of James and Kerber [1981] presents a proof that these definitions

specitj irreducible representations. Since the pairwise adjacent transpositions

generate S., it is enough to specify the representations of only these elements.It is useful to record this as

THEOREM. The matrices cr and w defined above generate Young’s seminonna[

and orthogonal representations, respectively. Wn”te o(m) and ti(~ ) for these

matrices at the permutation T = S..

Efficient Computation of Fourier Inversion for Finite Groups 55

In order to apply Theorem 3 for performing Fourier inversion, it is necessary

to be working with representations PA of S. such that when restricted to S._ I

the matrices become block diagonal as specified by the branching theorem. For

example consider the irreducible representation P(3,z, of S5. For n ~ S5 such

that 7(5) = 5, it should be the case that

where B K is the appropriate matrix representation of m- regarded as an

element of Sd. Similarly, if ~ fixes both 4 and 5, then the blocks B(3’ 1) and(s) B(Z. 1), and B(2, l), respectively,and so on. ForB(22, split further into B ,

Young’s orthogonal and seminormal forms, this block structure is automatic.

Specifically, in [Diaconis and Rockmore, 1990], it is shown

THEOREM ([DIACONIS AND ROCKMORE, 1990] PROPOSITION 4). Let A be a

partition of n and ph be the corresponding irreducible matrix representation of S,, in

Young’s seminormal (respectille~, o)~hogonal) form. Let m E S,, _, < S.. Then

pJl)(7r) o ““” o\

o pv(dm) ““” opA(m) = . >

0 0 . . . ipv([] m) /\

where the V(’) are partitions of n – 1 determined by the Branching Theorem andthe pvt,,(m ) are again gillen in Young’s seminormal (respectil’ely, orthogonal) fo~

as defined for S,, _ ~.

Thus, both Young’s seminormal and orthogonal forms satisfy the conditions

of Theorem 3. In addition, it is easy to see using the Branching Theorem that

restriction of any irreducible representation from S,, to S,l _ 1 is multiplicity-

free (i.e., at most one of each irreducible representation of S. _ ~ occurs).

Consequently, Theorem 6 specializes to the symmetric group yielding

LEMMA 1. Let S. denote the symmetric group on n letters and let S,, _ ~ be the

subgroup of S,, given by permutations jixing n. Fix the coset representatilles

{(1), (n – 1, n) ,..., (1, n)} for S,,/S. _,. Let{ U*}A_. and{ COA}*➤ ,, be the completeset of irreducible representations of S. in Young’s seminormal and orthogonal

forms. Let p + n – 1 with { p(’) I 1 s i < rP} the corresponding partitions of n

that occur when WV is induced to S.. Then the change of basis matrix A such that

I(@n) o . . . 0 Io q)(m) ““” o

A“ . .~-l = (ay~sn)(m)

o 0 . . . “(q,,) T )\

(respectively, w) for euery ~ E S,, maybe constructed directly from the entries of

the matrices { ffP,,l(i, n) I 1 5 i 5 n}. The column indexed by the element (j, n) 8 ek

56 D. N. ROCKMORE

is giuen by the tnmspose of the row

(respectiueb, o), where d, is the dimension of ~P,,.

Thus, assuming that the Fourier transform over S. is given with respect to

Young’s orthogonal (respectively, seminormal) form, no extra computation is

needed to form the change of basis matrices that permit the recove~ of the

restricted transforms on S,, _ ~ in Young’s orthogonal (respectively, seminormal)

form. To illustrate this construction, in the next section, an explicit example is

worked out and all change of basis matrices for going from Sj to Sz are

constructed. However, at this time, the running time of the algorithm on S. is

examined.

PROPOSITION 4. Let I(n) denote the number of operations needed to pe~orm

Fourier inuersion oler the symmetric group S.. Then

where a is the exponent of matrix multiplication.

PROOF. Applying Theorem 2 to the groups S,, _, s S., I(n) satisfies

Now use induction. ❑

COROLLARY. With notation as in Proposition 4, if a = 2, then

PROOF. From Proposition 4 and (2.1)

2

(-)I(n) <272! fi ‘(m – l)!=2n! ~ m <21z! ~ . ❑

~=z m! ~=1

In [Diaconis and Rockmore, 1990], the running time for computing the

Fourier transform over S. is examined in detail. There it is assumed that the

Fourier transform is computed with respect to either Young’s seminormal or

orthogonal form. It is shown that

PROPOSITION ([DIACONIS AND ROCKMORE, 1990] PROPOSITION 3). Let T(n)

denote the number of operations required to compute the Founier transform over

Wmmetric group S.. Let a denote the exponent of matrix multiplication. Then

(5.2)

Thus, with all computations being performed in either the seminormal or

orthogonal forms, Propositions 3 and 4 may now be applied in combination

with (5.2) and a bound for computing convolutions on S. given.

Eficient Computation of Fourier Irulersion for Finite Groups 57

PROPOSITION 5. Let a denote the exponent of matrix multiplication. Then at

most

operations are needed to compute the cormolution of two functions defined on the

symmetric group S..

Letting a = 2 and applying (2.1) gives a rough estimate on the number of

operations for convolution as

COROLLARY. With notation as in Proposition 6, if a = 2, then the number of

operations needed to compute the convolution of two functions defined on S. is at

most

n!+ 2n!(nz – 1).

Using Vershik and Kerov’s result (5.1), more careful analysis of both I(n)

and T(n) may be carried out. For T(n) it may be shown that

THEOREM ([DIACONIS AND ROCKMORE, 1990] THEOREM 1). Let T(n) denote

the number of operations required to compute the Fourier transform over the

symmetric group S.. If a >2 denotes the exponent of matrix multiplication, then

n 2–(~11/2)

() ]+exp(–~c) ~ > (5.3)

where ~ = ( a – 2)/2 and c = 0.2312 and all error terms are uniform in a.

For I(n), a similar bound is now shown,

THEOREM 4. Let I(n) be the number of operations required to pe~orm Fourier

iru~ersion ouer the symmetric group S,,. If a >2 denotes the exponent of matrix

multiplication, then

where P = ( a — 2)/2 and c = 0.2312, and all error terms are uniform in a.

PROOF. Proposition 4 gives

I(n)<2~~ ~d:.

n!,,, =2 ‘! A+rr-1

58 D. N. ROCKMORE

This may be rewritten as

Explicit rewriting of (5.4) bounds l(n)\n ! by 2(n – 1)! 6 times

[

(7Z - I)a-’na-lexp(–c~JFT) +

(7Z – l)@exp(–c~~-)

(n – 2)a-1+

((n - 1)(72 - Z))pexp( –c~~-)

.Za-1

+ “.. +(n–l)!~ Iexp(–c~)

Split the sum at the term for m = n/2. Note that CY– 1 – /? = cr/2. Then

the first half is bounded by 2(rz – 1)! ~ times

Efficient Computation of Fourier Irw’ersion for Finite Groups

Meanwhile, the second half is bounded by

59

exp’-’(;)[;:;;:~]=exp(p’~’(;r”r2)+)(pn(2()

~ 2–(p,l/2)

SnPexp( -c/3) ~ .

Combining terms give the theorem. ❑

Note that this essentially gives a bound of

Thus, as a ~ 2+, ~ ~ O and Theorem 12 reduces to

Z(n) < O(nz(n!)).

Combined with the careful estimate for T(n) in (5.3) a more detailed bound

for the number of operations needed to compute the convolution of two

functions over S,l may now be given.

THEOREM 5. Let all notation be as in Theorem 4. Then the number of

operations needed to compute the convolution of two fimctions dejined on S,, is at

most

2(n!)B+l

2–(/.3n/2)

() ]+2exp(–c~) ~

+(n!)p+l exp(–c~fi).

PROOF. Again using (5.1), ZA, .df may be bounded by

~ d: = n!,~,, $df”zAbn

< n!(n!)Pexp(–c~&). (5.5)

Applying the estimates of (5.5), Theorem 4, and (5.3) to Proposition 5 yields the

theorem. ❑

Once more, as a ~ 2+, ~ ~ O so that the bound for convolutions goes to

O(nz(n!)).

Remark. Clausen [1989b] also studies in detail the problem of efficient

Fourier analysis on the symmetric group. To compute the Fourier transform,

the basic approach is similar to that found in [Diaconis and Rockmore, 1990]

but in addition, Clausen further takes advantage of the fact that in Young’s

orthogonal and seminormal forms, the representations of the coset representa-tives (j, n) for S,, _ ~ < S. can be factored as a product of a total of O(nz )

sparse matrices. The representations at the pairwise adj scent transpositions are

sparse. Consequently, Clausen derives ([1989b], Theorem 1.4) that the Fourier

transform on S. can be computed in at most 0(n3 “ n!) = 0( 1s,,llog3 IS,, 1)

operations.

60 D. N. ROCKMORE

Thus, a similar bound is then obtained by viewing the problem of Fourier

inversion as the computation of the product of an arbitrary input vector by the

inverse of the Wedderburn transform. Here, Clausen is able to take advantage

of the fact that the inverse is “almost” the transpose of the Wedderburn

transform. Consequently, similar advantages are obtained through the use of

sparse matrices. Hence, it can also be shown ([ Clausen, 1989bl, Theorem 1.6)

that Fourier inversion on S. may be performed in at most 0(1S. llog31S,l 1)

operations as well.

In more recent work with Baum, which appeared after this paper was

submitted [Baum and Clausen, 1991], Clausen exploits the close relationship

between the inverse of the Wedderburn transform and its transpose to obtain a

bound for the complexity of inversion in terms of the bound for the complexity

of the Fourier transform.

6. An Explicit Example

As an explicit illustration of the methods of Sections 4 and 5, the change of

basis matrices for reducing Fourier inversion on S~ to Fourier inversion on Sz

are constructed now. This will be done with respect to Young’s seminormal

form.

Briefly recall the “algorithm” given in Section 4. Let H be a subgroup of G

and q a fixed matrix representation of H in the vector space W with respect to

the basis {el, ..., e~}. (q T G) gives a representation of G in a vector space V

with respect to a basis in which the matrices may be written in some block

diagonal form. Call this representation p (it is equivalent to (q T G)). Then, V

has a decomposition into G-irreducible subspaces

v=v, @””” @~.

The block diagonal form of p means that matrices p(s) (for s = G) are given

with respect to a basis for V of

(fl>...>f:>f:>>f:r)..f:r)where {~,’} are basis for ~.

Viewed as an H-space ~ is denoted as l?es(~). There is now an additional

assumption on the basis {~,’} of Res( ~): there is a decomposition as H-in-

variant (not necessarily irreducible) subspaces

with distinct subsets of basis vectors {fit} spanning the various ~’ (i.e., thematrix has block diagonal structure). Furthermore the decomposition is such

that either ~’ contains no occurrences of q or it is identical to rjI. Then forevery ~’ which is identical to q the associated element in Hom~( W, Res(q))

is the identity matrix and for the others it is the zero matrix. From this, the

appropriate matrix in Honz~ ( W, Res( V )) may be constructed and finally the

matrix in Hom~(lnd(W), V).

Thus, consider the group S3 with subgroup Sz, the set of permutations that

fix 3. Sz has 2 irreducible representations, ~ (the trivial representation) and m

(the sign representation) corresponding to the partitions (2) and (1, 1), respec-tively. Both representations are one dimensional, so in either case let e denote

E#icient Computation of Fourier Inuersion for Finite Groups 61

the basis vector with respect to which the representation is given in Young’s

seminormal form.

The transpositions {(l), (2, 3), (1, 3)} are taken as coset representatives for

S~/Sz. Sq has three irreducible representations corresponding to the three

partitions of three, (3), (2, 1), and (1,1, 1). In Young’s seminormal form, the

corresponding representations of the coset representatives are given by

‘1 3

5:P(Z,1)(23) = 1

1 .—

\ 2

and

P(1,1,1)(23) = P(1,1,1)(13)

p(~,l)(13) =

–1.

Let A, be the change of basis used for (e ? S~) and AU the change of basis

for (~ ~ S~). Let ~ be the irreducible representation space for PA. (All

representations are taken to be in Young’s seminormal form.) By the 13ranch-

ing Theorem, ( e T S~) contains one copy of each of p(~j and P(z,, ~. Thus, let V

be the vector space space

and let { fO} span V(S) and {f ~, f~} span V(z, ~, with their union spanning V.

The Branching Theorem also gives

( P(s)$ s,) = ~

and

(P(2,1)$S2) = a@ e.

So that for m ● Sz

P(3) @ P(2,1)(~) =

C(m-) o 0

0 )(T(7T) o .

0 0 E(’i’f)

That is, Res(V~)) = ~2). Consequently, the associated element in

\Hom~s~2), lles(~~, ) is determined by the map sending

Consequently, the associated element in Hom~J~2), Res(~z, ~))) is determined

by the map sending

62 D. N. ROCKMORE

So that “gluing” the above two maps together, the associated element inllcvn~~~z), Res(V)) is determined by the map sending

Now the matrix in Honz~$ lnd( ~ ), V) is determined. Column 1 of A, (which

is the image of the basis vector 1 @ e or e) is

[11

0.

1

The rest of A, is now determined,

100

1 30

Zz

o 1 –;

and

——

Thus

1

0

0

1

0

1

0

1

5–

l–/

1 1

33

i –i

1 1— . —

2 2

1

0

1

—

Similarly, consider the construction of Am. By the Branching Theorem,

( o ~ S3) contains one copy each of P(l,,, I, and P(z,~). Thus, here let V denote

the vector space

v=ql, ,) f3q21).,

and let {~0} span v (1,1,J) and {~1, ~z} span ~z, 1, with their union spanning V.

63Eflicient Computation of Fourier Iru)ersion for Finite Groups

The Branching Theorem gives

(P(l,l,l)J~2) = @

and

(p(2>1)Js2) = ~@ ~.

For t-r = S2

[

a(m) o 0

P(l,l,l) @ P(2,1)f77-) = o

)

a(m) o .

0 0 e(m)

That is, l?es(~l, I,IJ) = ~1 ~). Consequently, the associated element in

~omslql,l)>~e~(ql,l,l) )) k determined by the map sending

e ~fo.

As previously,

Re~(J%l)) = ~1,1) @ ~2) = V. @ V,.

Thus, an element in Honzs$~l, ~), Res(~z, ~))) is determined by the map sending

e ~f,.

SO that the associated element in Horn~J~l, ~), Res(V)) is determined by themap sending

e~fo+fl.

Now the matrix in Horn~( Znd( J(W), V) is determined. Column 1 of Am is

[)1

1.

0

The rest of Am is constructed as before:

Aa((23) C3e) = ~(1,1,1J(23) o P(Z,1)(23) “~a(l @ e)

and

–1 o 0

1 30

31

0 1 –:

–1

1

T

1

e)

64

Hence.

D. N. ROCKMORE

Am =

\

L L

o 1 –1

7. Implementation and Final Remarks

The ideas of the previous sections readily admit implementation on a com-

puter. In this section an algorithm for S. is presented in an Algol-like language

accompanied by some comments on its features. This is essentially the “guts”

of a program currently being written in “C” for a SUN4 system.

The program below performs Fourier inversion for a function f defined on

S,, whose Fourier transform (with respect to the seminormal form) is stored in

memory. It is assumed that for all m between 1 and n the matrices p(i, m) for

all irreducible representations p of S~ and all i between 1 and m ((m, m) is

the identity) also are stored. The startup cost of generating these matrices is

discussed in [Diaconis and Rockmore, 1990].

In brief, the algorithm is as follows: For any m between 1 and n – 1 assume

that all the restricted transforms to Sm+ ~ have been computed. In one pass, for

every representation q of S., the change of basis matrices AT can be

computed directly from the matrices { p(i. m + 1)} where p runs over the

irreducible representations of S~ + ~. Then, as T runs over the representations

of Sm, the restricted transforms on S~ are recovered by conjugation of

appropriate (as determined by the branching theorem) matrices that are

directly constructed from the restricted transforms on S., + ~. The process is

now iterated on S~ _ ~. moving all the way down to SI whereupon f has been

recovered.

Note that upon recovering the restricted transforms on S~, neither the

change of basis matrices A ~, nor the restricted transforms on Sn +, are needed

again. Thus, they may be discarded. The restricted transforms require storage

for n! floating point numbers. The change of basis matrices require

~((m + l)d~)’ = (m + I) ’m!

T

floating point numbers be stored and the transpositions, an additional (m +1)

(m + 1)! storage locations. Thus, essentially, storage for 2n . n! floating point

numbers is needed. As the change of basis matrices are only needed one at a

time, they of course could be generated and discarded in turn.

In combination with the algorithms for computing Fourier transforms on S.(see [Iliaconis and Rockmore, 1990], Section 6, and also [Baum and Clausen,

1992] for a survey) as well as actual implementations of these ideas [Rockmore,

1988b; Baum and Clausen, 1992] the algorithm given here now allows actual

fast computation of convolutions on S,,.

In the algorithm that follows let ~~ be the set of words

1~ ={i~, i~_l,..., im+l Il<i, <j}.

For a c 1,. let f. = f,n, ,,m+,, where if f’ e L(S~), then f, ● L(S~_l) is de-

fined by L(n) = f((i, n)n) and inductively, for 1 s j s n – 1, ~,, = (~),.

Efficient Computation of Fourier Inversion for Finite Groups 65

It is assumed that the following subroutines exist:

1. branch(~, branches) —Given a representation q, branch returns in the

variable branches (which is a list or array of representations) the irreducible

representations that occur in (q T G).

2. change-of-basis(A, q, branches)—This takes the representation q and the

information in branches and returns in the matrix variable A, the change of

basis matrix Aq.

3. build-matrix(diag, branches, a) —Returns in the matrix variable diag the

block diagonal form of ~(q ~ G).

4. conjugate(induced, diag, A) —Returns in the matrix variable induced the

matrix A-l “ diag -A.5. store-transforms( induced, q, a )—Picks out and stores from the matrix

variable induced the blocks giving f=(q), ..., f=(q).

6. free-storage—Discards all old transforms and change of basis matrices.

Ideas for data structures for representations and file management for

working with the symmetric group may be found in [Rockmore, 1988].

BEGINFOR m = n DOWNTO 2 DO

BEGINFOR all irreducible representations q of S,n , DO

BEGIN

branch(q, branches);change-of-basis( A, q, branches);FOR a 6 I. DO

BEGINbuild-matrix(diag, branches, a );conjugate(induced, diag, ,4);store-transforms( induced, q, a);

ENDEND

free-storage;END

END

NOTE ADDEDIN PROOF: Over the course of this paper’s review, there has been much newwork related to the development of generalized fast Fourier transforms. A good source forrecent work in FFT’s for finite groups is the new book of Clausen and Baum [1993]. IrI thecontext of an application to statistics, Driscoll et al. [1989] have developed fast algorithmsfor computing discrete polynomial transforms for orthogonal polynomial sets. In severalparticular cases, these algorithms have been implemented and techniques have beendeveloped for increasing numerical reliability at little cost in efficiency [Moore et al., 1993].

In the continuous setting, improved spherical convolution algorithms have recently beenobtained and implemented [Healy et al., 1993]. Computer vision, image processing, andclimate modeling are a few of the areas in which these algorithms may have applications (cf.the discussion and references in [Driscoll and Healy, to appear]). In further work, Maslen[1993] has built on the work of Driscoll and Healy and determined sampling theorems andfast algorithms for compact groups and their quotients.

ACKNOWLEDGMENTS. I would like to thank my thesis adviser, Persi Diaconis,for his encouragement and for reading an early draft of this work. I would also

like to thank Joseph Bernstein for his helpful conversations regarding the

change of basis constructions. Lastly, I would like to thank both referees for

their careful reading and helpful comments.

66 D. N. ROCKMORE

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RECEJVEDJUNE 1989; REVISED JUNE 1992: ACCEPTED JULY 1992

Jcoumal of the A.wu&mn for Computmg Mx-hinerv, Vol 41 No 1, l.muuy 1994