Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials ()
1. Preliminaries
Consider a finite group
, typically non-abelian, and let 
and 
be two functions in
, the finite Hilbert space of all complex valued functions (usual inner product
) for which elements of 
form the (standard) basis. We assume that this basis 
is ordered and make the natural identification, as vector spaces, with
, where
.
A 
-convolution of 
and 
is defined by the following action, 
Definition. A function 
is called a multiplicative character if
In the cyclic case multiplicative characters are eigenfunctions of the convolution operator and we have 
multiplicative characters, the Fourier complex exponentials, for example, see [1] for more details. The main problem with the non-abelian group, as opposed to the abelian one, is the lack of multiplicative characters. Multiplicative characters for any group 
are constant on its conjugacy classes.
Definition. A finite dimensional representation of a finite group 
is a group homomorphism
where 
denotes the general linear group of degree
, the set of all 
invertible matrices. We refer to 
as the degree of the group representation. The field of complex numbers is denoted by
.
Definition. Two group representations
are said to be equivalent if there exists an invertible matrix 
such that
for all
.
Every finite dimensional group representation is equivalent to a representation by unitary matrices. For more information on group representations see [2] for example.
Definition. Let 
be the set of all (equivalence classes) of irreducible representations of the group
. Let 
be of degree 
and let
. Then the Fourier transform of 
at 
is the 
matrix
The Fourier inversion formula, 
, is given by
We alert the reader to an involution switch 
in the summand functions. We refer the reader to [3] for more details. Let 
be the algebra of complex valued functions on 
with respect to 
-convolution. Let 
and identify the function 
with its symbol
Let 
and 
be two elements in
. We have a natural identification
understood with respect to the induced group algebra multiplication. We have a non-abelian version of the classical 
transform. The action of 
on 
through 
-convolution is captured by the matrix multiplication by the 
-circulant matrix
, in particular
The character of a group representation 
is the complex valued function
defined by
For all 
the quantity 
is a sum of complex roots of unity. Moreover, we have 
for all
. A character is called irreducible if the underlying group representation is irreducible. We define an inner product on the space of class functions, functions on 
that are constant its conjugacy classes
Note that a character is a class function. We have as many irreducible characters as there are conjugacy classes of
. If 
is abelian, then we have 
irreducible characters. With respect to the usual inner product we have
where 
is the Kronecker delta. Irreducible characters form a basis for the space of class functions on
.
Definition. Let
. The adjoint of
, denoted by
, in the group algebra 
is the element
Associate 
and 
with the corresponding functions 
and
. We collect a few simple facts. Let 
be the adjoint of the 
-circulant matrix
. Then we have
. The matrix 
is normal if and only if 
and selfadjoint if and only if
.
The Fourier transform gives us a natural isomorphism
where
with
. A typical element of 
is a complex valued function
and the typical element of 
is the direct sum of Fourier transforms
Fourier transform turns convolution into (matrix) multiplication
In the abelian setting the Fourier transform is a unitary linear transformation (proper scaling required). In the non-abelian setting we recapture this property if we define the right inner product on the space
. We will provide more details on this later on. Let 
and define for 
Note
. We are able to decompose a function 
into a sum of 
functions which is the number of conjugacy classes of
.
Every group 
admits a trivial irreducible representation 
for which 
for all
. For 
the Fourier transform of 
at 
is given by
and
Thus the constant mean function is always represented in our decomposition. The decomposition 
must be orthogonal. The following can be, for example, found in [4] . The notation 
refers to the Frobenius norm.
Proposition 1.1. Let 
be given as above. We have
Corollary 1.1. Let 
then
Equip the space 
with the following inner product. Let 
and
. Then
whete 
denotes the adjoint of
.
Corollary 1.2. The Fourier transform is a unitary transformation from 
onto
.
For more information of non-abelian Fourier transform see the works of [5] -[11] , for example.
2. Main Results
Consider for a moment
, where 
is a cyclic group of size
. Given a vector
, we can writ
where the Fourier complex exponentials 
are orthonormal vectors and 
are the Fourier coefficients. Being multiplicative characters for
, these functions 
are 
-decorrelated, in particular, for
,
This important property makes the Fourier exponentials vital in signal analysis. The need for time shift de-correlation or spatial shift de-correlation is reflected in the cyclic group structure of
.
We extend these observations to non-abelian groups
, recall that finite abelian groups are direct sums of cyclic groups. We say that two vectors
, 
in 
are 
-decorrelated if
We observe that even in the non-abelian case the linearly independent multiplicative characters are G-decorrelated as the following simple observation reveals
as linearly independent multiplicative characters are orthogonal.
Definition. For given vectors 
the 
cross-correlation function is defined by
Note that 
and 
are 
-decorrelated if and only if
. Recall we view the following three objects
, 
and 
as isomorphic vector spaces.
Lemma 2.1. Consider 
and the corresponding
. Then we have
Proof: We have
where
.
Corollary 2.1. Let 
be the Fourier transforms of 
and 
respectively. Then we have
Thus functions 
are 
-decorrelated if and only if
Corollary 2.2. Let
. Then 
and 
are 
-decorrelated if and only if
Observe that if 
is a multiplicative character then
However, these are not the only functions with this property, i.e. 
is a multiple of
. In fact, we have the following, note that 
below could be complex.
Lemma 2.2. Let
. Then
if and only if
with
, where 
is a projection matrix for all
. Note 
in independent of
, but 
can depend on
.
Proof: Using Corollary 2.1, the function 
has the property 
if and only if
for all
. Observe that, as a result, the matrix 
has to be normal. Therefore, we can orthogonally diagonalize 
with the diagonal matrix
. Now the above matrix equality translates to the following
which forces all the non-zero diagonal entries of 
to be the same. This is exactly the claim that 
is a multiple of some projection matrix 
that could depend of
.
We say a set of functions 
in 
is 
-orthogonal if
where 
is the Kronecker delta and
.
Theorem 2.1. Let 
and 
be the set of all irreducible characters of
. Then we have
where
and the set of functions 
is 
-decorrelated. If 
is a multiplicative character, 
, then
Note that if 
and 
are real valued then so is the corresponding
. Moreover, the set 
is G-orthogonal.
Proof: Recall
Now define an (orthogonal) projection 
on 
by the following action, 
The action of the linear operator 
in the Fourier domain is given by the (matrix) multiplication by the vector
where the 
idenity matrix 
is in the jth position. The inverse Fourier transform of this vector is the function (evaluated at
)
Therefore for all 
we have
where 
is the (inverted) character of the irreducible representation
. Now Proposition 1.1, Lemma 2.1 and Corollary 2.1 can be used to show that 
and 
are 
-decorrelated. Using Lemma 2.2 we conclude that the set 
is 
-orthogonal.
It is important to note that if 
is not a multiplicative character, then
in general.
In order to obtain the above 
-decorrelated decomposition one does not need to know explicitly the irreducible group representations, just the irreducible characters for the group
. For any group 
these (irreducible) characters are much easier to find than the corresponding irreducible group representations. This alone makes the above decomposition amicable for applications. Also note that in a case of multiplicative character
, the corresponding decomposition function 
is a multiple of the (inverted) character
.
However, in the case of (irreducible) character 
stemming from a higher dimensional irreducible representation, this is no longer the case. The intuitive interpretation of the function 
then becomes more difficult.
Corollary 2.3. Let 
be a subset of
. Then
where
In the cyclic case we can talk about frequencies in the context of the Fourier complex exponentials. As a result, we can design filters, that can isolate specific frequencies and block others. In the non-abelian case this becomes less clear as the concept of frequencies is lost in the irreducible characters.
We can go further and obtain a 
-decorrelated decomposition of any function 
that consists of 
summands. Moreover, this 
-decorrelated decomposition is obtained by orthogonal projections. However, the drawback is that we have to know the irreducible representations of the group 
and not just the irreducible characters.
Theorem 2.2. Let 
and let 
be the set of all irreducible representations of
, 
has size
. Let 
be the 
entry in the 
matrix of
. Consider the (involuted) function
. Then 
can be written as a 
-decorrelated sum of vectors 
where
where
Moreover, the (diagonal) set of functions 
is 
-orthogonal.
Proof: We invoke the Schur’s orthogonality relations, see [12] , for example. With notation as in Theorem 2.1, we observe, using the Schur’s orthogonality relations
and conclude, using Proposition 1.1, the functions
form an orthonormal basis for
. Therefore, we have
Note that
where 
is a matrix whose entries are all zero except
. Therefore
unless 
and
. Now using Lemma 2.2 we conclude that the (diagonal) set of functions 
is 
-orthogonal.
Note the (non-diagonal) set of functions.
is not necessarily 
-orthogonal. Also, unlike the irreducible characters, we have 
in general.
3. Example: The Symmetric Group S3
We will consider the symmetric group 
in our example. The group 
consists of elements
The group 
has three conjugacy classes
We have three irreducible representations, two of which are one dimensional, 
is the identity map, 
is the map that assigns the value of 1 if the permutation is even and the value of 
if the permutation is odd. Finally, we have
, the two dimensional irreducible representation of
, defined by the following assignment
The irreducible characters of 
are given by
where 
and 
are also multiplicative characters. Moreover, we have
Observe that all three irreducible characters are real valued and hence all the decomposition functions 
are also real valued if 
is real valued as well. The 
-convolution by a function 
can be induced by a 
-circulant matrix 
given by
and specifically, note that
,
Set 
and we obtain
4. Applications to Crossover Designs in Clinical Trials
The application of non-abelian Fourier analysis has been studied extensively; we refer the reader to the works of [13] for example. However, we believe that the property of 
-decorrelation among functions in 
has to be further investigated. We have to capture a natural scenario where the underlying group structure 
is relevant to the corresponding 
-decorrelation. One of the places this does appears is the crossover designs in clinical trials, in particular the William’s 
design with 3 treatments.
During a crossover trial each patient receives more than one treatments in a pre-specified sequence. Therefore, as a result, each subject acts as his or her own control. Each treatment is administered for a pre-selected time period. A so called washout period is established between the last administration of one treatment and the first administration of the next treatment. In this manner the effect of the preceding treatment should wear off, at least in principle. Still there will be some carry-over effects in all the specified treatment sequences, clearly starting with the second treatment. For more information on crossover designs in clinical trials see [14] or [15] for example.
In our example, we record the sum of all carry-over effects of the treatments in any given treatment sequence. We will follow the William’s 
design with 3 treatments
, 
and
. In particular we have 6 treatment sequences
, 
, 
, 
, 
,
. For example, suppose the order of treatment administration is
, with 
first. We decide to collect the sum of all carry-over effects of the treatments in this sequence (starting with the second one),
. We observe the sequence 
as a permutation of the sequence 
by the permutation
, an element of the group
. Thus we can write
. Similarly, a permutation sequence 
would result in
.
It is here where we can capture the essence of 
-decorrelation. We can start with some initial treatment order say 
and then administer crossover designs involving all 6 treatment permutations, denoted by
. Similarly, we could have started with a different initial combination of treatments, say 
and then administer all 6 treatment permutations, denoted by
. Now it is natural to request for the data sequences 
and 
to be 
-decorrelated, meaning that our data sequences are decorrelated even when we allow the initial treatment permutation to vary.
Let us be specific and give a hypothetical example. Suppose we obtain a carryover sequence
, with the order of the elements respecting the group structure. Assume that the 
values refer to the sums of all carryover effects of the treatments in the given sequence. For example 
and
. We now wish 
-decorrelate the vector 
over the group
. We obtain
Observe
, 
and
. Observe that the function 
is a multiple of the multiplicative character
, and similarly, the function 
is a multiple of the multiplicative character
. However, the function 
is not a multiple of the (irreducible) character
, recall, the (irreducible) character 
is not a multiplicative character.
In the 
-decorrelated sum the function 
represents the carry-over effect from the administration of the three treatments
, 
and
, reflecting the permutation independence in all of the carry-over effects from all 6 permutation options. The function 
reflects the sign permutation dependence, meaning how sensitive the carry-over effects are to switche from permuting two treatments versus three treatments. The interpretation of the function 
is more involved, the best is to view 
as
.
Let us now decompose the function 
further into a 
-decorrelated sum. Now we have to use the irreducible representations of 
themselves, in particular the two dimensional irreducible representation
. We obtain the following
As complex decompositions have little interpretation in our context, we can write a decomposition of 
with two vectors, 
and
,
