ABSTRACT

Let G be a non-abelian group and let be a finite dimensional Hilbert space of all complex valued functions for which the elements of G form the (standard) orthonormal basis. In our paper we prove results concerning G-decorrelated decompositions of functions in. These Gdecorrelated decompositions are obtained using the G-convolution either by the irreducible characters of the group G or by an orthogonal projection onto the matrix entries of the irreducible representations of the group G. Applications of these G-decorrelated decompositions are given to crossover designs in clinical trials, in particular the William’s design with 3 treatments. In our example, the underlying group is the symmetric group.

Keywords:Non-Abelian Fourier Transform, Group Algebra, Irreducible Representation, Irreducible Character, G-Circulant Matrix, G-Decorrelated Decomposition, 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,

References