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Let *G* be a non-abelian group and let l^{2}*(G)* 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 *l*^{2}*(G)*. These *G*-decorrelated 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 6×3 design with 3 treatments. In our example, the underlying group is the symmetric group *S*_{3}.

Consider a finite group

A

Definition. A function

In the cyclic case multiplicative characters are eigenfunctions of the convolution operator and we have

Definition. A finite dimensional representation of a finite group

where

Definition. Two group representations

are said to be equivalent if there exists an invertible matrix

for all

Every finite dimensional group representation is equivalent to a representation by unitary matrices. For more information on group representations see [

Definition. Let

The Fourier inversion formula,

We alert the reader to an involution switch

Let

understood with respect to the induced group algebra multiplication. We have a non-abelian version of the classical

The character of a group representation

defined by

For all

Note that a character is a class function. We have as many irreducible characters as there are conjugacy classes of

where

Definition. Let

Associate

The Fourier transform gives us a natural isomorphism

where

with

and the typical element of

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

Note

Every group

and

Thus the constant mean function is always represented in our decomposition. The decomposition

Proposition 1.1. Let

Corollary 1.1. Let

Equip the space

whete

Corollary 1.2. The Fourier transform is a unitary transformation from

For more information of non-abelian Fourier transform see the works of [

Consider for a moment

where the Fourier complex exponentials

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

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

Note that

Lemma 2.1. Consider

Proof: We have

where

Corollary 2.1. Let

Thus functions

Corollary 2.2. Let

Observe that if

However, these are not the only functions with this property, i.e.

Lemma 2.2. Let

if and only if

with

Proof: Using Corollary 2.1, the function

for all

which forces all the non-zero diagonal entries of

We say a set of functions

where

Theorem 2.1. Let

where

and the set of functions

Note that if

Proof: Recall

Now define an (orthogonal) projection

The action of the linear operator

where the

Therefore for all

where

It is important to note that if

in general.

In order to obtain the above

However, in the case of (irreducible) character

Corollary 2.3. Let

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

Theorem 2.2. Let

where

Moreover, the (diagonal) set of functions

Proof: We invoke the Schur’s orthogonality relations, see [

and conclude, using Proposition 1.1, the functions

form an orthonormal basis for

Note that

where

unless

Note the (non-diagonal) set of functions.

We will consider the symmetric group

The group

We have three irreducible representations, two of which are one dimensional,

The irreducible characters of

where

Observe that all three irreducible characters are real valued and hence all the decomposition functions

and specifically, note that

Set

The application of non-abelian Fourier analysis has been studied extensively; we refer the reader to the works of [

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 [

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

It is here where we can capture the essence of

Let us be specific and give a hypothetical example. Suppose we obtain a carryover sequence

Observe

In the

Let us now decompose the function

As complex decompositions have little interpretation in our context, we can write a decomposition of