Thangarajah & Zizler, Cogent Mathematics (2017), 4: 1313926
http://dx.doi.org/10.1080/23311835.2017.1313926

APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE

On wide sense stationary processes over finite
non-abelian groups
Pamini Thangarajah1* and Peter Zizler1
Received: 25 October 2016
Accepted: 26 March 2017
First Published: 05 April 2017
*Corresponding author: Pamini
Thangarajah, Department of
Mathematics and Computing, Mount
Royal University 4825, Mount Royal Gate
SW, Calgary, Alberta, Canada
E-mail: pthangarajah@mtroyal.ca
Reviewing editor:
Lei-Hong Zhang, Shanghai University of
Finance and Economics, China
Additional information is available at
the end of the article

Abstract: Let X be a real-valued wide sense stationary process over a finite nonabelian group G. We provide results on optimal orthogonal decomposition of X into
real-valued mutually orthogonal components and using this decomposition we
develop a test for correlation of X over the group G. Applications of these results to
the analysis of variance of the carry-over effects in the cross-over designs in clinical
studies are given. Our focus will be on groups S3 , S4, and A4.
Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Algebra; Pure
Mathematics; Applied Mathematics; Mathematical Modeling
Keywords: symmetric group; alternating group; non-abelian Fourier transform; wide sense
stationary process; irreducible characters; orthogonal decomposition
AMS (MOS) subject classifications: 20C30; 62M15; 62M99
1. Introduction
Applications of group representations to probability and statistics is a rich subject with Diaconis (1998)
as an excellent reference. In this paper, we will study some aspects of the spectral theory where the
underlying group G is finite non-abelian. Please see Giannakis (1999) for abelian case. In particular, we
will consider wide sense stationary processes over the group G. We refer the reader to Peccati and
Pycke (2005) for material on stochastic processes over non-abelian groups. We will consider finite nonabelian groups, provide simplified proofs of the relevant results in the finite setting, and give results on
the optimality of the orthogonal decompositions into real components. We also provide a classification
on the case of X being de-correlated over G. Applications of these results to carry-over effects in the
cross-over designs in clinical studies will be given later. Our focus will be on groups S3 , S4, and A4.

ABOUT THE AUTHORS

PUBLIC INTEREST STATEMENT

Pamini Thangarajah is an associate professor at
the Department of Mathematics and Computing,
Mount Royal University, Calgary, Alberta, Canada.
She has developed many mathematics courses
and two minor programs. She enjoys working with
students on undergraduate research, as well as
presenting at conferences. Her research interests
include algebra, representation theory, invariant
theory, and data envelopment analysis.
Peter Zizler obtained his PhD from the University
of Calgary in 1995. Currently, an associate
professor of mathematics at Mount Royal
University in Calgary, his research interests include
linear algebra and various projects in applied
mathematics and statistics. His spare time is
taken up by the hockey commitments of his two
young sons.

Applications of group representations to probability
and statistics is a rich subject. Our paper connects
the representation theory with the study of
stochastic processes over non-commutative
groups. The main results are applied to carry-over
effects in the cross-over designs in clinical studies.

© 2017 The Author(s). This open access article is distributed under a Creative Commons
Attribution (CC-BY) 4.0 license.

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Thangarajah & Zizler, Cogent Mathematics (2017), 4: 1313926
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Let G denote a finite non-abelian group and let X denote a zero mean real-valued wide sense stationary process (WSS) over the group G. In particular, we have E(X(t)) = 0 for all t ∈ G. The auto−1
correlation function RX of X is defined as follows, set 𝜏 = t s, where t, s ∈ G

RX (t, s) = E(X(t)X(s))
= RX (𝜏).
Suppose X and Y are two zero mean WSS processes over G. We define the cross-correlation function
of X and Y as follows:

RXY (𝜏) = E(X(t)Y(t𝜏)).
r

Let {Xi }i=1 denote a family of zero mean WSS processes over G. Such a family is said to be mutually
orthogonal if RX X (𝜏) = 0 for all 𝜏 ∈ G and i ≠ j. In such case, we refer to the WSS process over G as
i j
white. For more information of random processes in general, we refer the reader to Bartoszynski and
Niewiadomska-Bugaj (2008).
A natural place where WSS processes over non-abelian groups arise are the cross-over designs in
clinical trials. We will refer to this later on in this paper.
Non-abelian Fourier Transform. Let ℂ denote the n-dimensional vector space over the complex
n
numbers. The standard basis for ℂ is identified with the ordered group elements of G, where |G| = n.
n

A finite dimensional representation of a finite group G over ℂ is a group homomorphism

𝜌:G ↦ GL(dj , ℂ),
where GL(dj , ℂ) denotes the general linear group, the set of all dj × dj invertible matrices. We refer
to dj as the degree of the group representation.
Two group representations

𝜌1 :G ↦ GL(dj , ℂ) and 𝜌2 :G ↦ GL(dj , ℂ)
are said to be equivalent if there exists an dj × dj invertible matrix T such that

T◦𝜌1 (g)◦T −1 = 𝜌2 (g)
for all g ∈ G.
An irreducible group representation of G is a group representation 𝜌 of G, for which there is no nonj
trivial subspace W of ℂ for which

𝜌(g)W ⊂ W
for all g ∈ G.
Let ℂ[G] be the algebra of complex-valued functions on G with respect to G-convolution. Let
𝜓 = (c0 , c1 , … , cn−1 ) ∈ ℂn and identify the function 𝜓 with its symbol

Ψ = c0 1 + c1 g1 + ⋯ cn−1 gn−1 ∈ ℂ[G], where G = {g0 = 1, g1 , … , gn−1 }.
A G-convolution of 𝜓 and 𝜙 is defined by the following action, 𝜎 ∈ G

(𝜓 ∗ 𝜙)(𝜎) =

∑

𝜓(𝜎𝜏 −1 )𝜙(𝜏).

𝜏∈G

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̂ be the set of all (equivalence classes) irreducible representations of the group G. WLOG we
Let G
can assume these representations are unitary (please see Hazewinkel (2001) for further details). Let
̂ be of degree d and let 𝜙 ∈ ℂn. Then, the Fourier transform of 𝜙 at 𝜌 is the d × d matrix
𝜌∈G
j
j
j
̂ =
𝜙(𝜌)

∑

𝜙(s)𝜌(s).

s∈G

The Fourier inversion formula, s ∈ G, is given by

𝜙(s) =

(
)
1 ∑
̂ ) .
dj tr 𝜌j (s−1 )𝜙(𝜌
j
|G| ̂
𝜌j ∈G

Observe the the switch s → s in the above functions. We refer the reader to Stankovic, Radomir,
Moraga, and Astola (2005) for further reading on this subject.
−1

n

Let 𝜓 and 𝜙 be two elements in ℂ . We have a natural identification

𝜓 ∗ 𝜙 ↦ ΨΦ.
The action of 𝜓 on 𝜙 via G-convolution is delivered by the matrix multiplication by the G-circulant
matrix CG (𝜓)

𝜓 ∗ 𝜙 = CG (𝜓)𝜙.
Definition 1
R𝜓, 𝜙 (𝜏) =

∑

For given vectors 𝜓, 𝜙 ∈ l2 (G) the G cross-correlation function is defined by
𝜓(t)𝜙(t𝜏).

t∈G

We have, see Zizler (2014),
R𝜓, 𝜙 = Ψ∗ Φ = CG∗ (𝜓)𝜙.

The character of a group representation 𝜌 is the complex-valued function
𝜒:G → ℂ

defined by
𝜒(g) = tr(𝜌(g)), g ∈ G.

We call a character irreducible if the underlying group representation is irreducible. We define an
inner product on the space of class functions, functions on G that are constant on its conjugacy
classes

⟨𝜒, 𝜃⟩ =

1 �
𝜒(g)𝜃(g).
�G� g∈G

A character is a class function. There are as many irreducible characters as there are conjugacy
classes of G(please see Dummit (1999), p. 870 for details). Let r denote the number of conjugacy
classes of G and we have r irreducible characters {𝜒1 , 𝜒2 , … , 𝜒r } for the group G. We have, with
respect to the usual inner product,

⟨𝜒i , 𝜒j ⟩ = 𝛿ij , for all i, j ∈ {1, 2, … , r}
where 𝛿i, j is the Kronecker delta. The set of all irreducible characters form a basis for the space of
class functions on G, see Dummit (1999) for more details.
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The Fourier transform gives us a natural isomorphism

̂
ℂ[G] ⇒ M(G),
where

� =M
M(G)
d ×d (ℂ) ⊕ Md ×d (ℂ) ⊕ ⋯ ⊕ Md ×d (ℂ),
1

1

2

2

r

r

2
2
2
n
with d1 + d2 + ⋯ + dr = n. A typical element of ℂ is a complex-valued function

𝜓 = (c0 , c1 , … , cn−1 )
̂ is the direct sum of Fourier transforms
and the typical element of M(G)
� ) ⊕ 𝜙(𝜌
� ) ⊕ ⋯ ⊕ 𝜙(𝜌
� ).
𝜙(𝜌
1
2
r
Fourier transform turns convolution into (matrix) multiplication

̂
∗𝜙=
𝜓

r
⨁

̂
𝜓
̂j 𝜙̂j = 𝜓
̂ 𝜙.

j=1

̂ with the following inner product. Let 𝐯 = 𝜙(𝜌
� ) ⊕ 𝜙(𝜌
� ) ⊕ ⋯ ⊕ 𝜙(𝜌
� ) and
Equip the space M(G)
1
2
r
𝐰 = 𝜁�(𝜌1 ) ⊕ 𝜁�(𝜌2 ) ⊕ ⋯ ⊕ 𝜁�(𝜌r ).
Then

�
� d
�
�
�
�
d
̂ )𝜁̂∗ (𝜌 ) + 2 tr 𝜙(𝜌
̂ )𝜁̂∗ (𝜌 ) + ⋯ + r tr 𝜙(𝜌
̂ )𝜁̂∗ (𝜌 )
tr 𝜙(𝜌
1
1
2
2
r
r
�G�
�G�
�G�
∗
whete 𝜁̂ (𝜌) denotes the adjoint of 𝜁̂(𝜌).

⟨𝐯 ∙F 𝐰⟩ =

d1

n

Let 𝜙 ∈ ℂ and define for s ∈ G

𝜙j (s) =

dj
|G|

(
)
̂ ) .
tr 𝜌j (s−1 )𝜙(𝜌
j
∑r

Note 𝜙 = j=1 𝜙j. We are able to decompose the function 𝜙 into a sum of r functions which is the
number of conjugacy classes of G.
n

We define an (orthogonal) projection Pj on ℂ by the following action, 𝜙 ∈ ℂ

n

Pj (𝜙) = 𝜙j .
The action of the linear operator Pj in the Fourier domain is given by the (matrix) multiplication by
the vector

0 ⊕ ⋯ ⊕ 0 ⊕ 𝐈j ⊕ 0 ⊕ ⋯ ⊕ 0,
where the dj × dj identity matrix 𝐈j is in the jth position. The inverse Fourier transform of this vector
is the function (evaluated at g ∈ G)

dj
|G|

tr(𝜌j (g−1 )) =

dj
|G|

𝜒j (g−1 ) =

dj
|G|

𝜒j (g).

n

Therefore, for all 𝜙 ∈ ℂ , we have

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Pj (𝜙) =

dj
|G|

𝜒̃j ∗ 𝜙,

where 𝜒̃j (g) = tr(𝜌j (g )) is the (inverted) character of the irreducible representation 𝜌j. Let 𝜌j (k, l)(s)
−1
be the (k, l) entry in the dj × dj matrix 𝜌j (s) and consider the function defined by f (s) = 𝜌j (k, l)(s ),
s ∈ G. Observe the image and the kernel of Pj are given by
−1

Im(Pj ) = span{𝜌j (k, l) | k, l ∈ 1, … , dj }
and

Ker(Pj ) = Im(Pj )⟂ = span{𝜌i (k, l) | i ≠ j and k, l ∈ 1, … , di }.
(
)
2
Note dim Im(Pj ) = dj . Moreover, the functions
⎧�
⎫
dj
⎪
⎪
𝜌j (k, l) � k, l ∈ 1, … , dj ⎬
⎨
�G�
⎪
⎪
⎩
⎭
form an orthonormal basis for Im(Pj ). Also note Im(Pj ) ⟂ Im(Pi ) for i ≠ j. We also have

𝜒̃j ∗ 𝜙 =

�
⟨𝜙, 𝜌j (k, l)⟩𝜌j (k, l).
k, l

The important observation here is the fact that the space Im(Pj ) is invariant under the group circulant matrix C = CG (𝜓) and is also 𝜓 independent. We have an orthonormal basis for each Im(Pj ) ,
n
and thus for ℂ , but these vectors no longer need to be eigenvectors for the group circulant matrix
C = CG (𝜓). Still, this orthonormal basis is 𝜓 independent. For more information we refer the reader
to Zizler (2013). For more details on group representations, we refer the reader to Dummit (1999).
We refer the reader to Stankovic et al. (2005) or An and Tolimieri (2003) for more material on the
non-abelian Fourier transform and its applications.

2. Main results
We need the orthogonal components {Xj } to be real valued for real-life applications. An element of
a group G is said to be real if it is conjugate to its inverse. Recall that two elements a and b of a group
−1
G are said to be conjugate if there exists c ∈ G such that c ac = b. A conjugacy class is said to be
real if it has a real element. Note that if a conjugacy class has a real element then all the elements
of this conjugacy class are real. It is a known result, see James and Liebeck (1993), for example, that
the number of real irreducible characters of a group G is equal to the number of real conjugacy
classes of G. Therefore, a group G has all irreducible characters real if and only if all the elements of
that group are real. Therefore, the symmetric group Sn has all the irreducible characters real as all of
its elements are real. However, this is no longer true for the alternating groups.
r

Recall r denotes the number of conjugacy classes for G and {𝜒̃j }j=1 denotes the set of all the (inverted) irreducible characters of G. Let dj denote the degree of the irreducible representation 𝜌j.
Define a vector valued zero mean WSS process over G as

X = (X(g0 ), X(g1 ), … , X(gn−1 ))T ,
where G = {g0 , g1 , … , gn−1 } and each X(gi ) is a zero mean WSS process. Consider a decomposition
∑p
of X = j=1 Xj where {Xj } is a mutually orthogonal set of zero mean real-valued WSS processes. We
say that the value p is optimal if it is impossible to decompose X into more zero mean real-valued
mutually orthogonal WSS processes. Here, of course, the value p should be independent of X. Thus
this decomposition is requested for all X and is only group G dependent. Note that a specific process
X could potentially be decomposed into more mutually orthogonal components.
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Theorem 1 Let X denote a zero mean real-valued WSS process over the group G. Let p = r+s
where s is
2
the number of real conjugacy classes of G. Then, we can write

X=

p
∑

Xj ,

j=1

where {Xj }rj=1 is a mutually orthogonal set of zero mean real-valued WSS processes and the value p
is optimal. Here, we have
Xj =

dj
|G|

dj

𝜒̃j ∗ X =

|G|

CG (𝜒̃j )X if𝜒j is a real

and
Xj =

dj (

)
dj
𝜒j + 𝜒̃j ∗ X =
C (𝜒 + 𝜒̃j )X if 𝜒j is a complex.
|G|
|G| G j

Moreover, we have
E(X(t)X ∗ (s)) =

p
∑

E(Xj (t)Xj∗ (s))

j=1

for all t ∈ G.
Proof Define the Fourier transforms of the two zero mean WSS processes X and Y evaluated at the
irreducible representation 𝜌j
( ) ∑
( )
∑
̂ 𝜌 =
̂ 𝜌 )=
X
X(t)𝜌j (t); Y(
Y(t)𝜌j (t).
j
j
t∈G

t∈G

Consider the cross-correlation function RXY and let  denote the Fourier transform operator. Define
Cj =  (RXY )(𝜌j ).

We have
( )
∑
̂ 𝜌 Y ∗ (s)) =
E(X(t)Y ∗ (s))𝜌j (t)
E(X
j
t

=

∑

RXY (ts−1 )𝜌j (t)

t

=

∑

RXY (𝜏)𝜌j (𝜏s)

t

= Cj 𝜌j (s)

we obtain
( ) ( )
( )
∑
̂∗ 𝜌 ) =
̂ 𝜌 Y
̂ 𝜌 Y ∗ (s))𝜌 (s−1 )
E(X
E(X
j
j
j
j
s

=

∑

Cj 𝜌j (s)𝜌j (s−1 )

s

=

∑

Cj

s

= |G|Cj .

Thus
Cj =

( ) ( )
1 ̂
̂∗ 𝜌 ).
E(X 𝜌j Y
j
|G|

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Define
dj

Xj =

|G|

𝜒̃j ∗ X =

dj
|G|

CG (𝜒̃j )X; Yj =

dj
|G|

𝜒̃j ∗ Y =

dj
|G|

CG (𝜒̃j )Y.

Note, here the components Xj can be complex valued. Set Z = X −
correlation function for Z. Observe for any j ∈ {1, 2, … , r}
( ( ) ( ))
̂ 𝜌 X
̂ 𝜌
E X
= 0 if i1 ≠ i2 ,
i
j
i
j
1

∑r

i=1 Xi and let RZ denote the auto-

2

and consider, for any j ∈ {1, 2, … , r}
( ( ( ) ( )))
( )
1
̂ 𝜌 Z
̂∗ 𝜌
E Z
 RZ (𝜌j ) =
j
j
|G|
)
(
r
r
( ( ) ( ))
( ( ) ( ))
( ( ) ( ))
∑
∑
1
∗
∗
∗
̂
̂
̂
̂
̂
̂
E Xi 𝜌j Xi 𝜌j
E X 𝜌j Xi 𝜌j
E X 𝜌j X 𝜌j
+
−2
=
2
1
|G|
i1 , i2 =1
i=1
( ( ( ) ( ))
( ( ) ( ))
( ( ) ( )))
1
̂ 𝜌 X
̂∗ 𝜌
̂ 𝜌 X
̂∗ 𝜌
̂ 𝜌 X
̂∗ 𝜌
=
E X
− 2E X
+E X
j
j
j
j
j
j
|G|
= 0.

Now we have
(
)( )
 R X X 𝜌n = 0
i j

for all n = {1, 2, … , r} as long as i ≠ j. Thus, we have, for i ≠ j, RX X (𝜏) = 0 for all 𝜏 ∈ G. In this manner,
i j
∑r
we have a decomposition into potentially complex components. We have X = j=1 Xj as zero mean
WSS processes over G.
To analyze the real case, we make a key observation. If 𝜒 is an irreducible character of G then so
is it conjugate. Note that 𝜒(g) = 𝜒(g)
̃
= 𝜒(g−1 ) = 𝜒(g). Consider the complex mutually orthogonal
decomposition as above
X=

r
∑

Xj .

j=1

If 𝜒j is a real-valued irreducible character we set Yj = Xj in the sum. If 𝜒j is complex valued, we pair it
1
up with the irreducible representation 𝜒j which is another irreducible character in the list of irreducible characters of G, say 𝜒j . In this case we set
2

Yj = Xj + Xj .
1

2

Therefore, the value p equals to s, the total number of real irreducible characters of G, which equals
to the number of real conjugacy classes, plus half of the remaining complex-valued irreducible characters of G. Thus p = r+s
.
2
We now address the optimality of the decomposition. Observe that we are requesting, for all X and
for all n = {1, 2, … , r}
( ( ) ( ))
̂ 𝜌 X
̂ 𝜌
E X
= 0 for all j1 ≠ j2 .
j
n
j
n
1

2

̂
̂ with the following element in M(G)
We can associate X
j
� ≡0⊕…⊕0⊕X
� ⊕ 0 ⊕ … ⊕ 0.
X
j
j

Suppose we can decompose an arbitrary X further. This would mean there exist square matrices E
and F so that
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E + F = I and EQFQ = 0, for all square Q.

Assume Q is non-singular and we have EQ(I − E) = 0 which implies EQ = EQE. We re-write E as follows:
E = EQEQ−1 for all Q.

Now choose Q so that QEQ−1 is J, the Jordan canonical form of E. Now it is straightforward to see that
✷
the relation E = EJ is impossible unless E = 0 or E = I. The result now follows. 
Note that if all elements of the group are real then p = r. The above theorem can be used to decomposition the variance of X(t), in particular, for any t ∈ G, we have

E(X(t)X ∗ (t)) =

p
∑

E(Xj (t)Xj∗ (t)).

j=1

The interpretation of var(X1 ) = E(X1 (t)X1 (t)) is straightforward since 𝜌0 denotes the trivial group
representation. This quantity refers to the amount of variance of X(t) captured by the variance of the
∑r
mean of X over G. The above decomposition X = j=1 Xj into complex components will be referred to
∑p
as the optimal complex orthogonal decomposition, similarly, the above decomposition X = j=1 Xj
into real components will be referred to as the optimal real orthogonal decomposition.
Theorem 2 We have E(X(t)X(s)) = 0 for all t, s ∈ G such that t ≠ s if and only if for any t ∈ G, we have
var(Xj (t)) =

dj2
|G|

varX(t),

where {Xj }rj=1 is the optimal complex orthogonal decomposition.
∑r
Proof Consider the decomposition X = j=1 Xj into possibly complex valued components and suppose E(X(t)X(s)) = 0 for all t ≠ s. Then, we have
(
)
E Xj (t)Xj (t) = E
=

=

=

=

=

)
) dj (
)
dj (
CG (𝜒̃j )X (t)
CG (𝜒̃j )X (t)
|G|
|G|

(

dj2
|G|

2

dj2
|G|

2

dj2
|G|

2

dj2
|G|2
dj2
|G|

(
)
E CG (𝜒̃j )CG (𝜒̃j )X(t)X(t)
(
)
E CG (𝜒̃j )X(t)X(t)
(
)
tr CG (𝜒̃j ) varX(t)
|G| varX(t)

varX(t).
d2

Conversely, suppose var(Xj (t)) = |G|j varX(t) for all t ∈ G. Define n = |G| unknowns {x𝜏 }𝜏∈G, where
x𝜏 = E(X(t)X(s)) = RX (𝜏) with 𝜏 = t−1 s. Now consider the following linear equations,
(
)
{E CG (𝜒̃j )X(t)X(t) = |G| varX(t)}j={1, …, r}, t∈G

and note that the number of these equations is equal to rn. However, we have only as many linearly
independent equations as there are linearly independent columns in the matrices
{CG (𝜒̃j )}rj=1 .
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∑r
This number is equals to j=1 dj2 = n, the number of unknowns. The result follows with the unique
✷
solution E(X(t)X(s)) = 0 for t ≠ s. 

As an application of the above result we develop a hypothesis testing for correlation of the WSS
processes X(t) and X(s) based on testing equality of variances of the following possibly complexvalued random variables

{

}r
1
X
dj j

.
j=1

Testing the above random variables might be better that testing correlations of X(t) and X(s) due
to averaging process that could bring the random variables close to normality.

3. The group S3
We will consider the symmetric group S3 in our example. The group G = S3 consists of elements

g0 = (1); g1 = (12); g2 = (13)
g3 = (23); g4 = (123); g5 = (132).
The group S3 has three conjugacy classes

{g0 }, {g1 , g2 , g3 }, {g4 , g5 }.
We have three irreducible representations, two of which are one-dimensional, 𝜌1 is the identity map,
𝜌2 is the map that assigns the value of 1 if the permutation is even and the value of −1 if the permutation is odd. Finally, we have 𝜌3, the two-dimensional irreducible representation of S3, defined by
the following assignment

)
)
(
0 1
; g1 ↦
1 0
(
(
)
2𝜋i∕3 )
0
e−2𝜋i∕3
0
e
; g3 ↦
g2 ↦
e2𝜋i∕3
0
e−2𝜋i∕3
0
( 2𝜋i∕3
)
( −2𝜋i∕3
)
e
0
e
0
g4 ↦
; g5 ↦
.
−2𝜋i∕3
2𝜋i∕3
0
e
0
e
(

g0 ↦

1
0

0
1

The irreducible characters of S3 are given by

𝜒1 = (1, 1, 1, 1, 1, 1)T
𝜒2 = (1, −1, −1, −1, 1, 1)T
𝜙3 = (2, 0, 0, 0, −1, −1)T ,
where 𝜒1 and 𝜒2 are also multiplicative characters. Moreover, we have

𝜌3 (1, 1) = (1, 0, 0, 0, e−2𝜋i∕3 , e2𝜋i∕3 )T
𝜌3 (1, 2) = (0, 1, e−2𝜋i∕3 , e2𝜋i∕3 , 0, 0)T
𝜌3 (2, 1) = (0, 1, e2𝜋i∕3 , e−2𝜋i∕3 , 0, 0)T
𝜌3 (2, 2) = (1, 0, 0, 0, e2𝜋i∕3 , e−2𝜋i∕3 )T .
T

2

The G-convolution by a function 𝜓 = (c0 , c1 , c2 , c3 , c4 , c5 ) ∈ l (G) can be induced by a G-circulant
matrix CG (𝜓) given by

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⎛ c0
⎜
⎜ c1
⎜ c
CG (𝜓) = ⎜ 2
⎜ c3
⎜ c4
⎜ c
⎝ 5

c1
c0
c5
c4
c3
c2

c2
c4
c0
c5
c1
c3

c3
c5
c4
c0
c2
c1

c4
c2
c3
c1
c5
c0

c5 ⎞
⎟
c3 ⎟
c1 ⎟
c2 ⎟⎟
c0 ⎟
c4 ⎟⎠

and specifically, note that 𝜒̃j (g) = 𝜒 j (g) = 𝜒j (g). We have

⎛ 1
⎜
⎜ 1
⎜ 1
CG (𝜒1 ) = ⎜
⎜ 1
⎜ 1
⎜ 1
⎝

1
1
1
1
1
1

1 ⎞
⎟
1 ⎟
1 ⎟
.
1 ⎟⎟
1 ⎟
1 ⎟⎠

1
1
1
1
1
1

1
1
1
1
1
1

1
1
1
1
1
1

⎛ 1
⎜
⎜ −1
⎜ −1
CG (𝜒2 ) = ⎜
⎜ −1
⎜ 1
⎜ 1
⎝

−1
1
1
1
−1
−1

−1
1
1
1
−1
−1

−1
1
1
1
−1
−1

1
−1
−1
−1
1
1

1 ⎞
⎟
−1 ⎟
−1 ⎟
.
−1 ⎟⎟
1 ⎟
1 ⎟⎠

⎛ 2
⎜
⎜ 0
⎜ 0
CG (𝜒3 ) = ⎜
⎜ 0
⎜ −1
⎜ −1
⎝

0
2
−1
−1
0
0

0
−1
2
−1
0
0

0
−1
−1
2
0
0

−1
0
0
0
−1
2

−1 ⎞
⎟
0 ⎟
0 ⎟
.
0 ⎟⎟
2 ⎟
−1 ⎟⎠

Set

X = [X(g0 ), X(g1 ), X(g2 ), X(g3 ), X(g4 ), X(g5 )]T
= [X(0), X(1), X(2), X(3), X(4), X(5)]T
and we obtain

1
1
C(𝜒1 )X = ⟨X, 𝜒1 ⟩𝜒1 .
6
6
1
1
X2 = C(𝜒2 )X = ⟨X, 𝜒2 ⟩𝜒2 .
6
6
1
X3 = C(𝜒3 )X.
3

X1 =

Therefore, the set {X1 , X2 , X3 } is an optimal set consisting of real zero mean mutually orthogonal
WSS processes over S3. As a result we have

E(X(t)X ∗ (t)) =

r
∑

E(Xj (t)Xj∗ (t)) for all t ∈ G.

j=1

4. Carry-over effects in the cross-over designs
The above orthogonal decomposition of the WSS processes appear naturally in the cross-over designs in clinical trials, in particular, the William’s 6 × 3 design with 3 treatments. During a cross-over
trial every patient receives more than one treatment in a certain pre-specified sequence. Thus, each
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subject then acts as his or her own control. Each treatment is administered for a pre-selected time
period. In these experiments a washout period is established between the last administration of one
treatment and the first administration of the next treatment. The intention is for the effect of the
preceding treatment to wear off during the trial. However, there will be some carry-over effects in all
the specified treatment sequences, clearly starting with the second treatment. For more information
on cross-over designs in clinical trials see Senn (2002) or Chow and Liu (2013), for example.
Consider a certain sequence of treatments. By a carry-over effect within these treatments we
understand the total sum of all effects arising from all the treatments within this sequence. In our
example, we follow the William’s 6 × 3 design with 3 treatments A, B, and C, in particular, the underlying group is the symmetric group of three elements S3. In particular, we have six treatment sequences ABC, ACB, BAC, BCA, CAB, CBA. For example, suppose the order of treatment administration
is BCA, with B first. We decide to collect the sum of all carry-over effects of the treatments in this
sequence (starting with the second one), XBCA. We observe the sequence BCA as a permutation of the
sequence ABC by the permutation g4 = (123), an element of the group S3. Thus, we can write
XBCA = X(g4 ) = X(4). Similarly, a permutation sequence ACB would result in XACB = X(g3 ) = X(3).
We treat X as a zero mean WSS process, if necessary we can subtract the mean to ensure zero
mean WSS process. In particular, we assume the variance of the cross-over effect is the same for all
−1
treatment sequences. Moreover, we assume E(X(gi )X(gj )) only depends on the sequence gi gj, in
another words, depends only on the permutation that gets us from the sequence gi to the sequence
gj. However, note that the random processes X(gi ) and X(gj ) are in general dependent and, in general, might have different distributions for i ≠ j.
We can now decompose the variance of the zero mean WSS process X over S3 as follows. For all

t ∈ G we have

var(X(t)) = var(X1 (t)) + var(X2 (t)) + var(X3 (t)).
In this context, the real zero mean WSS processes over S3, {X(t)}t∈S , are un-correlated if and only if
3

}
{
1
X1 , X2 , X3
2

have the same variance.

5. The group S4
The symmetric group of four elements has five conjugacy classes represented by

(1); (12); (123); (1234); (12)(34)
with sizes 1, 6, 8, 6, and 3, respectively. We have two irreducible characters of degree 1, namely the
principal one and the one whose value at a permutation is its sign. There is one-degree two irreducible character 𝜒3 and two-degree 3 irreducible representations 𝜒4 and 𝜒5. The value of 𝜒4 at a permutation is the number of fixed points in that permutation minus one. Finally, we have 𝜒5 = 𝜒4 𝜒2. For
reference, see Dummit (1999). Find below a character table for S4.

conj. classes
𝜒1
𝜒2
𝜒3
𝜒4
𝜒5

(1)
1
1
2
3
3

(12)
1
−1
0
1
−1

(123)
1
1
−1
0
0

(1234)
1
−1
0
−1
1

(12)(34)
1
1
2
−1
−1

Suppose we have a cross-over design experiment with 4 treatments where we have all permutations of treatments allowed. We study the variance of the carry-over effects from the sequence of
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treatments. Once, again, we define the carry-over effect for the treatment sequence as the sum of
all the carry-over effects in that specific sequence. We can now decompose the variance of the zero
mean WSS process X over S4 as follows. We define

⟩
1⟨
1
C(𝜒1 )X =
X, 𝜒1 𝜒1 .
24
6
⟩
1
1⟨
X2 =
C(𝜒2 )X =
X, 𝜒2 𝜒2 .
24
6
1
C(𝜒3 )X.
X3 =
12
1
X4 = C(𝜒4 )X.
8
1
X5 = C(𝜒5 )X.
8
X1 =

and obtain

var(X(t)) = var(X1 (t)) + var(X2 (t)) + var(X3 (t)) + var(X4 (t)) + var(X5 (t)).
In this context, the zero mean WSS processes {X(t)}t∈S are un-correlated if and only is
4

}
{
1
1
1
X1 , X2 , X3 , X4 , X5
2
3
3
have the same variance.

6. The group A4
The case of the alternating group A4 is a little different as r ≠ p. Here, we would still have 4 treatments, but only even permutations of the treatments are allowed in the cross-over design. We have
three irreducible representations of degree 1 and one of degree 3, see Dummit (1999) for reference.
We have 4 conjugacy classes represented by

(1); (12)(34); (123); (132)
of sizes 1, 3, 4, and 4, respectively. The irreducible characters are not necessarily real, find below
the character table.

conj. classes
𝜒1
𝜒2
𝜒3
𝜒4

(1)
1
1
1
3

(12)(34)
1
1
1
−1

(123)
1
𝜓
𝜓2
0

(132)
1
𝜓2
𝜓
0

2𝜋i

where 𝜓 = e 3 . We can now decompose the variance of the zero mean WSS process X over A4 as
follows. We do not have 4 summands in the decomposition as we do not have 4 real irreducible
characters. We start with complex decomposition

X = X1 + X2 + X3 + X4
with

var(X(t)) = var(X1 (t)) + var(X2 (t)) + var(X3 (t)) + var(X4 (t)).
where

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⟩
1 ⟨
1
C(𝜒1 )X =
X, 𝜒1 𝜒1
12
12
⟩
1
1 ⟨
X2 =
C(𝜒2 )X =
X, 𝜒2 𝜒2
12
12
⟩
1 ⟨
1
C(𝜒3 )X =
X, 𝜒3 𝜒3
X3 =
12
12
1
X4 = C(𝜒4 )X.
4
X1 =

We now create a real decomposition of X consisting of three zero mean real-valued WSS processes
as follows:

var(X(t)) = var(Y1 (t)) + var(Y2 (t)) + var(Y3 (t)).
where, letting 𝜒 = 𝜒2 + 𝜒3,

1
C(𝜒2 + 𝜒3 )X
12
1
=
C(𝜒)X
12
1
=
⟨X, 𝜒⟩ 𝜒
12

Y2 =

and Y1 = X1, Y3 = X4. Observe the values of 𝜒 on the respective conjugacy classses of A4 are given
as follows:

conj. classes
𝜒

(1)
2

(12)(34)
2

(123)
−1

(132)
−1

In this context, the zero mean WSS processes {X(t)}t∈A are un-correlated if and only if

{

X1 , X2 , X3 ,

1
X
3 4

}

4

have the same variance.
Funding
The authors received no direct funding for this research.
Author details
Pamini Thangarajah1
E-mail: pthangarajah@mtroyal.ca
Peter Zizler1
E-mail: pzizler@mtroyal.ca
1
Department of Mathematics and Computing, Mount Royal
University, 4825, Mount Royal Gate SW, Calgary, Alberta,
Canada.
Citation information
Cite this article as: On wide sense stationary processes
over finite non-abelian groups, Pamini Thangarajah & Peter
Zizler, Cogent Mathematics (2017), 4: 1313926.
References
An, M., & Tolimieri, R. (2003). Group filters and image
processing. Psypher Press.
Bartoszynski, R., & Niewiadomska-Bugaj, M. (2008). Probability
and statistical inference (2nd ed.). New York, NY: Wiley.
Chow, S.-C. & Liu, J.-P. (2013). Design and analysis of clinical
trials: Concepts and methodologies. Hoboken, NJ: Wiley.

Diaconis, P. (1998). Group representations in probability and
statistics. Hayward, CA: Institute of Mathematical
Statistics.
Dummit, D. S., & Foote, R. N. (1999). Abstract algebra. New
York, NY: John Wiley and Sons.
Giannakis, G. B. (1999). Cyclostationary signal analysis, digital
signal processing handbook. Boca Raton, FL: CRC Press LLC.
Hazewinkel, M. (Ed.). (2001). Peter-Weyl theorem. Encyclopedia
of mathematics: Springer. ISBN 978-1-55608-010-4.
James, G., & Liebeck, M. (1993). Representations and characters
of groups. New York, NY: Cambridge University Press.
Peccati, G., & Pycke, J.-R. (2005). Decompositions of stochastic
processes based on irreducible group representations. (p. 27).
Senn, S. (2002). Cross-over trials in clinical research (2nd ed.).
Queensland: Wiley.
Stankovic, Radomir S., Moraga, Claudio, & Astola, Jaakko T.
(2005). Fourier analysis on finite groups with applications
in signal processing and system design. Hoboken, NJ: IEEE
Press, John Wiley and Sons.
Zizler, P. (2013). On spectral properties of group circulant
matrices. PanAmerican Mathematical Journal, 23(1), 1–23.
Zizler, P. (2014). Some remarks on the non-abelian fourier
transform in crossover designs in clinical trials. Applied
Mathematics, 5, 917–927.

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