559
I ntnat. J. Mh. & Math. Si.
Vol. 3 No. 3 (1980) 559-574
REPRESENTATION THEORY OF
FINITE ABELIAN GROUPS APPLIED TO A
LINEAR DIATOMIC CRYSTAL
J.N. BOYD
Department of Mathematical Sciences
Virginia Commonwealth University
Richmond, Virginia 23284 U.S.A.
P.N. RAYCHOWDHURY
Department of Mathematical Sciences
Virginia Commonwealth University
Richmond, Virginia 23284 U.S.A.
(Received November 21, 1979)
ABSTRACT.
After a brief review of matrix representations of finite abellan groups,
projection operators are defined and used to compute symmetry coordinates for
systems of coupled harmonic oscillators.
The Lagrangian for such systems is
discussed in the event that the displacements along the symmetry coordinates are
complex.
Lastly, the natural frequencies of a linear, dlatomic crystal are deter-
mined through application of the Born cyclic condition and the determination of
the symmetry coordinates.
KEY WORDS AND PHRASES. Abelian groups, Born cyclic condition, Group reprentn
H-armonc osrs Lagrangian mechanic, Nat Frequenci, Projection operators.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 20C35, 70J10.
560
I.
J. N. BOYD AND P. N. RAYCHOWDHURY
INTRODUCTION.
A classic application of abstract algebra is found in the use of the
representation theory of finite groups in the separation of the equations of
motion for symmetrically
coupled harmonic oscillators.
on the subject is extensive
[3], [4], [5], [6], [9],
review of certain pertinent facts.
Although the literature
we shall present a brief
We shall define projection operators in the
event that the group under consideration is abelian and use the projection
operators to compute the symmetry coordinates for a physical system.
Thereafter
we shall discuss the effect of the transformation to symmetry coordinates upon
a Lagrangian function.
Finally, we shall find the natural frequencies of a one
dimensional, diatomic crystal.
2.
GROUP REPRESENTATIONS AND PROJECTION OPERATORS.
S
Let
let
G
denote a system of finitely many coupled, harmonic oscillators, and
be a finite symmetry group acting on
multiplicative group of
F: G
M(n)
F(G)
{F(g)
group,
n
n
is a homomorphism
gsG}
S
M(n)
Then, if
denotes the
and if
matrices over the complex numbers,
of groups, the image of
F
given by
is a matrix representation of dimension
n
for the symmetry
G
If
F(G)
is unitary and if there exists no similarity transformation to
convert
F(G)
into the block diagonal form
k
F.(G)
where
F.(G)
is a matrix
j=l
representation of
G
with dimension less than
n
for each
said to be a unitary, irreducible matrix representation of
j
G
then
there are precisely as many of these as there are conjugacy classes in
conjugacy classes in
G
In fact,
If
G
and if the dimensions of the nonequivalent,
unitary, irreducible matrix representations are
know that
is
The number of
nonequivalent, unitary, irreducible matrix representations is finite.
there are
F(G)
n(1)
n(2)
n()
we
REPRESENTATION OF FINITE ABELIAN GROUPS
(n(j))
2
IGI
j=l
GI
where
If
G
is the order of the finite group
GI
is abelian, then
561
[6].
n(j)
and
implying that all
i
irreducible representations will be one dimensional.
Indeed, if the full symmetry group of the system,
This fact is quite useful.
S
is nonabelian, it is
often desirable to choose an abelian subgroup of the larger group to act as
G
thereby obtaining significant advantages in the computational procedures.
EXANPLE 2.1.
by
R, a 90
Let
{E, R, R
G
2, R3}
be the cyclic four-group generated
rotation about a fixed axis.
There are four irreducible, unitary
repr esentat ions
F
I(G)
F
I(E)
F
I(R)
F
2(G)
F
2(E)
F
2(R2)
F
3(G)
F
3(E)
I
F
F 3 (R
F4(G)
3)
F
I(R 2)
I
3(R)
F
F
I(R3)
2(R)
i
F
F
I
2(R3)
3(R 2)
I
I
i
F4(E)
F4(R)
i
F4 (R3)
i
F4 (R2)
i
i
Suppose that there is associated with each oscillator in
m
i
If the displacement of the
i-th
mass is taken to be
S
v
i
a point mass,
the smallest
vector space containing all linear combinations of these displacement vectors
for all particles is the solution space,
Now for each element,
exactly one linear operator,
g
of the physical system,
of the symmetry group,
(g)
G
S
.
there is induced
which acts on the basis vectors of
562
J. N. BOYD AND P. N. RAYCHOWDHURY
.Gag (g)l
A( (G))--{
Furthermore, it can then be shown that
ge
a e}
is an algebra
g
over
In
A((G))
there is a subset of operators upon which we fix attention.
DEFINITION 2.1.
Let
be abelian and let
G
Fi(g)
the i-th unitary, irreducible, one dimensional representation of
linear operator
{Pill
P.eA( (G))
IGI}
1,2
P
is defined by
i
I
(F (g)) (g)
gG
i
G
Then the
and
is the set of projection operators in the algebra
We emphasize that, in this definition,
G
A((G)).
The reader may easily
is abelian.
find the general definition of the projection operators when
to be abelian
Fi(G)
belong to
G
is not required
[I], [9], [i0].
EXANPLE 2.2.
{E, R, R2,
G
R
3}
as in Example 2.1.
Then the projection
operators are
P1
P2
P3
P4
(E) + (R) + (R 2) + (R
(E)
(R) + (R
{i’
3)
(R
i(R
3)
(E)
(R 2) + i(R
3)
i(R)
2)
and
has the orthonormal basis
Then the symmetry coordinates for the physical system can
u2’’’’’u n
be obtained from
(R
(E) + i(R)
Now suppose that the solution space,
B
2)
3)
B
DEFINITION 2.2.
as indicated in the next definition.
Suppose that
P
i()
is not the zero vector.
distinct unit vectors of the form
Pi(u)
are the symmetry coordinates of the physical system.
Then the
REPRESENTATION OF FINITE ABELIAN GROUPS
There are exactly
distinct nonzero vectors which can be obtained
dim
n
563
from this definition, and these are mutually orthogonal [i].
matching the symmetry of
establishes a new basis for
S
This set of vectors
and determines the
transformation of coordinates which will separate the equations of motion to the
greatest extent possible through the use of symmetry alone.
basis of symmetry coordinates by
{
el,e2,...,en}
We denote the new
and to exhibit their use, we
give a simple example.
EXAMPLE 2.3.
Four identical masses,
are uniformly spaced and inter-
m
connected by identical springs on a frictionless, circular loop fixed in a plane.
k
The spring constant for each spring is
to the loop.
SOLUTION:
All motion of the masses is confined
We find the natural frequencies of motion [I], [2].
The physical system is shown below.
be the rotation group
G
{E, R, R
2
R
The symmetry group is taken to
3}
The basis for the solution space is the set of unit tangents
The projection operators are given in Example
are
{Ul,^
^u 2, ^u 3,
.
2.2 and the symmetry coordinates
J. N. BOYD AND P. N. RAYCHOWDHURY
564
PI(I
el
i__
-2
e
i
2
i
e3
i
e4
(i
+
[(E) + (R) + (R
u
2
+
u3 + u4)
2 + 3 4)
(Ul + i2 -3 i4)
(u
u
I
(Ul
+ (R3)]u I
2)
i2
3
and
i4)
+
U
These equations serve to define the unitary, transformation matrix
such
that
cl(l 2 3 4
col( I
U
u
2
u
3
4
Thus
u
1
i
I
1
-i
I
-i
i
i
1
-i
-
-i
If the displacements of the masses along the unit tangents are
x
I
x
2
x
3
x
4
1/21 2 3 4
and their velocities are
the displacements and velocities as the column vectors
i
cl(l
234
Then the Lagrangian of the system is
L
i
mr
are row vectors obtained by taking the transposes of
X
col(x
we can write
I
x
2
x
3
I kXVX where X
and
X
x
4)
and
and
X
REPRESENTATION OF FINITE ABELIAN GROUPS
i
0
0
0
0
i
0
0
T
0
0
1
0
0
0
0
1
2
-i
0
-i
-i
2
-i
0
0
-1
2
-1
-1
0
-1
2
V
and
565
We next transform the Lagrangian to obtain a function of the displacements,
n l’
n2’ n3’ 4’
I’ 2’ 3’4
and velocities
respect to the symmetry
with
Thus
ooordinates, e l, e 2, e 3, e 4
L= mU-IuTu-Iu
i
1
2
kU-Iuvu-Iux
-:1 aN* (UTU -1) -:I k* (UVU -I) N
where
N
col(n
col(
I n2
posed, complex conjugates of
UTU
-i
T
and
N
UVU
i 2 3 14)
respectively.
-i
*
and
N*
are the trans-
Since
0
0
0
0
0
4
0
0
0
0
2
0
0
0
0
2
we have
L
1
-*-
m(in I
-*- + -*- +
+ 22
33
I
*
--2 k(422 +
.
.*-
4n4
*
*
2n33 + 2n44)
The completely separated equations of motion are given by
(’_j)_dt
d__
0
(Equation 2.1)
566
J. N. BOYD AND P. N. RAYCHOWDHURY
or equivalently by
(Equation 2.2)
for
j{l, 2, 3, 4
It follows immediately that the four natural frequencies for this
system are
3.
THE LAGRANGIAN WITH COMPLEX COORDINATES.
In the preceding example, the transformation to symmetry coordinates yielded
a real-valued Lagranglan function in the complex variables
similarity transformations of the
T
V
and
nj*
nj
The
matrices leave their eigenvalues
As these elgenvalues, in fact, determine the natural frequencies,
undisturbed.
such transformations are quite permissible.
However, we also wish
to show that
in the equations of motion as given by Equations 2.1 and 2.2, the variables
and their complex conjugates,
nj*
nj
To that end
will never appear together.
let us consider the following exercise in partial differentiation.
.
Let
F
be a function of the independent complex variables
Furthermore, 6uppose that
If
Yi
Yt
0
x.3
noting that
j*
F
xj + lyj
qj
x.
is differentiable with respect to
y
and
3
and if we apply the chain rule for partial derivatives while
x
j
we obtain
iyj
3F
x]
F +
j
3F
and
F
y]
Formally, we write
8x
J
8nj
+
and
Yj
Inverting these operational equations, we have
i-.3
i
n;
i
3F
o,.-i
.1
.
F
REPRESENTATION OF FINITE ABELIAN GROUPS
l(____S
2
8
x.3
j
i
__B
567
and
YJ
Thus, whenever the Lagrangian of a physical system of harmonic oscillators
i
i
,To k
*VoN
m
the form
CIS k or C2Sn k
is written in the matrix notation
expanded Lagrangian will be of
L
each summand of the
where
C
I
and
C
2
are constant.
Thus the differential equation
only the complex conjugates
*k
together with their second derivatives with
respect to time and
0
only the coordinates
nk
and
k
contains
That is, in neither equation will
coordinates and their complex conjugates be mixed.
The proper frequencies can
be obtained from equations of either form.
4.
THE ONE-DIMENSIONAL DIATOMIC CRYSTAL.
We now consider a circular loop of alternating masses of two sorts,
M
connected by identical springs having force constant
2N
masses in all with
N
of each kind.
If
N
B
is taken to
[2].
m
and
Let there be
be very large, we
have essentially applied the Born cyclic condition to a linear, diatomic crystal
J. N. BOYD AND P. N. RAYCHOWDHURY
568
-
We will now find the symmetry coordinates and natural frequencies of
[I], [8]
the system.
The system is +/-nvariant with respect to rotations by
2
and, in the
neighborhood of the 2n-th and (2n+l)-th particles, the circular arrangement
appears as shown.
In the calculations to follow, two facts concerning the complex roots of
unity are required.
First, let
n
I
and
n
exp
2i
be the primitive complex
2 {0,1,2,...,K
Addition for Complex Numbers
I}
K-th root of unity.
Then let
By application of the Parallelogram Law of
-- .
-
569
REPRESENTATION OF FINITE ABELIAN GROUPS
exp n
I
.----. +
2i3
nl-n2
2i
n2(---)
exp
2 cos(-
In the event that we need to consider
we write
exp Oi
+ exp
2
n
nl+n2
2i
exp(-------)
+ exp
exp 2i
n-l
cos
(-)
K
complex
Secondly we recall that the sum of the
the complex number zero.
2
-2
exp
(Equation 4.1)
0 in
where
K
1
.----_
K-th roots of unity is
That is
K-I
exp n
2i
+
0
(Equation 4.2)
K > I
0i
n=0
C
The symmetry group of the circular arrangement is taken to be
of rotations
{R(2-)I
N}
i, 2, 3
The
N
N
the group
one dimensional represent-
ations of the symmetry group are given by
r k (G)
N
i, 2, 3
for
.
F
k
.2kr,
exp t--’--)
(2r)
(R.---if-.)
i
[i].
Thus the projection operators are of the form
N
Pk
=i
.2kr,,,
))
(exp(---)i)(R(
for
The symmetry coordinates are found by letting
tangent
2N-I
and then upon
and
Pk
*act first upon the unit
U2N
I
e2k-i
N
i, 2, 3,
k
N
.2k.
---/- Pk U2N_l --/- =i (exp(----)i)u2
i
J. N. BOYD AND P. N. RAYCHOWDHURY
570
.
and
_I
/
e2k
ek
i
/
U2N
2k
(exp(__)i)u2
=i
for k=l, 2, 3,...,N
-- - -- - - - -
The transformation matrix
fexp
2i
0
0
0
0
is
4i
0
0
0
0
0
exp
8i
exp
6i
exp
4i
exp
8i
exp
4i
exp
U
exp
2i
exp
4i
exp
i
N
0
exp
12i
N
0
6i
0
exp
12i
0
1
1
0
0
1
i
0
i
0
i
0
i
0
0
i
0
i
0
i
0
1
In terms of the displacements and velocities along the unit tangents, the
Lagrangian of the system is
L
BX
X
col(x
col(XlX 2
2N
where
I
x
2
X2N)
REPRESENTATION OF FINITE ABELIAN GROUPS
0
M
0
0
571
0
0
0
0
0
0
0
i
0
0
0
0
M
m
0
and
0
0
0
0
i
0
0
0
0
0
0
0
0
-i
0
M
-i
2
-i
0
0
0
0
-i
2
-i
0
0
0
0
-I
2
0
0
0
0
0
0
2
-i
-i
0
0
0
-I
2
V
J. N. BOYD AND P. N. RAYCHOWDHURY
572
The transformed Lagrangian is
N
where
col(nl2...2N
easily shown that
21--B* (UVU-I)
i
--m
* (UTU-I)
L
UTU -I
with respect to the symmetry coordinates.
-i
is
straightforward but tedious.
--- -
use of the identities given in Equations 4.1 and
2
-2 cos
-2 cos
exp(--)
It is
T
UVU
The computation of
N
exp
2
i
However, by the
4.2, the matrix product becomes
0
0
0
0
0
0
0
0
0
0
0
0
27
2
i3
exp(---
0
0
0
0
0
0
0
0
2
-2
0
0
0
0
-2
2
The k-th
2 x 2
2
-2 cos
2
exp (-
-2 cos
2k-i
2i)
2
-
block along the principal diagonal of the matrix
-2 cos k
2
Thus, for
-2 cos
and
k exp(-
2k
ki
N
we have
2
ki
exp(---)
I
/
UVU
-I
is
573
REPRESENTATION OF FINITE ABELIAN GROUPS
---1
B (k-lk)
-2 cos
i2
-k-i 2k-I + M-k
#
,In*2k_in2k_l
cos
k
-2 cos
2
cos
k
k
exp(-
---.
ki)
2
exp(
2k-
2k
2k)
exp(-
,
ki
N "n2kn2k-1
.ki.
expt---)k-ln2k + qk
k#
The equations of motion are
1
m
2
2k-i + 82k-I
IM
2 nk +Bn 2k
8cos
8 cos
kz
k
exp(-
.ki.
N 2k
0
expt----)
kni)
---" n2k-i
0
The frequencies satisfy the secular determinant
2 +
m
det
42f2
2__
m
2 8
k
cos
exp(-
Thus
(+)
f
k
exp
ki
2_ + 4 2 f 2
S
ki
N
+
cos
+
)
4
Mm
sin
2k
N
Recognizing that, in solid state physics, the wave number for the vibration
is
K
where
a
is the separation of neighboring particles, the result
J. N. BOYD AND P. N. RAYCHOWDHURY
574
may be rewritten as
/,1 +
/
a
K
The value
lattice space.
K--
=-za
f
i
)
4
Mm
defines the boundary of the first Brillouln zone in reciprocal
The familiar forbidden frequency gap appears since, for
will have the separated values
or
27
2n
[7]
REFERENCES
i.
"An Application of Projection Operators to
Bulletin of the Institute of Mathematics Ac.ad. emi.a
Boyd, J.N. and Raychowdhury, P.N.,
a One Dimensional
Crystal’",
Sinica 7(1979), 133-144.
2.
Boyd, J.N. and Raychowdhury, P.N., "Projection Operator Techniques in
Lagrangian Mechanics: Symmetrically Coupled Oscillators", to appear in
International Journal of Mathematics and Mathematical Sciences.
3.
Duffey, G.H., Theprtica$ Physi_cs
4.
Gamba, A., "Representations and Classes in Groups of Finite Order", Journal
of Mathematical Physics 9 (1968), 186-192.
5.
Hamermesh, M., G_roup Theory and Its Application to Physical Problems, Addison
Wesley Publishing Company, Reading, Mass., 1962.
6.
Joshi, A.W., Elements of Group .Theory for Physic.ists
Houghton Mifflin Company, Boston, 1973.
Halsted Press, New York,
1973.
7.
Kittel, C., Introduction to Solid State Physics (Second Edition), John Wiley
and Sons, New York, 1959.
8.
Mariot, L., (translated by A. Nussbaum), Group The.ory and Solid State Physic
Prentice-Hall, Englewood Cliffs, N.J., 1964.
9.
Nussbaum, A., "Group Theory and Normal Modes", American Journal of Physics
36, 529-539.
I0.
Wilde, R.E., "A General Theory of Symmetry Coordinates", American Journal of
32 (1964), 45-52.