Burnside Rings: Application to Molecules of Icosahedral Symmetry

Abstract

The Burnside ring, B(G), of a group G is the set of isomorphism classes of orbits of G together with the operations of addition and product. The addition is defined as the disjoint union, and the product as the Cartesian product. This paper describes basic facts about this algebraic structure and develops some applications in chemistry, as the labelling of atoms in molecules of high symmetry and the construction of symmetry-adapted functions. For illustrating such applications, the concept of Burnside ring is applied to the icosahedral symmetry. Sets of points which are isomorphic to the orbits of the I group are described and the multiplication table of B({I}) is obtained from the table of marks. This multiplication table allows us to obtain an elegant labelling of the atoms of the buckminsterfullerene which is consistent with the icosahedral symmetry. Also, we obtain complete sets of symmetry-adapted functions for the buckminsterfullerene which span the Boyle and Parker's icosahedral representations.

Full text

Journal of Mathematical Chemistry Vol. 35, No. 2, February 2004 (© 2004)
Burnside rings:
Application to molecules of icosahedral symmetry
Emilio Martínez Torres
Department of Physical Chemistry, Escuela de Magisterio, University of Castilla-LaMancha,
Ronda de Calatrava, 3, 13003 Ciudad Real, Spain
E-mail: Emilio.MTorres@uclm.es
The Burnside ring, B(G), of a group Gis the set of isomorphism classes of orbits of G
together with the operations of addition and product. The addition is defined as the disjoint
union, and the product as the Cartesian product. This paper describes basic facts about this
algebraic structure and develops some applications in chemistry, as the labelling of atoms in
molecules of high symmetry and the construction of symmetry-adapted functions. For illus-
trating such applications, the concept of Burnside ring is applied to the icosahedral symmetry.
Sets of points which are isomorphic to the orbits of the Igroup are described and the multi-
plication table of B(I) is obtained from the table of marks. This multiplication table allows
us to obtain an elegant labelling of the atoms of the buckminsterfullerene which is consistent
with the icosahedral symmetry. Also, we obtain complete sets of symmetry-adapted functions
for the buckminsterfullerene which span the Boyle and Parker’s icosahedral representations.
KEY WORDS: Burnside rings, group theory, symmetry-adapted functions, icosahedral
group, buckminsterfullerene
1. Introduction
The concept of G-set (a set whose elements are interchanged by the transforma-
tions of a group G) is widely applied in several branches of chemistry; for example, con-
struction of symmetry-adapted functions, classification and determination of the sym-
metric coordinates of a vibrating molecule, enumeration of compounds, etc. One of the
first task in working with G-sets is the enumeration of their orbits, i.e., the decomposi-
tion of the G-sets into subsets whose elements are equivalent with respect to the group
action. The table of marks is an useful tool for decomposing a G-set into orbits since it
allows us to know the number and type of orbits which are contained in a G-set start-
ing from the numbers of elements that are invariant with respect to the subgroups of G.
AG-set can be considered as the sum of its orbits. The direct product of two orbits is a
G-set which can be decomposed into orbits. Thus, we can define the multiplication table
of the orbits of a group as that containing the decomposition of the products into orbits.
Table of marks is an essential tool for the attainment of such multiplication table.
Burnside ring is an algebraic structure which raises when we introduce the op-
erations of product and sum in the complete set of isomorphism classes of orbits of a
105
0259-9791/04/0200-0105/0 2004 Plenum Publishing Corporation

106 E.M. Torres / Burnside rings
finite group. It is an interesting concept which until now has not been used beyond the
boundaries of pure algebra.
2. Burnside rings
Let Gbe a finite group. The set Sis said to be a G-set if to each g∈Gand
each x∈Sthere corresponds an element gx ∈S, such that 1x=x(where 1 is the
identity element of G)andf(hx) =(f h)x for all f, h ∈G[1]. Two G-sets Sand T
are isomorphic with respect to the group action (denoted by S∼
=T) if there exists a
bijection φ:S→Tsuch that gφ(x) =φ(gx) for all x∈Sand g∈G.
For each element xin S, its orbit Gx ={gx:g∈G}is the smaller G-set contain-
ing x. Two elements xand yof Sbelong to the same orbit if there exists an element g
of Gsuch that y=gx.EveryG-set Scan be partitioned into a disjoint union of orbits.
AG-set consisting of a single orbit is called transitive.
The stabiliser Gxof an element xof Sis the subset of Gwhich fix x, i.e., Gx=
{g∈G:gx =x}.Foranyx∈S, the stabiliser Gxis a subgroup of G.Ifxand yare
two elements of Sbelonging to the same orbit, the stabilisers Gxand Gyare conjugated
subgroups, Gy=gGxg−1,wherey=gx. The stabilisers of the elements of an orbit
form a complete conjugacy class of subgroups of G.
For any subgroup Hof G, the set of left cosets of Hin G,givenbyG/H =
{gH:g∈G}, is a transitive G-set under the G-action given by f(gH) =(fg )H .In
this case, the stabiliser of gH is gHg−1. For any element xof a G-set Sthe orbit of xis
isomorphic to G/Gx.
Let Hand Kbe subgroups of G, the sets G/H and G/K are isomorphic G-sets if
and only if Hand Kare conjugated subgroups.
Let ={G1(={1}), G2,G
3,...,G
s(=G)}be a full set of nonconjugated sub-
groups of G. The set of transitive G-sets {G/Gi:i=1,2,...,s}is a complete set of
orbits. This means that every G-set Sis isomorphic to a disjoint union of such orbits:
S∼
=•

i
ai(G/Gi), (1)
where Giranges over all elements of and aiis the number of times that the orbit G/Gi
appears in the decomposition of S. The coefficients aiare uniquely determined and can
be obtained as solutions of the system of linear equations [2]:
s

i=1
Mjiai=bj,j=1,2,...,s. (2)
Here Mji is the number of elements in G/Giwhich are fixed points of the subgroup Gj,
and bjis the number of elements in Swhich are fixed points of Gj, where both Giand
Gjrun through the set .

E.M. Torres / Burnside rings 107
The square matrix of dimension sformed by the numbers Mji is called the table
of marks of the Ggroup. This matrix is nonsingular, hence we can to obtain its inverse
M−1which is known as the Burnside matrix. From equation (2) we obtain:
ai=
s

j=1M−1ij bj,i=1,2,...,s, (3)
where (M−1)ij is the ij entry of M−1.
Let Sand Tbe two G-sets. The Cartesian product of Sand T, denoted by S×T,
is the set of all ordered pairs (x , y) where x∈Sand y∈T;i.e., S×T={(x, y):
x∈S,y ∈T}. The action of Gon S×Tis given by g(x,y) =(g(x), g(y)),forany
g∈Gand any (x, y) ∈S×T.Sinceg(x) ∈Sand g(y) ∈T, S ×TisaG-set.
The Cartesian product of the G-sets G/Giand G/GjisaG-set, then it is isomor-
phic to a disjoint union of orbits:
(G/Gi)×(G/Gj)∼
=•

k
nij ,k (G/Gk), (4)
where Gkranges over all elements of .IfGlis a subgroup of Gthe number of fixed
points of Glin G/Giand G/Gjare Mli and Mlj , respectively. Then the number of fixed
points of Glin (G/Gi)×(G/Gj)is MliMlj . By applying equation (3), we obtain:
nij,k =
lM−1klMliMlj .(5)
The Burnside ring B(G) of the group Gis defined by [3]
B(G) =s

i=1
ai(G/Gi):ai∈Z,(6)
where Zis the set of integer numbers. The Burnside ring is a commutative ring with
identity G/Gs, where the sum (G/Gi)+(G/Gj)is the disjoint union of G/ Giand
G/Gj, and the product (G/Gi)·(G/Gj)is the Cartesian product of G/Giand G/Gj,
i.e.,
(G/Gi)+(G/Gj)=(G/Gi)˙
∪(G/Gj),
(G/Gi)·(G/Gj)=(G/Gi)×(G/Gj). (7)
From equations (6), (8) and (9) we conclude that every G-set Sis isomorphic to an
element of B(G):
S∼
=
i
ai(G/Gi). (8)

108 E.M. Torres / Burnside rings
3. Representations of the group Ggenerated by G-sets
By the action of the elements of G,aG-set Saffords a permutation representa-
tion of G. According to equation (1), can be reduced as
=
i
aii,(9)
where iis the transitive permutation representation generated by G/ Gi. Each ele-
ment g∈Gis represented in iby a permutation matrix of dimension |G|/|Gi|whose
elements are given by
g(i)
xy =1,if gx =y,
0,if gx = y, (10)
for any x,y ∈G/Gi. The representations sand 1are the identity and the regular
representations, respectively.
The elements of an orbit G/Gican be combined linearly in order to obtain basis
functions of the irreducible representations (IR) of the group Gwhich are contained in
the representation i. Thus, if the IR is contained in i, we can write the γ-basis
vector of as
|γ =
x|xx|γ ,(11)
where the summation extends over all the elements x∈G/Gi. Usually the coefficients
x|γ are obtained by using the projection operator method [4]. However, if the number
of elements of Gis high, this method is very tedious. In addition, if an IR is contained
more than once in i, the different sets of basis functions of such IR obtained with the
projection operator method are, in general, no-orthogonal.
The Cartesian product (G/Gi)×(G/Gj)affords a representation i×j,which
according to equation (4) can be reduced as
i×j=
k
nij ,k k.(12)
Each element g∈Gis represented in i×jby a permutation matrix whose
elements are
g(i×j)
(u,v),(w,x) =1,if g(u, v) =(w, x),
0,if g(u, v) = (w, x), (13)
for any u, w ∈G/Giand v,x ∈G/Gj. According to equation (10), equation (13) is
equivalent to
g(i×j)
(u,v),(w,x) =1,if g(u) =wand g(v) =x,
0,if g(u) = wor g(v) = x. (14)

E.M. Torres / Burnside rings 109
From equations (14) and (10) we obtain:
g(i×j)
(u,v),(w,x) =gi
uwgj
vx.(15)
Hence, i×jis the Kronecker product of the representations iand j[3].
If the orbit G/Gkis contained in the product (G/Gi)×(G/Gj)we can obtain
symmetry-adapted functions for G/Gkby coupling the symmetry-adapted functions for
G/Giand G/Gj. In fact, let , and  be irreducible representation of Gsuch that
∈k,
∈i,
 ∈jand ∈×, then the γ-basis vector of can be written
as γ ;,
=
γ,γ  
x,y (x, y)xγyγγγγ ,(16)
where xand yrun over the elements of G/Giand G/Gj, respectively, and γγ|
γ are Clebsch–Gordan coefficients.
4. Orbits of the icosahedral rotation group (I )
Figure 1 shows the subgroup lattice for the Igroup. As we see, there exist nine
icosahedral orbits. In order to simplify the notation, the icosahedral orbit I/G
k(where
Gkis a subgroup of I) will be denoted as (Gk). The icosahedral orbits are isomorphic
to I-sets which can be obtained from a regular icosahedron (see table 1). The orbits
(D2), (D3), (D5)and (T ) cannot be isomorphic to sets containing single elements [5]
and, hence, for such orbits we have used I-sets whose elements are sets containing
more than one element. Figures 2 and 3 show the orbits (C5)and (T ), respectively.
Figure 1. Subgroup lattice of the Igroup.

110 E.M. Torres / Burnside rings
Table 1
Orbits of the Igroup.
Orbit Description
(I ) A single point in the origin of coordinates
(T ) The set of three orthogonal pairs of antipodal edge midpoints of the icosahedron
(D5)The set of pairs of antipodal vertices of the icosahedron
(D3)The set of pairs of antipodal face midpoints of the icosahedron
(D2)The set of pairs of antipodal edge midpoints of the icosahedron
(C5)The set of vertices of the icosahedron
(C3)The set of face midpoints of the icosahedron
(C2)The set of edge midpoints of the icosahedron
(C1)The set of vertices of a truncated icosahedron
Figure 2. Numbering of the icosahedral vertices.
Table 2contains the characters of the representations of the icosahedral group generated
by the above orbits and the reductions into irreducible representations. The permutation
representation of the generators of the Igroup which are spanned by the orbits (T ) and
(C5)are shown in table 3.
From the table of marks for the Igroup [6] and its inverse, shown in tables 4 and 5,
respectively, and using equations (4) and (5), we have obtained the multiplication table
for the icosahedral Burnside ring B(I) shown in table 6.
A simple application of such multiplication table is the labelling of elements of
I-sets. For example, (T ) ·(T ) ∼
=(T ) +(C3), where it is evident that
(T ) ∼
=(a, a):a∈(T ),
(C3)∼
=(a, b):a= b;a, b ∈(T ).

E.M. Torres / Burnside rings 111
Figure 3. Elements of the orbit (T ).

112 E.M. Torres / Burnside rings
Table 2
Characters and reduction of the transitive permutation representations of the Igroup.
IE12C512C2
520C315C2Reduction
I11111A
T50021A+G
D561102A+H
D3100012A+G+H
D2150003A+G+2H
C5122200A+T1+T2+H
C3200020A+T1+T2+2G+H
C2300002A+T1+T2+2G+3H
C1600000A+3T1+3T2+4G+5H
Table 3
Permutation representations C5and Tfor the generators of
the Igroup.
C5T
C1,12
5(1)(26543)(71110987) (14523)
C1,4,3
3(1 4 3)(2 5 8)(6 9 7)(10 12 11) (1)(2 4 3)(5)
C1,2
2(1 2)(3 6)(4 11)(5 7)(8 10)(9 12) (1)(2 3)(4 5)
Table 4
Table of marks of the Igroup.
(I ) (T ) (D5)(D
3)(C
5)(D
2)(C
3)(C
2)(C
1)
I10 0 0 0 0 0 0 0
T11 0 0 0 0 0 0 0
D510 1 0 0 0 0 0 0
D310 0 1 0 0 0 0 0
C510 1 0 2 0 0 0 0
D211 0 0 0 3 0 0 0
C312 0 1 0 0 2 0 0
C211 2 2 0 3 0 2 0
C115 6101215203060
This means that each face of the icosahedron can be labelled by (a, b),where
a= b, 1a5and1b5 (see figure 4).
According to table 6, the triple product of (T ) ·(T ) ·(T ) can be decomposed into
five orbits: (T ) ·(T ) ·(T ) ∼
=(T ) +3(C3)+(C1), where it is easy to see that
(T ) ∼
=(a,a,a):a∈(T ),
(C3)∼
=(a,a,b):a= b;a, b ∈(T ),
(C3)∼
=(a,b,a):a= b;a, b ∈(T ),

E.M. Torres / Burnside rings 113
Table 5
Inverse matrix of the table of marks of the Igroup.
IT D
5D3C5D2C3C2C1
(I ) 1000 0 0000
(T ) −1100 0 0000
(D5)−1010 0 0000
(D3)−1001 0 0000
(C5)00−1/20 1/20000
(D2)0−1/30 0 0 1/30 0 0
(C3)1−10−1/20 01/20 0
(C2)20−1−10−1/20 1/20
(C1)−11/31/21/2−1/10 1/6−1/6−1/41/60
Figure 4. Labelling of the faces of the icosahedron.
(C3)∼
=(b,a,a):a= b;a, b ∈(T ),
(C1)∼
=(a,b,c):a= b= c;a= c;a, b, c ∈(T ).
Thus, each vertex of the truncated icosahedron can be labelled by (a,b,c),wherea=
b= c, a = c, 1a5,1b5and1c5 (see figure 5). According to this,
the regular representation of I,C1can be obtained from T.
At last, the orbit (C1)is isomorphic to the product (T ) ·(C5). Then, since (T ) ·
(C5)={(a, b):a∈(T ), b ∈(C5)}, every vertex of a truncated icosahedron can be
labelled by (a, b),where1a5and1b12 (see figure 6).

114 E.M. Torres / Burnside rings
Table 6
Multiplication table of the Burnside ring (I).
(I ) (T ) (D5)(D
3)(C
5)(D
2)(C
3)(C
2)(C
1)
(I ) (I ) (T ) (D5)(D
3)(C
5)(D
2)(C
3)(C
2)(C
1)
(T ) (T ) +(C3)(C
2)(C
3)+(C2)(C
1)(D
2)+(C1)2(C3)+(C1)(C
2)+2(C1)5(C1)
(D5)(D
5)+(C2)2(C2)(C
5)+(C1)3(C2)2(C1)2(C2)+2(C1)6(C1)
(D3)(D
3)+(C2)+(C1)2(C1)3(C2)+(C1)(C
3)+3(C1)2(C2)+4(C1)10(C1)
(C5)2(C5)+2(C1)3(C1)4(C1)6(C1)12(C1)
(D2)3(D2)+3(C1)5(C1)3(C2)+6(C1)15(C1)
(C3)2(C3)+6(C1)10(C1)20(C1)
(C2)2(C2)+14(C1)30(C1)
(C1)60(C1)

E.M. Torres / Burnside rings 115
Figure 5. Labelling of the elements of (C1) by using the elements of (T ).
Figure 6. Labelling of the elements of (C1) by using the elements of (C5)and(T).
5. Symmetry-adapted functions for the C60 molecule
Because of the high order of the icosahedral group I, the attainment of icosahedral
symmetry-adapted functions by direct application of the projection operators method is
a laborious task. Therefore, any alternative method introducing some simplification is
well received. Here we show how the symmetry-adapted functions for the C60 molecule
can be obtained by using the concepts developed in the above sections.

116 E.M. Torres / Burnside rings
Table 7
Symmetry-adapted functions for the orbit (C5).
ψAa =(1/2√3)(v1+v2+v3+v4+v5+v6+v7+v8+v9+v10 +v11 +v12)
ψT1x=(1/2√+2)(v1−v2+v4+v5−v7+v9−v11 −v12)
ψT1y=(1/2√+2)(v3+v4−v5−v6+v7+v8−v10 −v11)
ψT1z=(1/2√+2)(v1+v2+v3+v6−v8−v9−v10 −v12)
ψT2x=(1/2√+2)(v1−v2−v4−v5+v7+v9+v11 −v12)
ψT2y=(1/2√+2)(−v3+v4−v5+v6+v7−v8+v10 −v11)
ψT2z=(1/2√+2)(−v1−v2+v3+v6−v8+v9−v10 +v12)
ψHϑ =(1/2√2)(v1+v2−v3−v6−v8+v9−v10 +v12)
ψHε =(1/2√6)(−v1−v2−v3+2v4+2v5−v6+2v7−v8−v9−v10 +2v11 −v12)
ψHx =(1/2)(v3−v6−v8+v10)
ψHy =(1/2)(v1−v2−v9+v12)
ψHz =(1/2)(v4−v5−v7+v11)
Note. =(1+√5)/2 is the golden number.
Table 8
Symmetry-adapted functions for the orbit (T ).
ϕAa =(1/√5)(v1+v2+v3+v4+v5)
ϕGa =(1/2√5)(4v1−v2−v3−v4−v5)
ϕGx =(1/2)(v2−v3+v4−v5)
ϕGy =(1/2)(−v2+v3+v4−v5)
ϕGz =(1/2)(v2+v3−v4−v5)
Using equation (16) and bearing in mind the relation (T ) ·(C5)∼
=(C1)(see
table 6) we can obtain mutually orthogonal symmetry-adapted functions for (C1)
by coupling the symmetry-adapted functions of (T ) and (C5). For this purpose we
have obtained symmetry-adapted functions for (T ) and (C5)by using the results ob-
tained by Boyle and Parker in their paper on a vibrating icosahedral cage [7] (see
tables 7 and 8). In order to use equation (16) we have employed the coupling coefficients
which were obtained by Fowler and Ceulemans [8] for the single-valued irreducible
representations of the Igroup based on the symmetry functions of Boyle and Parker.
The symmetry functions for the C60 molecule thus obtained are basis functions of the
matrix representation of the Igroup given in the appendix to the work by Boyle and
Parker [7]. Table 9contains the functions for the representation Hobtained by coupling
the symmetry-adapted functions for (C5)and (T ) which are base of the representations
T1and G, respectively. By reasons of space the resting functions are not shown in this
paper, but are available upon request.
Bearing in mind the relation (T ) ·(T ) ·(T ) ∼
=(T ) +3(C3)+(C1),where(C1)∼
=
{(a,b,c):a= b= c;a= c;a, b, c ∈(T )}, we could obtain symmetry-adapted
functions for (C1)from those of (T ). However, the functions thus obtained have the
disadvantage that, as occurs with the projection operator method, the different sets of
functions belonging to the same irreducible representation are non-orthogonal.

E.M. Torres / Burnside rings 117
Table 9
Functions |Hγ;H, G,γ =ϑ,ε, x, y, z.
|Hϑ;H,G=(1/8√15)(−6v1
1−v2
1+4v3
1+4v4
1−v5
1−6v1
2+4v2
2−v3
2−v4
2+4v5
2+6v1
3
+v2
3−4v3
3+v4
3−4v5
3−5v2
4−5v3
4+5v4
4+5v5
4+5v2
5+5v3
5−5v4
5−5v5
5
+6v1
6−4v2
6+v3
6−4v4
6+v5
6+5v2
7+5v3
7−5v4
7−5v5
7+6v1
8−4v2
8+v3
8−4v4
8
+v5
8−6v1
9+4v2
9−v3
9−v4
9+4v5
9+6v1
10 +v2
10 −4v3
10 +v4
10 −4v5
10 −5v2
11
−5v3
11 +5v4
11 +5v5
11 −6v1
12 −v2
12 +4v3
12 +4v4
12 −v5
12)
|Hε;H,G=(1/8√5)(2v1
1−3v2
1+2v3
1+2v4
1−3v5
1+2v1
2+2v2
2−3v3
2−3v4
2+2v5
2+2v1
3
−3v2
3+2v3
3−3v4
3+2v5
3−4v1
4+v2
4+v3
4+v4
4+v5
4−4v1
5+v2
5+v3
5+v4
5+v5
5
+2v1
6+2v2
6−3v3
6+2v4
6−3v5
6−4v1
7+v2
7+v3
7+v4
7+v5
7+2v1
8+2v2
8−3v3
8
+2v4
8−3v5
8+2v1
9+2v2
9−3v3
9−3v4
9+2v5
9+2v1
10 −3v2
10 +2v3
10 −3v4
10 +2v5
10
−4v1
11 +v2
11 +v3
11 +v4
11 +v5
11 +2v1
12 −3v2
12 +2v3
12 +2v4
12 −3v5
12)
|Hx;H,G=(1/4√30)(−5v3
1+5v4
1+5v2
2−5v5
2+4v1
3−v2
3−v3
3−v4
3−v5
3−5v4
4+5v5
4
−5v2
5+5v3
5−4v1
6+v2
6+v3
6+v4
6+v5
6−5v2
7+5v3
7−4v1
8+v2
8+v3
8+v4
8+v5
8
+5v2
9−5v5
9+4v1
10 −v2
10 −v3
10 −v4
10 −v5
10 −5v4
11 +5v5
11 −5v3
12 +5v4
12)
|Hy;H,G=(1/4√30)(4v1
1−v2
1−v3
1−v4
1−v5
1−4v1
2+v2
2+v3
2+v4
2+v5
2−5v3
3+5v5
3−5v2
4
+5v3
4+5v4
5−5v5
5+5v2
6−5v4
6+5v4
7−5v5
7+5v2
8−5v4
8−4v1
9+v2
9+v3
9+v4
9
+v5
9−5v3
10 +5v5
10 −5v2
11 +5v3
11 +4v1
12 −v2
12 −v3
12 −v4
12 −v5
12)
|Hz;H,G=(1/4√30)(−5v2
1+5v5
1−5v3
2+5v4
2+5v2
3−5v4
3+4v1
4−v2
4−v3
4−v4
4−v5
4−4v1
5
+v2
5+v3
5+v4
5+v5
5+5v3
6−5v5
6−4v1
7+v2
7+v3
7+v4
7+v5
7+5v3
8−5v5
8−5v3
9
+5v4
9+5v2
10 −5v4
10 +4v1
11 −v2
11 −v3
11 −v4
11 −v5
11 −5v2
12 +5v5
12)
6. Supplementary material
A print-out of the complete symmetry-adapted functions for the C60 molecule is
available upon request.
References
[1] J.S. Rose, A Course on Group Theory (Cambridge University Press, Cambridge, 1978).
[2] W. Hässelbarth, Theor. Chim. Acta 67 (1985) 339.
[3] C.W. Curtis and I. Reiner, Methods in Representation Theory (Wiley, New York, 1981).
[4] M. Hamermesh, Group Theory and its Applications to Physical Problems (Addison-Wesley, Reading,
MA, 1962).
[5] S. Fujita, Theor. Chim. Acta 76 (1990) 45.
[6] F. Oda, J. Algebra 236 (2001) 29.
[7] L.L. Boyle and Y.M. Parker, Mol. Phys. 39 (1980) 95.
[8] P.W. Fowler and A. Ceulemans, Mol. Phys. 54 (1985) 767.