The Relation between Generator Theory and the Theory of Matrix Representations in Symmetry Point Groups of Finite Orders I. Baraldi and A. Carnevali Dipartimento di Chimica, Universita' di Modena, Via Campi 183, 41100 Modena, Italy
In textbooks and articles on group theory and molecular symmetry destined for chemists, generator theory of a group and its relation to the theory of representations of a group with matrices receives a rather scanty treatment. Some of these texts feature the concept of generator sets in order to present point groups (1,2), whereas other texts use generators to determine the symmetry species of molecular wave functions ( 3 , 4 ) . The article by Burrow and Clark on the graphs of point groups (5)is interesting, but it is confined to generator theory. Thus, as far as we can determine, some thing vital is lacking: material that connects the generator theory of symmetry point groups and the theory of representations of these groups with matrices. In this article we develop for chemists these aspects of the theory of representations of a group that are less well known to them. We assume that the reader is familiar with the concepts of a generator set, a word, and a set of defining relations of a group (6, 7).First, we briefly apply the generator theory to the finite symmetry point groups. Then we reelaborate the theory of matrix representations of these groups using generator theory In view of the importance of the representations of a finite point group with matrices for molecular spectroscopy and theories of chemical bonding, this article takes on special interest. A simple example is given by way of demonstration. Generator Sets and Defining Relations for Finite Symmetry Point Groups Let G be a finite group of order g and elements
GI, G2. .... Gg
Generator Sets (S) and Set of Defining Relations (qS) = E) for Finite Symmetry Point Groups
O=E Symmetry S = (GI,G2, ....G,j R ( G ,G2,.... G point groups Cn
1cd
G =E
cma
(cn,ov)
G=&E
(u.c~)~ =E
Cnh(n even) (Cn,oh)
G=+E
(ohCn)"= E
(n odd)a (cn,oh)
G=&E
U~C~OZ'
cnh
s2nb
(s2n1
&=
Dn
I Cn, Cil
G=&=E
Dnh
G'=
E
( c Z C ~ )=~E
(n odd)
Dnd
T Td Th
0 Oh
Then we write G = (GI,Gz, ..., Gpl
Y
Because each group is identified using isomorphisms from one of its generator sets S = (GI,..., GJ and from a set of defining relations R(Gl, Gz, ..., GJ =R(S)= E the group can be described in the following way, G = (SIR(S)=El where E is the identity element.' In finite point groups the various symmetry operations -E, C., u, S,, and i-that make up the elements of the group can be expressed as combinations of one or more elementary transformations, namely C. rotation and u reflection. Thus, each of their generators will be nothing other than arotation, a reflection, or a product of these two 'To be precise, the last relation should be written using angular brackets ( E >).
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Journal of Chemical Education
operations. The words of these groups are therefore seauences of rotations and reflections belongina to the set S, and each element of the group can be represented as product of rotations and reflections belonging to S. The presentation of symmetry point groups using generators is well-known. (See, for example, ref 2 and the table of this article.) Usually generator sets contain a maximum of three elements, and different generator sets can be assigned to the same point group (8).Each symmetry point group is completely defined when we add the set of defining relations of G to one of its generator sets S.
a
The table shows the generator set and the set of defining relations for all finite symmetry point groups. I n this table the defining relations of the point groups with a principal axis have been obtained intuitively by generalizing the relations known for some groups (5,6). For the cubic groups the reader is referred to ref 5. For the icosahedral groups, the defining relations of Y have been taken from the book by Grossman and Magnus (6). The relations of Y h are
because Y r = Y x C; The tabi; includes a11 the infomation needed tn describe finite ooint erouos. It also hiehliehts the isomorohism between'the groups and the reiations not only beiween the powers of the generators but also between all the group elements because each group element can be written as a word. For example, groups Cq,, Dq, and DM are isomorphous; they can be considered a s different realizations of the same abstract non-Abelian group G of order 8, generated by the abstract elements GIand Gz of orders 4 and 2, and with the following set of definitions.
- .
Relation between the Generator Theory and the Theory of Matrix Representations in Finite Groups A representation of a group G is a matrix group (squares, nonsingular, all of the same order n) onto which the group to be represented is homomorphic (9).If this homomorphism is isomorphism, the representation is said to be faithful. Let
This notation will hold also when the representation is irreducible. The foregoing definition of matrix representation leads to the following important result: The properties of the matrices of the set Ds that are not destroyed by the matrix multiplication and by the operation of inversion are also ~ r o ~ e r t i of e sthe matrices of D. Therefore, if all the matri&sbflls are unitary miltriceri~ormiltriceswith same block dii~coni~l form,, then the matrices of D will also be unitary (or have the same block diagonal form a s those of Ds). The demonstration of these properties is based on the fact that the omduct and the inversion of unitarv matrices (or block diagonal matrices with the same formjgenerates unitarv matrices (or block diazonal matrices with the same form).~owever,the matricesof D can be expressed a s matrix words. showine the validitv of the ~ r e v i o u statement. s reduced, In particular, if the-matrices o i are~completely ~ then so are those of D. Reducibility If A is a matrix that commutes with all the matrices of Ds, then A wmmutes with all the matrices of D. On the basis of this, the well-known theorem connected with the lemma of Schur can be reformulated a s follows. Lemma of Schur If there is a matrix A that is not a multiole of the unit matrix and that commutes wnh all the mntnrea ofD. then rhls wprc sentatton
D = ID(G,), MGz), ... 1 be a matrix representation of G. From the definitions of generator set, set of defining relations, and homomorphism between groups, we get the following.
IS
wduclble
{See W l p w Y
.Smlrnov $10,
Reformulation If the matrix A that commutes with all the matrices of Ds is not a multiple of the unit matrix, then D is reducible.
If S = IG,, Gz, ..., G,I is a generator set of G, then Ds = ID(Gl),D(Gz),...,D(GJ1 constitutes a generator set of D.
Retaining Block Diagonal Form Let D(X) be a nonsingular matrix of order n such that the similarity transformation D(x)-' Ds D(X) = Ds
'If
produces a set Dp= ID'(Gl), D'(Gz), ..., D'(Gi)I
is a relation of G. In other words, if (with G,, ..., G, then it follows that
E
Sand na, nb, ..., nz = +1)
D(G,)~D(G~)" *... D(GJN = D(E)
(where D(E) is the n x n unit matrx) is a relation of D. If D is a faithful representation and if the relations R,,Rz, ..., R, constitute a set of defdng relations of G, then D(R,), D(Rz), ...,D(R,) will constitute a set of defining relations of D. However, if D is unfaithful, in order to obtain a set of defining relations of D, other relations must be added. The complete set of defining relations of D is indicated by R(Ds) = D(E) In concise form the representation D can be indicated as D = IDs IR(Ds) =ME))
composed of block diagonal matrices all of the same form. Thus. the same similaritv transformation amlied to the matrices of D transforms them into matricesif the same block diagonal form corresponding to those of Dp, that is
D~XP D an = D' where D'= ID'(Gl), D'(Gz), ... 1
contains matrices that all have the same form a s those of Dp. These two theorems can easily be demonstrated using the concept of matrix word. Simplifying the Reduction of a Representation Thus, the theoretical procedure of reduction of a representation (reducible) in its reduced fonn will be considerably simplified. Indeed, in this case it is sufficient to find the matrix D(X) that simultaneously transforms into block diagonals all the matrices of Ds, and n o t a s in classical procedure--all the matrices of D. Because in the finite symmetry point groups the maximum number of generators is 3, the set Ds will contain a t most three matrices of D. The convenience of our procedure for the reduction of D into irreducible components appears evident. It is suffiVolume 70 Number 12 December 1993
965
cient to find the similarity transformation that reduces Ds to irreducible comuonents. The matrices of D~may always be formed of unitary matrices because the matrices of D can be ~ ui nt the form of unitary matrices after appropriate similarity transformations have been carried out. A unitary matrix can always be put in the form of a diagonal matrix using a similarity transformation. Even two or more unitary matrices that commute two and two can always be simultaneously put in the form of diagonal matrices using the same similarity transformation. Thus, the set Ds composed of unitary matrices that commute two by two can always be put simultaneously in the form of diagonal matrices, that is, formed of one-dimensional irreducible representations. This is no longer possible, in general, for set Ds composed of matrices that do not commute. In this we are dealing also with multidimensional irreducible representations. Generator Sets and Defining Relations for Matrix ReDre~entati~I'S of Finite Svmmetw Point G~OUDS In this I;ist section we discuss the generator sets and the defining relations for faithful representations of finite symmetry point groups. In addition, a n example will be given. Faithful and Unfaithful Representations
From the isomorphism existing between groups and faithful representations, a table analogous to the full table giving the generator sets and set of defining relations can easily be derived, though operating in the field of (faithful) matrix representations. This table is easily obtained from the table by substituting the symmetry operations Gj E S with the corresponding matrices D(G,) E Ds of the faithful representation. Due to this resemblance the entire table is not given, and only two examples follow.
,., Some symmetry elements and AO's Zp, of CH
due to isomorphism-are similar to those obtained between the generators. For example, let us consider the four-dimensional representation of the symmetry point group D4h, which is obtained by taking as basis the four atomic orbitals (AO's) 2p, of C4H4(see the figure). This faithful representation can be indicated as follows. D(D,) = ID(C4),D(C2),D ( ~ , ) I D ( c=~D(C212 ) ~ = D(shl2 = (D(CZ)WC~))~ = (D(o~)D(cJ)~ = (D(o~)D(cz))' = D(E)t
where
The re~resentativematrices of all the other elements of thegroup U4h were obtained by multiplication ofthree maFor example, because trices: D C,), DIG,,, and Dcnh~.
and due to isomorphism, we get
Let The matrices belonging to Ds of a faithful representation are periodic matrices in which the period coincides with the order of the corresponding symmetry operation. When this does not hold, the representation is unfaithful. I n other words, for the unfaithful representations the relations corresponding to those of the table continue to hold, except that they are no longer the fundamental relations between the matrices of Ds. Moreover, the defining relations of faithful representation also imply another set of relations between the powers of generator matrices that-
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Journal of Chemical Education
which is the orthogonal matrix for study of the reducibility of Ds = lD(C4L D(C21, D(adt The similarity transformation D(D'
Ds D(X) = D,
gives the set Ds = lD'(C4), D'(C2), D'(ah)I, whose matrices are
The four AO's 2p, of the CaH4 give rise to one MO of a2, species, one of bzu species, and a degenerate pair of eg species. Conclusion
that is, they are completely reduced matrices. On the basis of what was said in the theoretical part, the similarity transformation
The approach to the theory of matrix representations in finite symmetry point groups expounded here highlights certain aspects that, in our o~inion,tend to remain in the background when following'the method of presentation trnditimally used in texthooks on group theory intended for chemists. Perhaps the most important point is that it is sufficient to deal with an extremely limited number of matncri;: onlv those heloncineto D ~ a n dnot to the total set D. The probcm of the redYucgon OFDthus becomes the mere reduction of Ds, a problem normally containing one, two, or-at m o s t t h r e e matrices. Acknowledgements
gives a completely reduced representation D Y D d We now check the matrix D(i) previously calculated. As expected, we get
The authors are grateful to G. Z ~ DofDthe ~ De~artment Dini" of the u&ersity of ~ l o r e n c efor of Mathematics, '6 his valuable sueeestions and stimulating discussions. ~, Also, I' Mimne i f t h e 1)epanment of ~ h e m i s t r L.niversity of Modena, suggested some useful changes to the text. Literature Cited
-.
In particular, the representation D'(Da) is a n example of unfaithful representation because D'(C4I2 = D(E). From comparison of the character table of group D4h to the characters of the block diagonals forming the set of matrices of Ds., we get D(Dph)
= Azu + Bzu + Eg
Ekcfmnlc Sfrudum of ~ d y x o i y o fMohcuks; ~~i~ Van ~ o s t ~ & d~nneeton: : NJ, 1966. 4. Barchewitz, P Spectroscopic Atorniqup at Moleculoim, Tome 11; Sp~cfroscopm Mokculoim; Masson: Paris, 1971. 5. Burnun. E. L.:Clsrk, M. J. J. Cham. Educ. 1974.61.87-W. 6. Gmrsman, I.; Magnus, W . Gmups and Thair Graphs: New Mathematical Library RandomHouse: New Yark, 1966. 7. Ledemann, W introduction to Gmup Theory: Longman, 1973. 8. Kouyoumdjian, E. R. J ChemZdue 1989.60.643-644 9. Wigner, E. P. Gmup Thmry;Academic Piera: New York. 1959. 10. Smrnou, V.Course de Mothamtiquas SupROums, T o m 1ii;Prpmiwe Pol?k,Editions MIR: Moscow, 1970.
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