Well known properties of induced unitary representations are generalised for induced PUA representations. They are applied to obtain an expression for the multiplicities in the Clebsch-Gordon series of induced PUA representations. Finally we show how these multiplicities can be used for the determination of selection rules

Broek, P.M. van den

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van den Broek, P.M.

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Induced PUA representations and selection rules

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J. Phys. A: Math. Gen., Vol. 11, No. 5, 1978. Printed in Great Britain. @ 1978 Induced PUA representations and selection rules P M van den Broek Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, Nijmegen, the Netherlands Received 29 September 1977, in final form 19 December 1977 Abstract. Well known properties of induced unitary representations are generalised for induced PUA representations. They are applied to obtain an expression for the multi- plicities in the Clebsch-Gordon series of induced PUA representations. Finally we show how these multiplicities can be used for the determination of selection rules. 1. Introduction In the quantum mechanical description of a physical system with symmetry group G the Hilbert space 2 of state vectors carries a projective unitary-anti-unitary (PUA) representation T of G, according to the well known results of Wigner (1939) and Bargmann (1954). As a basis for 2 we may choose vectors 4V, transforming under G as basis vectors of irreducible PUA representations Dp of G: Here p labels the irreducible PUA representations of G which have the same factor system as T, the index a denotes that there may be more than one set of basis functions transforming according to Dp, and m and n have values from one up to the dimension of Dp. All relevant physical information, such as transition probabilities, is contained in matrix elements (4V, O&5Tb), where 0’ is an irreducible tensor opera- tor transforming according to the irreducible UA representation D’ of G: In order to study the properties, due to the symmetry group G, of these matrix elements one has to study the PUA representations of G. Analogous to the well known selection rules for unitary representations the matrix elements ( 4 2 ,  OE4Tb) are zero for each m, k, I, a and b if the trivial UA representation of G does not occur in the triple direct product Dp’ 0 D’ 0 D”. This will be shown in 0 5 .  An important example of a physical system where one has to use projective unitary representations instead of ordinary unitary representations is an electron in a crystal- line solid with a uniform magnetic field (Brown 1964, Tam 1969). Moreover, whenever an element g of G contains the time reversal operation, T(g)  must be an anti-unitary operator. This shows that the study of PUA representations is of general importance for physics. 813 814 P M van den Broek It often occurs, for instance if G is a Shubnikov space group, that one has to obtain the PUA representations of G with the help of an induction procedure. The induction procedure for unitary representations, which goes back to Frobenius, is by now well known. (See for instance Coleman (1966) and Bradley (1966).) This induction procedure has recently been generalised for PUA representations by Shaw and Lever (1974). The aim of this paper is to study general properties of induced PUA represen- tations and to apply these properties to the study of selection rules. In § 3 we generalise some well known properties of induced unitary representations, such as for instance the Frobenius reciprocity theorem and Mackey’s subgroup theorem. In § 4 we use these results to obtain an expression for the multiplicities in the Clebsch- Gordan series of induced PUA representations. Finally in § 5 we discuss selection rules, using the results of the preceding sections. 2. Preliminaries Let G be a finite group and GO a subgroup of index 1 or 2. A PUA representation of G with respect to GO is a mapping D from G into the operators on some Hilbert space %? such that: (i) D ( g )  is unitary if g E GO and anti-unitary if g &  Go. (ii) D ( g )  D ( g ‘ )  = v ( g ,  g ‘ ) D ( g g ’ )  for all g,  g’ E G for some mapping v: G x G + (iii) D ( e )  = I, where e is the identity of G and I is the identity operator on %?. U(1). The mapping v is called a factor system of G with respect to Go and satisfies d g ,  e > = & ,  g ) =  1 V g c  G (2.1) d g ,  g”g’, g ” )  = c+(g, g ’ g ’ ’ ) d ( g ’ ,  g “ )  (2.2) and where A ’ is defined by V g ,  g ’ ,  g”E G if g E  Go if g &  Go and the asterisk denotes complex conjugation. In the following we shall drop the phrase ‘with respect to . . .’ wherever it is clear which the appropriate subgroup of index 1 or 2 is. Suppose 2 is finite-dimensional and let { e l , .  . . , e , }  be an orthonormal basis of 2. For each operator D ( g )  a unitary n x n matrix 9 ( g )  is defined by B i i ( g )  = (ei, D ( g ) e j ) .  The matrices 9 ( g )  form a PUA matrix representation or PUAM representation of G :  ~ ( g ) ~ ’ ( g ’ )  = a(s7 g ’ P ( g g ’ )  (2.4) where 9’ is defined by if g E  Go if g &  Go. A PUA representation D of G is reducible if there is a proper subspace 22“ of 2t which is invariant under D(G);  otherwise D is irreducible. The direct product D1 X D2 of PUA representations D1 and D2 of G with factor systems v1 and v2 respectively is a Induced PUA representations and selection rules 815 PUA representation of G with factor system ( ~ 1 ~ 2  and is defined by (91 9 2 k ) i j , k I  = g l ( g ) i k 9 2 ( g ) j l *  (2.6) Two PUA representations D1 and D2 of G are equivalent if there exists a unitary matrix S such that (2.7) or, equivalently, if Tr@a(g)) = Tr{92(g)I V g  E Go. (2.8) In the following we shall identify equivalent PUA representations, and also identify a PUA representation with the corresponding PUAM representation. Suppose H is a subgroup of G, and let g l  = e, g 2 ,  . . . , g, be left coset representatives of G with respect to H. Then G = giH. Let c be a factor system of G and let A be a PUA representation of H with respect to Ho with factor system (T, where Ho = H n Go. Then ALG is the PUA representation of G with factor system (T which is induced from A and is defined by It is straightforward calculation to verify that A G does not depend on the choice of the left coset representatives. By convention, A G will be an induced UA represen- tation. 3. Properties of induced PUA representations The number of times an irreducible PUA representation D” occurs in a PUA represen- tation D, which we will denote by i(D”’, D) ,  is given by where x” and x are the characters of D” and D respectively. This property has been given by Karavaev (1965) for UA representations and can easily be shown to hold for PUA representations as well, independent of the factor system involved. An immedi- ate consequence is where Do is the trivial UA representation of G and (Go( denotes the order of Go. In the remainder of this section w and (T will be factor systems of the finite group G, H will be a subgroup of G, and Do(H) and Do(G) will be the trivial UA representations of H and G respectively. Theorem 1 .  Let D be a UA representation of H. Then IHI P o l  l H o l  /GI i (Do(H),  0 )  = -~ i(Do(G), D t G) .  (3.3) 816 P M van den Broek Proof. Define S ( g )  by S ( g ) =  1 if gEH and S ( g ) = O  if g & H ;  let rL and x be the characters of D and D t G respectively. Then Theorem 2. Let A be a PUA representation of H with factor system U and r a PUA representation of G with factor system W .  Then r @ ( A t G ) = [ ( r J H ) O A ]  t G. U OJU (3.4) Proof. We have to show that Tr{[T O (ALG)](g)} and Tr{[((T .1 H ) O  A)L,G](g)} are equal for each g E  Go. Evaluating both terms we see that t h i v i s  true if x(g)w*(g, g j )w(gj ,  g i ' g g j )  = xgj(gY'ggj)  for each j and each g E GO for which g j ' g g j  belongs to H, if x is the character of r. But this follows from r(gj)rg ' (gT'ggj)rg ' (gi '  ) = m(gj, gT'ggj)W(ggj, g T ' F ( g )  if we take the trace of both sides and use the equations (2.1) and (2.2). Theorem 3 (Frobenius reciprocity theorem). Let r and A be irreducible PUA represen- tations of G and H respectively, both with factor system U, and let their characters be x and $ respectively. Then 1 1 - i ( r ,  A t G )  C x(g)x*(g)=iTJi i(A, r .1 W C cL(gM*(g). (3.5) /GI gcGo g C H o  Proof. From equation (3.2) it follows that the left-hand-side is equal to (IGol/lGl)i(Do(G), (A  ,! G ) O  r*) and the right-hand-side is equal to (IHol/lHl)i(Do(H), (r .1 H ) O  A*). But this is, from theorem 1,  equal to (IGol/lG~)i(Do(G), [(rJ H ) O  A*] .T G). Finally theorem 2 completes the proof. Theorem 4 (Transitivity of induction). Let M be a subgroup of G such that H is a subgroup of M. Let A be a PUA representation of H with factor system (T. Then ( A T M ) T G = A t G .  (3.6) U U  U Proof. We have to show that Tr{((A L M )  L G)(g)} = Tr{(A G)(g)}  for all g E Go. But this is, after evaluating the traces, a straightforward calculation using the equations (2.1) and (2.2). Let K be a subgroup of G. Then we can factorise G into double cosets: G = XiHdiK. Let Ki be defined by Ki = diKdT' and Li by Li = H n K i .  If D is a PUA representation of K with factor system U, then the PUA representation Di of Ki with Induced PUA representations and selection rules 817 factor system U is defined by Di (dikd 7' ) = u(dikd 7', di)u*(di, k ) D d i  (k  ) t lk E K. (3.7) Theorem 5 (Mackey's subgroup theorem). Let D be a PUA representation of K, then Proof. Let H = X j  h!Li be a left coset decomposition for H with respect to Li. Then a left coset decomposition for G with respect to K is given by G = Zi Z j  h{diK (Bradley 1966). After evaluating the traces of both sides of equation (3.8) with these coset representatives the theorem is proved in a straightforward way using the equations (2.1) and (2.2). Theorem 6. Let r be a PUA representation of H with factor system w and A a PUA representation of K with factor system U. Then ( A  t G )  0 (r t G) = 1 {((Ai .1 Li) 0 (r .1 Li)) t GI. (3.9) U " i UJU Proof. This theorem follows from the preceding theorems in the same way as for the case of unitary representations (Bradley 1966 and Coleman 1966). 4. Clebsch-Gordan series of induced PUA representations We assume that all irreducible PUA representations of G are known, and are obtained with the generalised induction procedure. So each irreducible PUA representation of G with factor system U may be written as A'"'" G, where A'""" is an irreducible PUA representation of a subgroup HW3" of G with factor system U. In this section we will consider the Clebsch-Gordan series (As'u t G )  0 (A''w t G )  = 2 C(w, Y, p ;  U, w)(AP'"" f G). (4.1) U w P "U Here the multiplicity C(@, v, p ;  U, w )  can be written, according to equation (3.2), as The U A  representation (Apsu" 2, G)* 0 (A'"'" G) 0 (Aysw L G )  can be decomposed by applying theorem 6 twice. Let us factorise G into double cosets as follows: G =  H Y  n H",". If AY' is defined according to equation (3.7) by ZiH"'wgiHW'u, let HF' be given by HY" = giH"*"g;' and KY";"." by KY*';".u = Af'"(giggi ')  = u(gigg;',  gi)c+*(gi, g )AwSugi (g)  V g  E H (4.3) (4.4) then we have (Arpu t G )  0 (A"@ G) = 1 {((A,""'J K ~ ~ ' ; " ~ u )  @ (Aysw  J K'"sy;"*u)) t G}. U w i w u  Now suppose G is factorised into double cosets for each i as follows: G =  K~"~"3ug., ,Hp*"u I t  and further and K!';"J';O*u are defined by HX$" = is defined according to g,,iH"'-g~/ and K ~ ~ ' ' p ; o ' u  = He..- I C  n Krv'"*u. If finally 818 P M van den Broek = {((A!;" .1 K )  0 (A?" 3. K )  0 (Av@ J. K ) )  t G )  (4.6) i i  where we wrote Kj,i instead of Kr;",p'OJ,U for brevity. Combining the equations (4.2) and (4.6) gives W O  Finally, using theorem 1 and the fact that, if K i i  = K,,i n GO (4.7) for each UA representation D of Kj,i, we obtain 5. Selection rules Let us consider a physical system with symmetry group G and suppose that the eigenfunctions of the Hamiltonian of the system transform under g as basis functions of a PUA representation T of G. An example of such a physical system is an electron in a crystalline solid with a uniform magnetic field; the symmetry group in this case is a Shubnikov space group (Brown 1964, Tam 1969). We are interested in matrix elements (c$L, OPc$;). Here the functions {d?, i = 1, . . . , dim Dy} form a basis for the irreducible PUA representation Dy of G with factor system U: and 0" is an irreducible tensor operator transforming according to the irreducible UA representation D" of G: T(g)OPT(g)-' D"lk(g)O?. (5 .2 )  1 Analogous to the well known selection rule for unitary representations the matrix Induced PUA representations and selection rules 819 elewent (47, O$$i‘) is equal to zero due to the symmetry if the trivial UA represen- tation of G does not occur in the triple direct product Dp* 0 D” 0 D”. This can be seen as follows. If D” and D p  are two inequivalent irreducible UA representations of G then we have the following orthogonality relation (Dimmock 1963): for each UA representation that does not contain the trivial UA representation. Since the operators T ( g )  are unitary if g E Go and anti-unitary if g& Go we have and the right-hand-side is equal to zero due to equation (5.4) if Dp’ 0 D’” O D” does not contain the trivial UA representation of G. Now suppose, as in 0 4, that Dp, D” and D” are obtained with the generalised induction procedure: D p  = (Apsu G); From the results of § 4 we finally obtain the result that ($,“m, Ot4;) is equal to zero for all m, k and 1 if (A:;“ .1 K;,)O (AY J. K;,)O (A”,, .1 Kiqi) (notation as in § 4) does not contain the trivial UA representation of K;i for any i and j .  Note that the order of APsu’, A” and A”,“ may be chosen arbitrarily. In a forthcoming paper we will examine this selection rule for the case where G is a Shubnikov space group. D” = (Au,“ t ,G) and D” = A ”  t G. Acknowledgment I thank Dr T Janssen for reading the manuscript. References Bargmann V 1954 Ann. Math., N Y  59 1-46 Bradley C J 1966 I. Math. Phys. 7 1145-52 Brown E 1964 Phys. Rev. A 133 1038-44 820 P M van den Broek Coleman A J 1966 Queen's Papers in Pure and Applied Mathematics No. 4 (Kingston, Ontario: Queen's Dimmock J 0 1963 J. Mark Phys. 4 1307-11 Karavaev G F 1965 Sou. PhysAolid St. 6 2943-8 Shaw R and Lever J 1974 Commun. Math. Phys. 38 257-77 Tam W G 1969 Physica 42 557-64 Wigner E P 1939 Ann. Math., N Y  40 149-204 University) 

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