Applications of S-Functional Analysis to Continuous Groups in Physics P.H

Applications of s-functional analysis to continuous groups in physics P.H. Butler, B.G. Wybourne

To cite this version:

P.H. Butler, B.G. Wybourne. Applications of s-functional analysis to continuous groups in physics. Journal de Physique, 1969, 30 (10), pp.795-802. 10.1051/jphys:019690030010079500. jpa-00206842

HAL Id: jpa-00206842 https://hal.archives-ouvertes.fr/jpa-00206842 Submitted on 1 Jan 1969

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LE JOURNAL DE PHYSIQUE TOME 30, OCTOBRE 1969, 795.

APPLICATIONS OF S-FUNCTIONAL ANALYSIS TO CONTINUOUS GROUPS IN PHYSICS (1)

By P. H. BUTLER and B. G. WYBOURNE, Physics Department, University of Canterbury, Christchurch, New Zealand. (Reçu le 14 mars 1969.)

Résumé. 2014 Les fonctions S, telles qu’elles ont été développées par Littlewood, sont passées en revue dans le but de simplifier l’algèbre des groupes continus. La division de la fonction S est définie et la théorie a été développée jusqu’au point où un programme pour calculatrice électronique (computer) a été établi, ce qui permet le calcul des produits Kronecker, des lois dérivées, des pléthysmes sur deux variables, et des produits internes du groupe symétrique.

Abstract. 2014 S-functions, as developed by Littlewood, are reviewed with the aim of simplifying the algebra of continuous groups. S-function division is defined and the theory developed to a stage where a computer programme has been written that performs Kronecker products, branching rules, plethysms on two variables, and inner products of the symmetric group.

1. Introduction. - In the past decade, theoretical the characters of representations of the unitary, sym- physicists have shown an unprecedented interest in plectic, orthogonal and rotation groups. and its the theory of continuous groups application Methods for calculating outer products of S-func- to a wide of range physical problems. Notable, tions are well known [7] and these are developed so the has been the use of among many applications, as to give simply and unambiguously the Kronecker the continuous to describe the compact groups sym- products for all the above groups. A method for metry transformation properties of N-particle atomic determining the inner product of S-functions without and nuclear wave functions the work following early the usual recourse the character tables, is used together of Racah have tended to concen- [1, 2]. Physicists with a more recent development, that of plethysm, to trate on the of continuous primarily development give us a general method of uniquely determining in the tradition of Elie Cartan and groups, [3] Sophus branching rules between the above groups and their Lie [4] by considering the properties of infinitesimal subgroups. transformations. Hermann Weyl’s book on "The This paper describes the relevant and how Classical [5] has undoubtedly exercised a theory Groups" it is used to a set of to considerable influence in these give computer programmes developments. the An alternative approach to the theory of continuous perform complete algebra. groups, which complements the earlier work of Cartan 2. S-Functions and Continuous - In this and Lie, has been developed by D. E. Littlewood as Groups. we shall outline the mathematical a natural consequence of Schur’s original thesis [6] section, briefly and notations in the on the properties of invariant matrices. Littlewood’s concepts required subsequent of this treatment circumvents the study of infinitesimal trans- development paper. formations the of by considering properties special Partitions. - A set of r whose sum functions of the roots of the matrices that characterize positive integers is n is said to form a partition of n. An ordered parti- the elements of the continuous groups. This approach tion is one where the are ordered from obviates the need to obtain the characters integers largest group to smallest (or vice versa). All partitions henceforth, These functions, known as Schur-func- explicitly. will be so ordered with the notation that a Greek letter or as have been used tions, simply S-functions, by a so : or will denote general partition (X) (À1, À2, ..., x,)., Littlewood to find relatively simple formulae relating and this will be assumed to be a partition of n. A Latin letter will denote a partition into one part only. (1) Research sponsored in part by the Air Force A is associated with each Office of Scientific Research, Office of Aerospace Re- Young diagram partition. r dots in the search, United States Air Force, under AFOSR Grant It is the graph of rows, Ài (or squares) No 1275-67. i th row, with each row left justified. A conjugate

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019690030010079500 796 partition is formed by interchanging the rows and the s and t being subscripts for the row and column columns of the graph, and will be denoted by (À) : respectively. We extend the definition of the homogeneous product sums to include ho = 1 and h. = 0

Some partitions are self-conjugates e.g. (332). Partitions with repeated parts are often written with a the number of times the superscript denoting part Littlewood has shown how to express any S-function occurs, thus (33221) = (32 221 ) . in terms of mononomials by appropriately labelling Frobenius’ notation is sometimes of use. The the Young diagram corresponding to partition (X) [8]. leading diagonal of the Young diagram is defined to The - Under the of be the one that starts in the top left hand corner. For Symmetric Group. operations the the with n ! elements each dot on the we write down the number group, symmetric group Sn diagonal is into classes with elements. Each class is of dots to the right of it, and below this, the number split p hp the complete set of conjugates of a given element. The irreducible representations of the group may be placed into a one to one correspondence with the partitions on n. Because conjugate matrices have the same characteristic (i.e. spur or trace) there exists a unique number xP(03BB), the characteristic for a particular class of a representation. The set of characteristics of a representation is known as the character of the representation. If we define a function :

we may prove that it is in complete correspondence with the representations of the symmetric group. Littlewood [9] has shown that this definition is entirely equivalent to the definition of the S-function given earlier, to give us a complete isomorphism between operations on S-functions and operations on representations of the symmetric groups. The corres- Functions. - A symmetric function on Symmetric pondence between the symmetric groups and the full k variables ai is one that is unchanged by any permu- linear group paved the way for Littlewood to express tation of the variables. Two such functions are of the of the continuous in terms of in this algebra groups interest paper. S-functions. This result is of key importance in the

- work. 1) Mononomial Symmetric Functions. If (p) is a following partition, we define the mononomial Sp such that : 3. Dimensions of Representations of Groups. - Di- mensions of representations are used extensively to check all and where the summation is over all different permutations branching rules, products plethysms. of the a’ s. For example, if k = 3 : Several authors [10] give formulae for the dimension (degree) f{À} of a representation {X}. Symmetric Group S n : 2) Homogeneous Product Sums, hn. - The homogeneous product sum hn is defined to be the sum over all of the mononomials Sp, p being a partition of n :

S-functions. - If CA) is a partition of n, the S-function where {À} is a partition of n into r parts. A for- (x) is the determinant of the h¡ s defined as follows : mula more suited to hand calculations is give . by Robinson [11]. 797

The Unitary Group The principal part of the product is the term obtai- ned when the partitions are simply added, i.e. the partition {Àl + [03BC1’ À2 + [1.2’ ..., Ài + [1.i, ...}. It corresponds graphically to putting all the (x’ s in the second so on. The Symplectic Group,> first row, the P’s in the and The other terms in the product may be systematically produced by removing the last element and trying it on the next lower line, then the next, etc. When it will fit nowhere else remove the second to last element also. Try to fit this in the same fashion, if no place is found remove the third to last element, when one is found to fit try to replace the elements in their highest positions. Note The Orthogonal and Rotation Groups On, R. : also that, for example, for the third y we may place it a) For odd dimensions n = 2v + 1 : on the same line, or below, the second y but not above it. This operation is checked dimensionally by use of the equation :

where (X) is a partition of n, and (03BC) of m. b) For even dimensions For example {31} {21} :

except that for the orthogonal group in the case of X, =,4 0 when the dimension is twice the above. The Exceptional Group G2 :

4. Outer Multiplication of S-functions. - The product of two S-functions on different sets of variables - corresponding to the product of representations of different symmetric groups - is known as the outer or ordinary multiplication of S-functions. It is of key importance in the algebra. The rules for performing the operation using the Young diagram representation note that the graphs : have been given many times [7] although, so far as it is known, it has not suggested how to systematically cover all possibilities for the graphs. The S-functions appearing in the product : are those which can be built by adding to the graph of {À} f1-1 symbols a, f1-2 symbols P, 03BC3 symbols y, etc., in this order and in the ways specified by the following : all break the rules above. 1) No two identical symbols appear in the same given column of the graph. Thus we obtain the expansion : 2) If we count the oc’ s, 3’ s, y’ s, etc., from right to left, starting at the top, then at all times while the count is being made, the number of a’ s must be not less than the number of P’s which must not be less than the number of y’s, and so on. 3) The graphs we obtain after the addition of each symbol must be regular, i. e. the corresponding partition must be ordered. 798

5. S-function Division. - Frequently the algebra S-function division may also be performed by use requires the evaluation of the sum of S-functions {v} of the properties of isobaric determinantal forms [8] : which when multiplied by a particular S-function {03BC} give a particular S-function {À}, the coefficient of {v} being the coefficient of {À }in the outer product. Hence where 03BBs, is the s-th part of the partition (03BB) and 03BCt, is we define the (outer) division of S-functions {À} /{03BC} the t-th part of the partition (03BC) and thus s labels the to be : rows and t the columns of the determinant. In the case of the preceding example, we have : where r 03BCv03BB is the same as the coefficient in the outer product : (’1)

The evaluation of the division is somewhat easier than each product, thus considerably simplifying the calculation. We have the and wish to know all graph of {03BC} as before. In practice, the preceding method is to possible ways of adding elements to form the graph be prefered for machine calculation. The nota- the rules for the To evaluate of {À} given product. tion {À} / {fL} is to be prefered over {À/fL} since we the draw the for with division, graph (h) squares may easily show : instead of dots, and fill up the left hand top corner with the graph corresponding to {03BC}. Graph {03BC} must fit entirely inside {À} or the result must be null. The remaining squares are then labelled by a’s, 3’s, Y’ s, etc., by rows, starting in the top left, as given by rules 1 and 2 of the product and also with : 3) The symbols must not decrease when reading left to right across a row, i.e. there must not be an a to the of a etc. right 03B2, 6. Expression of an S-function in Symmetric Parts. The resultant S-function must be ordered. 4) - When an S-function is defined from character For example {4211}/{211} : theory the problem of expressing it in terms of sums and differences of symmetric functions is a little difficult. However, the equivalence between that definition and our definition renders the problem trivial since our defining determinant is in terms of the hr s (eq. 3) :

Sometimes it is useful to know the expansion of {À } in terms of products of the antisymmetric representations {1 r}. Because hr == {1r}, we may take conjugates of the above relation. Alternatively we define the elementary symmetric function ar as the mononomial corresponding to the partition ( 1 r) ::

note that the graphs : leading to the result [9] :

7. Inner Multiplication of s-functions. - The evaluation of the inner product {À} 0 { tL} is considerably more complex than the outer product. Numerous attempts have been made to simplify the problem, are not allowed the rules. by with varying degrees of success. Most of these attempts have used the character table for the group in question, however Littlewood [12] has developed a much more flexible method where the character 799 tables are not required. The key theorem is stated of degree n all occur as subgroups of Un. Little- without proof : wood [13] has expressed the characters of the orthogonal and symplectic groups in terms of S-functions : where r p03C303BD has the usual meaning for the outer product. The symmetric representation is the identity element for this operation : where (y) and (8) are partitions of p and occur in the Frobenius series : (X) a partition on n. Inner multiplication is distributive with respect to addition, hence all that is required for the evaluation of any inner product is for one S-function to be expressed in terms of symmetric parts, The character theory for the rotation groups are essen- and the appropriate outer multiplications performed. tially the same as for the orthogonal groups except when To evaluate : the group dimension is even (n = 2v) and X, 0 0. In most of these cases, it is necessary to resort to the method of difference characters [8] though in the the are : steps particular cases of the group R, and R6 it is possible to use simpler methods as will be discussed later. The exceptional group G2 occurs as a subgroup of R7 and is important in the classification of the states of electrons or nucleons in equivalent orbitals. The character of is discussed in a later where (vi) is a partition of i, the sum is over all such theory G2 partitions, and rVavb.Vc03BC is the coefficient of 03BC in section. the outer product {va} { Vb}, ..., { Vc}. 9. Reduction of the Number of Parts of an s-function. We check our result may [9] : - by noting Under the operations of the restricted groups an S-function defined on n variables, where n = 2v or n = 2v + 1, and having more than v parts is equiva- Various relations among the coefficients are of lent to a series of S-functions on the same n variables importance [9] : but not having more than v parts [13]. The S-function is expressed in the form :

The conjugate relations are of importance in listing tables of inner products, reducing the number by a factor of four. Relations (21 b) express not only that the operation is Abelian, but that the coefficient of {v } in the product {À} 0 {03BC} equals the coefficient of {03BC} in the product I X I o {v 1. This means that if we define a of for some transfor- inner division analogous to outer division, we merely Ignoring possible change sign have the operation of inner multiplication : mations, this S-function is independent of r and will be denoted {À: 03BC}. It is expanded to give a series of S-functions using the relation : Trivial consequences of this that are often relevant to problems are that the identity I n I is contained only where the sum is over all in products of the type {À}o{À}, and the antisymme- S-functions {a}, being parti- tric representation {1 n} is only contained in those of tions of p. This relation is used as often as necessary to reduce all terms to those of no more than v parts. the type {X}o{X}. Two special cases of this equivalence relation are often useful. In n variables for unitary transforma- 8. Characters of Subgroups as S-functions. - The tions, we have : set of all non-singular matrices of order n2 form a group, the general linear group GL.. The subset of all unitary matrices also forms a group, the unitary Ignoring the change of sign when n = 2v only for group Un. The compact representations of these two transformations of negative determinant, we have also : groups have the same characters and Littlewood has shown [8], [9] that the S-functions on n variables are the simple characters of the groups. This latter relation gives the well known particle-hole The full orthogonal, rotation and symplectic groups correspondence. 800

10. Branching Rules. - Under restriction to a the term of highest weight. The terms of lower weights subgroup, the characters of a group decompose into may be systematically removed by subtraction. a sum of characters of the subgroup [13]. For the For example, in the case of (21), we may derive : unitary group in n variables, the characters are expressed as S-functions of up to n parts but for the restricted groups on n variables, into only v parts. Prior use of relation (26) allows us to use the following relations without producing non-standard symbols. may be expressed in S-functions to give : For the Orthogonal Group :

For even dimensional rotation groups, products in two groups have been where the sum is over all S-functions of even parts only separated satisfactorily, only : the groups in four and six [14] dimensions. In six dimensions, Littlewood [15] has shown that the group is isomorphic with the four dimensional full linear and where the terms of the division are taken as group and the correspondences : orthogonal group characters, i.e. :

The Symplectic Group : For the symplectic group the result is the same as for eq. (31 ) apart from the repla- may be established, thus allowing us to perform the products easily. For the four dimensional rotation there is a 2:1 with the double The Rotation Group : For odd dimensions the charac- group homomorphism binary full linear group and the correspondences : ters are the same as for the orthogonal group but for even dimensioned groups the characters with [03BCv #- 0 decompose into two conjugate characters : may be made. We may readily deduce that the separation of the Kronecker product in R4 is given by : The Exceptional Group G2 : The group G2 is a proper subgroup of the seven dimensional rotation group and Judd [10] has derived the branching rules by using the infinitesimal operator approach, to yield the result :

where the sum is over all integral values of i, j, k satis- 12. Plethysms for GL2 and R3. - Plethysms o fying the relations : characters on two basis variables are often of use in physics. Plethysms for GL2 may be evaluated by restricting the results for the same plethysm on an unrestricted number of variables. In the absence of such a table, we may generate the plethysm required by use of the recursive relation [16] : is used to remove characters which do not give regular representations of Gz. 11. Kronecker Products for the Continuous Groups. - Since we the characters of the may express unitary, The due to Little- and in terms of S-func- following polynomial expansion symplectic orthogonal groups wood for the of a the reduction of the Kronecker of these [17], plethysm R3 representation tions, products and an S-function into one is more suited be done in terms of the outer only part, groups may product to machine calculation : of S-functions and then performing the appropriate branching to get back the characters of the group. Kronecker products for G2 are done in the same where Kr is the coefficient of P-r in the expansion of : manner by a two stage process, expressing the characters of G2 in terms of those of R7, thence into S-functions and so on. The expression of the characters of G2 in terms of those of R7 is performed by noting that in This expression holds also for spin representations. the reduction R7 --> G, [u1 u2 0] contains (UIU2) as A similar expression for the plethysm of a GL2 repre- 801

sentation allows us to make the following isomorphism method used is influenced to a large extent by the between GL2 and R3 : machine, and since our machine is a fixed word length binary one, an IBM 360/44, with a Fortran IV compi- ler, all partitions and coefficients have been stored at

-r are used 13. Branching Rules to R3’ The use of the algebra half-word integers (16 binary bits). Vectors and matrices for sums of of plethysm [9], [14], [8] allows us to calculate, directly for storing partitions parti- in allow and unambiguously, the branching rules between any tions. Variable dimensions the subroutines us to store the coefficient and as a column compact group and a subgroup of lower dimension or partition vector of the of the vector to be a direct product of such groups [19], [20]. Quite length longest likely waste is generally, given that the unary character [1] of the required for the problem. Any storage quite the for the larger group decomposes to a sum of characters A tolerable, since storage required partitions is under the restriction, then any character X will decom- required to evaluate inner products of Slo of the for the instructions for the pose into the sum of characters given by A 0 X. order of that required The branching rules for any group R3 may therefore Kronecker product subroutine. be easily calculated after defining our unary de- 2. BASIC SUBROUTINES. - Ordered partitions of all composition. numbers up to a given maximum may be quickly For to calculate the of example, branching [210] produced in lexigraphical order, by a small routine if of to we note that of branches to a R7 R3 [100] R7 we remember which is the last part greater than one F i.e. the Hence single state, representation [3] of R3. and the current sum of the parts. the of is 0 decomposition [210] given by [3] [210]. A routine to store a partition and coefficient in a We express [210] in terms of S-functions, and then these matrix of other partitions is useful. Ordering the into sums and of into one products partitions part, partitions means we may quickly find its equal, if it giving : has one, and add the coefficients. Routines to perform outer products and divisions of S-functions are a little more difficult to write since Using the result [9] that the plethysm is distributive we must remember the labels for the squares. When on the with to both addition and multi- right respect this is the rest of the is plication, by using equation (38), and by multiplying accomplished programming more straight forward, with one rather weighty excep- together the R3 representations in the usual manner, tion, how to get the results out of the computer in a we obtain : tidy format. The system cannot print a 10 in a single decimal space and nor are blanks sightly. With the continuous group characters we know the size of the as it is with zeros, but with S-func- This method of the plethysm on the partitions padded performing tions the brackets and other in symmetric parts of the right hand side is not always signs vary position. This was overcome use of an assembler the easiest method. Often it is easier to make use problem by of the relation [16] : output routine. This routine had several output buffers so we could print on-line with a typewriter with very little loss in machine efficiency. by expanding the S-functions in terms of their antisymmetric parts. 15. Conclusions. - The results in this paper have The saving in labour for classifying the orbital states resolved many of the ambiguities associated with the of maximum multiplicity is immense. For example, representation theory as applied to the problems in in the t shell we use representations of R29. The unary both atomic and nuclear physics of the classification is -> Thus of states and and the derivation of selection decomposition [100, ..., 0] [14] = { 28 }. operators the states of maximum multiplicity for the quarter rules. Computer programmes have been written to filled shell labelled by [114] are easily found to be just encompass these results and relieve much of the tedium the terms in : of calculation. Extensive tabulations of results are being published [20]. The addition of general methods for the machine This has been expanded using the computer and there calculation of plethysms and the handling of difference are found to be 34,670 states of total angular momen- characters and spin representations for the orthogonal tum of a result that is difficult to obtain the 23, by and rotation groups should remove many of the remai- usual methods of determinantal states. ning problems. These problems will be discussed in a separate paper. 14. Programming Considerations. -1. STORAGE OF PARTITIONS. - When one sits down to programme 16. Acknowledgements. - We are grateful to Julian any of the operations of the algebra, one immediately Brown for assistance with certain aspects of the pro- strikes the difficult problem of how to efficiently gramming, and to the University of Canterbury for remember a partition or a sum of partitions. The the use of their computer.

LE JOURNAL DE PHYSIQUE. - T. 30. N° 10. OCTOBRE 1969. 51 802

REFERENCES

[1] RACAH (G.), Phys. Rev., 1949, 76, 1352. [11] ROBINSON (G. de B.), Representation Theory of the [2] RACAH (G.), Group Theory and Spectroscopy, Symmetric Group, Edinburgh University Press, Ergeb. der exakten Naturwiss., 37, Springer- Edinburgh, 1961. 1965. Verlag, Berlin, [12] LITTLEWOOD (D. E.), J. Lond. Math. Soc., 1956, 31, [3] CARTAN (E.), Sur la structure des groupes de trans- 89. formation finis et continus, Thesis, Nony, Paris, LITTLEWOOD Trans. Soc., 1894. [13] (D. E.), Roy. London, 1943, A 239, 389. [4] LIE (S.) and SCHEFFERS (G.), Vorlesungen über Continuieliche Gruppen, Teubner Verlagsgesell- [14] WYBOURNE (B. G.) and BUTLER (P. H.), J. Phy- schaft, Leipzig, 1893. sique, 1969, 30, 655. [5] WEYL (H.), The Theory of Groups and Quantum [15] LITTLEWOOD (D. E.), Lond. Math. Soc., 1948, 50, Mechanics, translated by H. P. Robertson, Dover 349. Publications, New York. [16] MURNAGHAN (F. D.), The Unitary and Rotation [6] SCHUR (I.), Über eine Klasse von Matrizen die sich Groups, Spartan Books, Washington D.C., 1962. einer Matrix zuordnen Lassen, Inau- gegebenen LITTLEWOOD Lond. Math. Soc., 1936, Dissertation, Berlin, 1901. [17] (D. E.), J. gural 11, 49. HAMERMESH [7] (M.), Group Theory, Addison-Wesley Trans. Publishing Co. Inc., Reading Mass., 1962. [18] LITTLEWOOD (D. E.), Roy. Soc., London, 1943, A 239, 305. [8] LITTLEWOOD (D. E.), The Theory of Group Charac- ters, 2nd ed., Oxford University Press, Oxford, [19] SMITH (P. R.) and WYBOURNE (B. G.), J. Math. 1958. Phys., 1968, 9, 1040. [9] LITTLEWOOD (D. E.), A University Algebra, William [20] WYBOURNE (B. G.), Symmetry Principles and Ato- Heinemann Ltd, London, 1950. mic Spectroscopy, with an appendix of tables [10] JUDD (B. R.), Operator Techniques in Atomic by P. H. Butler, John Wiley & Sons, New York, Spectroscopy, McGraw-Hill Co. Inc., 1963. 1970 (in press).