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 simpli- fying 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 homo- geneous 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 bet- ween 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.