E 7 ⊂Sp(56,R) irrep decompositions of interest for physical models
aa r X i v : . [ m a t h . R T ] M a r E ⊂ Sp(56, R ) irrep decompositions ofinterest for physical models L.Fortunato , W.A. de Graaf
Abstract
In this note we show how to obtain the projection matrix for the E ⊂ C chain and we tabulate some decompositions of the symplectic algebra C representations into irreps of the E subalgebra that are important for variousphysical models. The symplectic Lie groups and the corresponding algebras emerge in several phys-ical models as symmetry group or dynamical symmetry groups. Among them, inparticular, the group Sp(56, R ), despite having an unusually high rank, shows adistinctive trait, namely it contains E as a subgroup. It appears, for example, inthe study of symplectic extensions of N = 8 gauged supergravity theories [ ], thatare extensions of the SO(8) gauged supergravity, as the group of transformationsthat act on the electric and magnetic gauge fields vectors. These theories in par-ticular need to look at the way in which certain representations branch into irrepsof smaller algebras of distinguished importance and this fact motivated our effortto look for this particular branching that, due to its peculiarity, does not conformitself to known algorithms and requires a computational approach.High-rank symplectic algebras appear also prominently in various models of nu-clear and atomic physics (cfr. for instance Ref. [ ]) and in many-body quantumtheories [ ]. The Sp(2n, R ) group is the dynamical group of the n-dimensional Dip. Fisica e Astronomia “G.Galilei”, Univ. Padova and INFN-Sezione di Padova, via Marzolo8, I-35131 Padova, ItalyDip. Matematica, Univ. Trento, via Sommarive 24, I-38123 Povo (Trento), Italy E ⊂ SP(56, R ) IRREP DECOMPOSITIONS OF INTEREST FOR PHYSICAL MODELS isotropic harmonic oscillator, that has a fundamental importance in all realms ofquantum physics[ ]. While we are not interested here in the physical models basedon these groups, nor what is their precise physical interpretation, they rely on theclassification of the corresponding Lie subalgebras, a very interesting mathematicaltopic by itself (see, for example, Ref. [ ]), and on the solution of the branchingproblem, that is very challenging and might become computationally demandingwith increasing rank of involved algebras.We will investigate here the corresponding Lie algebras. The exceptional Liealgebra E is a special subalgebra of the C symplectic algebra, i.e. C ⊃ E .In order to study physical models based on this algebra chain, it is of fundamentalimportance to be able to determine the branching problem from the higher to thelower algebra.Notice that this particular branching rule does not appear in Ref. [ ], wheremany other important branching rules are derived. To the best of our knowledgethe first treatment of the present subalgebra chain is given in the Ph.D. thesis ofM. Lorente [ ]. The problem of finding the branching rules amounts to finding the projection ma-trix. In order to get this matrix we first describe how we realize the embedding E ⊂ C . We start with the Lie algebra of type E , construct its lowest dimen-sional representation, and then build the Lie algebra of type C around that.First we recall the following fact. Let M be a 56 × C , such thatdet( M ) = 0 and M T = − M . Set g = { a ∈ gl (56 , C ) | a T M = − M a } . Then g isa simple Lie algebra of type C .Let a be the simple Lie algebra of type E over C . The computer algebra system GAP ]) contains a function for constructing the irreducible representationsof a . A description of the underlying algorithm can be found in Ref. [ ]. Weused it to compute the 56-dimensional representation ρ : a → gl (56 , C ). Next weconsidered the space M consisting of all 56 × M such that M T = − M and ρ ( x ) T M = − M ρ ( x ), where x runs through a basis of a . Note that this yields aset of linear equations for the entries of M ∈ M . From the solution of this systemof equations, it turned out that dim M = 1 and we let M be a basis element of M . It also turns out that det( M ) = 0. So we constructed g as above, which is thesimple Lie algebra of type C containing ρ ( a ).In order to describe the construction of the projection matrix we recall thefollowing. Let h be a semisimple Lie algebra over C of rank n . Let C be the Cartan .FORTUNATO , W.A. DE GRAAF matrix of its root system. Then there are elements x i , y i , h i ∈ h such that[ h i , h j ] = 0[ x i , y j ] = δ i,j h i [ h j , x i ] = C ( i, j ) x i [ h j , y i ] = − C ( i, j ) y i . These elements generate h and are said to form a canonical generating set of h .We enumerate the nodes of the Dynkin diagrams of the Lie algebras a and g asfollows: ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ . . . ❡ ❡ < Correspondingly, we get canonical generating sets x a i , y a i , h a i , 1 ≤ i ≤ a and x g j , y g j , h g j , 1 ≤ i ≤
28 of g . Here the h a i , h g j are constructed so that each h a i lies in thespace spanned by the h g j . In other words, there are a i,j such that h a i = P j =1 a i,j h g j .Now the matrix ( a i,j ) is called the projection matrix, and it is displayed below:(1) This projection matrix is one of the main inputs to the algorithm for computingbranching rules, see [ ] and also [ ], Section 8.13. Given this matrix the algorithmthen uses the combinatorics of weights and root systems to obtain the decompositionof a given irreducible representation of g , when restricted to a . This algorithm hasbeen implemented in the SLA package ([ ]) inside the computer algebra system GAP ]). Using this we have investigated the branching problem for the smallestrepresentations. E ⊂ SP(56, R ) IRREP DECOMPOSITIONS OF INTEREST FOR PHYSICAL MODELS Table 1.
Branching C → E for irrep of dimension smaller thanone million.h.w. Dim. Branching Dimensions[1 , ] [0 , [0 , , ] [0 , , [2 , ] [0 ,
2] + [1 , ] + [0 , , ] [0 , , ] [3 , ] [0 ,
3] + [1 , ,
1] + [0 , + + [1 , , ] [0 , ,
1] + [1 , ,
1] + [0 , , ] + + [0 , , ] [0 , , ] [0 , , ] [0 , ,
0] + [1 , , + +[0 , , ,
1] + [2 , ] + [0 , , + + We enumerate in table 1 the branching of all the irreps of C of dimensionsmaller than one million. We give the C highest weight vectors in the first col-umn, the dimension of the corresponding representation in the second column, thebranching into E representations in the third column and the corresponding di-mensions in the fourth column. All these representations appear with multiplicityone. The notation 0 k in the highest weight vectors simply stands for · · · , , · · · repeated k times. We find, for example, that the 1596-dimensional adjoint rep-resentation of C is decomposed into the sum of the 1463-dimensional represen-tation and 133-dimensional adjoint representation of E , a most natural findingthat excludes, however, the possibility to have other partitions of 1596 into the set { , , , , , } . In general, one might get an idea of the combinatorialcomplexity of the problem from the solution of the Frobenius Diophantine equa-tion: excluding singlets, one gets three such possibilities (namely =12 × ,19 × +4 × and + ) and the number of solutions raises to 240 possiblepartitions when including the one-dimensional representation (i.e. [0 ]). Of coursenot all of these partitions are admissible from a purely algebraic perspective andthe branching rules indicate how to accomplish the task with the correct algorithmthat embeds representations of the smaller algebra into representations of the largerone. Due to the dimensions of the algebras involved, these numbers might growvery rapidly for higher representations. Notice that, among the 27 possible repre-sentations of E that have smaller dimension than 1 million, only 14 appear in thedecomposition. (A more complete list of these representations can be found in thetables of Ref. [ ]).It is also interesting to know how tensor products of the C algebra representa-tions decompose into sum of C irreps: for example the × (3136-dimensional)representation decomposes into the sum of + + . This kind of decompo-sitions are easily accomplished with GAP when the projection matrix is known. Weenumerate the smallest in Table 2 because they might be useful to the investigationof physical models based on the E ⊂ C subalgebra chain. .FORTUNATO , W.A. DE GRAAF Table 2.
Decomposition of tensor products of C irreps of di-mension smaller than one million. We have used here the dimen-sional notation, coefficients in the last two lines stand for multi-plicities.Tensor Product Dimension Decomposition × + + × + + × + + × × )+ ++ +3( ) Acknowledgments
We thank G.Dall’Agata (Padova) for having called our attention to this problemand for useful discussions.
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