J. A. Castilho Alcarás, J. Tambergs, T. Krasta, J. Ruža, and O. Katkeviius
Citation: Journal of Mathematical Physics 44, 5296 (2003); doi: 10.1063/1.1611265 View online: http://dx.doi.org/10.1063/1.1611265
View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/44/11?ver=pdfcov
Plethysms and interacting boson models
J. A. Castilho Alcara´s
Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01405-900 Sa˜o Paulo, Sa˜o Paulo, Brazil
J. Tambergs, T. Krasta, and J. Ruzˇa
Laboratory of Radiation Physics of Institute of Solid State Physics, University of Latvia, LV-2169, Salaspils, Latvia
O. Katkevicˇius
Institute of Theoretical Physics and Astronomy, 2600 Vilnius, Lithuania
共Received 25 February 2003; accepted 2 July 2003兲
A short review of the plethysm technique aiming to its application in finding branching rules for the reduction of an irreducible representation of a group under the restriction to one of its subgroups is given. The algebraic structure of the interacting boson model and some of its extensions is given together with the branching rules needed to classify their basis states, obtained by the use of
plethysms. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1611265兴
I. INTRODUCTION
In the study of irreducible representations共irreps兲 of the full linear group GL(n) in n dimen-sions an important role is played by the so called Schur functions.1–3In a given irrep兵其 of GL(n) the character of each of its elements A is the Schur function兵其 evaluated with the eigenvalues of A.
A Schur function is expressed in terms of fundamental symmetrical quantities ai, hi, and si, 4 polynomials in n unknowns that are left invariant under permutations of these unknowns. The plethysm of Schur functions turned out a powerful tool to determine branching rules for the
reduction of irreps of GL(n) subgroups under restriction to some of their subgroups.4,5 The
plethysm operation of Schur functions was discovered by Littlewood6as a third way of combining
two Schur function to obtain a linear combination of Schur functions of a same degree. With few exceptions,4,7,8it remainded almost unknown to physicists due to the great difficulties involved in its calculation. With the appearence of powerful computers the tedious labor of computing plethysms was no more a problem and new efforts were made in order to find algorithms for computing them.5,9–13
In most applications to physical problems such as in nuclear structure5,7 and in the present work only a particular class of plethysm is needed, namely that in which the left factor is a symmetric Schur function and in the expansion only those Schur functions with no more than a given number of rows are considered. In that case, using an induction formula for computing plethysms with both factors being symmetric Schur functions given in Ref. 13, we developed an
algorithm5 to compute plethysms with a symmetric Schur function in the left and all Schur
functions of a given degree at right.
A field in which the plethysm technique can show all its power is the interacting boson model 共IBM兲 and its generalizations.
The IBM when originally introduced by Arima and Iachello14 is 1975 takes the nucleons
outside a core of an even–even nucleus couple then into pairs to form bosons with angular
momentum 2(d-bosons兲 and 0(s-bosons兲, no other degree of freedom, besides their z-component
being taken into account. To work in the second quantization formalism they introduce 5 共for
d-bosons兲 ⫹1(for s-bosons)⫽6 creation and annihilation boson operators. The space of states is
taken as polynomials of degree N 共number of boson pairs兲 in creation operators acting on a
vacuum realizing in this way the basis states of irreducible symmetric representations of U共6兲.
5296
These bosons interact among themselves by interactions that preserve angular momentum and number of boson pairs so that their Hamiltonian can be written in terms of Casimir invariants of U共6兲 subgroups.
This original verson is nowadays referred to as IBM-1. Some extensions of the model appeared15in order to account for other degrees of freedom and the inclusion of bosons with other angular momenta. The unitary group is enlarged and a very rich algebraic structure arises. The basis states of the irreps of these unitary groups are labeled by labels of irreps of their subgroups
in chains ending with O⫹(3), the rotation group in three dimensions. To this end one needs to
know how an irrep of a group branches into irreps of some of its subgroups. We will show in this paper how plethysms can be used to find these branching rules. Besides, the cases here studied can serve as examples for applications in other areas.
II. SUMMARY OF PLETHYSMS
A partition ()⬅(1,2, . . . ,n) is a set of nonnegative integers 共parts兲 i such that 1
⫹2⫹¯⫹n⫽n. If, in addition, they satisfy 1⭓2⭓¯⭓n, the partition is called standard. Since we will deal only with standard partitions we will omit the word standard. Usually in a partition the null parts are omitted and the repeated ones are exponentiated. We will use greek letters to denote a partition and italic letters to denote a single part of a partition. To each partition
one associates a Young diagram, an array of n boxes with 1 boxes in the first row, 2 in the
second and so on. Due to that the nonzero parts of a partition are referred to as rows and the conjugate partition of a given partition (1,2, . . . ,p,0, . . ., 0) is defined as the partition whose
Young diagram is obtained from that of共兲 by interchange of rows and columns, i.e.,
共˜兲⫽共pp,共p⫺1兲p⫺1⫺p, . . . ,22⫺3,11⫺2兲. 共1兲 Given a set of n variables x1,x2, . . . ,xn and a partition 共兲 of r, the Schur function 兵其 associated to共兲 is defined as4,6
兵其⫽ 1 r!兩Zr兩
[], 共2兲
where Z is the matrix
Zr⫽
冉
s1 1 0 0 . . . 0 s2 s1 2 0 . . . 0 s3 s2 s1 3 0 . . . 0 . . . .. . . 0 sr⫺1 sr⫺2 . . . s1 r⫺1 sr sr⫺1 . . . s2 s1冊
共3兲and si⬅si(x1,x2, . . . ,xn) is the sum of ith powers of each variable x1,x2, . . . ,xn. In this notation of Schur function the variables and their number n are implied while r, called its degree, is
obtained by r⫽1⫹2⫹¯⫹r.
In共2兲, 兩Z兩() is the immanant of Z, an extension of the concept of determinant, given by 兩Z兩()⫽
兺
P []共P兲z
1 p1z2 p2¯zr pr, 共4兲
where the sum is over all permutations P⫽(p1, p2, . . . , pr) of the integers 1,2, . . . ,r and[]( P) is the character of permutation P in the irrep关兴 of the symmetric group S(r).
As a consequence of definition共2兲 a Schur function is an homogeneous polynomial of degree r in the variables x1,x2, . . . ,xn, being identically null for partitions共兲 of r into more than n nonzero parts.
Expression共2兲 can be worked out to produce an alternative definition6,9of the Schur function
兵其⫽兩共x兩共xi兲j⫹n⫺ j兩
i兲n⫺ j兩
, 共5兲
where兩 fj(xi)兩 denotes the determinant of a matrix M with elements Mi, j⫽ fj(xi).
A pair of Schur functions兵
⬘
其,兵⬙
其of degrees r⬘
and r⬙
can be combined into three differentways to produce linear combination of Schur functions 兵
其 of degree r
: inner 共or direct兲product, outer product and plethysm. These three operations will be denoted, respectively, as
兵
⬘
其⫻兵⬙
其⫽兺
␣共兵⬘
其⫻兵⬙
其→兵
其兲兵
其, 兵⬘
其兵⬙
其⫽兺
␣共兵⬘
其兵⬙
其→兵
其兲兵
其, 共6兲 兵⬘
其丢兵⬙
其⫽兺
␣共兵其丢兵⬙
其→兵
其兲兵
其,where ␣共¯兲 is a non-negative integer denoting the multiplicity of 兵
其 in the expansion. For clarity we attach to it an argument denoting the kind of operation that produced it.In the inner product the degrees of the Schur functions involved are all equal, i.e., r
⫽r⬘
⫽r⬙
⫽n, and the expansion coefficients ␣ are the coefficients of reduction of the Kronecker product of S(n) irreps关⬘
兴 and 关⬙
兴.In the outer product one has r
⫽r⬘
⫹r⬙
and the coefficients␣ are obtained by making the product of a Schur function in variables (x1,x2, . . . ,xn⬘) by another in variables (y1,y2, . . . ,yn⬙) and expressing it as a linear combination of Schur functions in variables (z1,z2, . . . ,zn) withzi⫽xi for 1⭐i⭐n
⬘
and zn⬘⫹i⫽yi for 1⭐i⭐n⬙
. Littlewood obtained a procedure to find the coefficients of the outer product known in the literature as ‘‘Littlewood’s rules.’’To define plethysm one needs first to introduce the concept of invariant matrix.
Let T(A) be an m⫻m matrix whose elements ti j are given homogeneous polynomials of
degree r in the elements of A. Let T(B) be a matrix built with the same polynomials ti jnow in the elements of B. If
T共A兲T共B兲⫽T共AB兲 共7兲
for any nonsingular m⫻m matrices A,B then the matrix T(A) is called an invariant matrix共of
degree r) of A.
It follows from共7兲 that, once the set of polynomial ti j is fixed, the set of matrices DT(A)
⬅T(A) is a representation of GL(n).
As the Kronecker product of two representations of a group is also a representation of this group, the Kronecker product of invariant matrices is also an invariant matrix, in general
reduc-ible. Schur1 demonstrated that if A is an n⫻n matrix, there are as many irreducible invariant
matrices of A of degree r as are the partitions of r with no more than n nonzero parts and the trace of them are the Schur functions of degree r in the eigenvalues of A. These irreducible invariant matrices are then labeled by those partitions and denoted by A[]. The details of con-struction of irreducible invariant matrices can be found in Refs. 16 and 17.
Since an invariant matrix of an invariant matrix is also an invariant matrix of the original matrix, it can be decomposed into irreducible components
关A[]兴[]⫽
兺
A[]. 共8兲
Let us denote by r, r, and r, the degrees of兵其, 兵其, and 兵其, respectively. Since the elements of A[] are polynomials of degree r in the elements of A and those of关A[]兴[] are polynomials of degree r in the components of A[], it follows that r⫽rr.
Equation共8兲 led Littlewood6to define a third composition rule of Schur functions denoted by
the symbol 丢 and defined as
兵其丢兵其⫽
兺
兵其, 共9兲
where the Schur functions 兵其 and the numerical coefficients are those given in 共8兲. This
operation was later on named plethysm.
The plethysm operation has the following properties:4,6,8
兵其丢共兵其丢兵其兲⫽共兵其丢兵其兲丢兵其, 共10兲 兵其丢共兵其⫾兵其兲⫽兵其丢兵其⫾兵其丢兵其, 共11兲 共兵其⫹兵其兲丢兵其⫽
兺
⬘⬙␣共兵⬘
其兵⬙
其→兵其兲共兵其丢兵⬘
其兲共兵其丢兵⬙
其兲, 共12兲 共兵其⫺兵其兲丢兵其⫽兺
⬘⬙共⫺兲 r⬙␣共兵⬘
其兵⬙
其→兵其兲共兵其 丢兵⬘
其兲共兵其丢兵˜⬙
其兲, 共13兲 兵其丢共兵其兵其兲⫽共兵其丢兵其兲共兵其丢兵其兲, 共14兲 共兵其兵其兲丢兵其⫽兺
⬘⬙␣共兵⬘
其⫻兵⬙
其→兵其兲共兵其丢兵⬘
其兲共兵其丢兵⬙
其兲, 共15兲 关兵其丢兵其兴T⫽再
兵˜其丢兵其 for r even, 兵˜其丢兵˜其 for r odd. 共16兲The sum in Eqs.共12兲, 共13兲, and 共15兲 includes the cases兵
⬘
其⫽兵0其⬅1,兵⬙
其⫽兵其 and兵⬘
其 ⫽兵其, 兵⬙
其⫽兵0其⬅1. Also, r⬙
and rare the degrees of兵⬙
其 and兵其.In Eq.共16兲 we used the notation
冋
兺
i ai兵其(i)册
T ⫽兺
i ai兵g其(i), 共17兲where ai are numerical factors,兵其(i)Schur functions and兵g其(i)their conjugate.
A. Special plethysms
The plethysm calculation is, in general, a hard and tedious task. Nevertheless there are special cases with closed and simple expressions
兵其丢兵1其⫽兵1其丢兵其⫽兵其, 共18兲
兵r其丢兵2其⫽
兺
i⫽0 [r/2] 兵2r⫺2i,2i其, 共20兲 兵r其丢兵12其⫽兺
i⫽1 [(r⫹1)/2] 兵2r⫺共2i⫺1兲,共2i⫺1兲其, 共21兲 兵2其丢兵r其⫽兺
兵其even, 共22兲 兵12其丢兵r其⫽兺
兵g,其even 共23兲 兵1r其丢兵2其⫽兵12r其⫹兺
i⫽1 [r/2] 兵22i,12r⫺4i其, r even, 共24兲 兵1r其丢兵12其⫽兵12r其⫹兺
i⫽1 [r/2] 兵22i,12r⫺4i其, r odd, 共25兲 兵1r其丢兵12其⫽兺
i⫽1 [(r⫹1)/2] 兵22i⫺1,12r⫺2(2i⫺1)其, r even, 共26兲 兵1r其丢兵2其⫽兺
i⫽1 [(r⫹1)/2] 兵22i⫺1,12r⫺2(2i⫺1)其, r odd. 共27兲 Equation共18兲 follows from plethysms definition while Eq. 共19兲 is set for consistency. In Eq. 共22兲兵其evenmeans partition of 2r with all parts even. Equations共23兲–共27兲 follow from conjuga-tion of Eqs. 共22兲, 共20兲, and 共21兲. In Ref. 13 there are formulas for the calculation of plethysms兵其丢兵其 when both Schur functions are symmetric or/and antisymmetric. To explain them we
need the following definition: a k-border strip of a Young diagram associated to a given partition 共兲 is a sequence of k squares in which the first of them is the last one of the first line of 共兲 and the next square to a given one is the one below it, if it exists, or the one to its left, otherwise.
For example, the three-border strips of (412), 共321兲, and (2212) are the squares with the symbol" in the figures below, respectively,
When兵其 and 兵其 are both symmetric, one has
兵n其丢兵m其⫽1 mk
兺
⫽1 m 兵n其共xk兲共兵n其丢兵m⫺k其兲, m⭓1, 共28兲 with 兵n其共xk兲⫽兺
Cn,k,兵其. 共29兲In共29兲 the 兵其’s are all Schur functions of degree nk. The coefficients Cn,k,are obtained from the Young diagram associated to共兲 removing, in sequence, n k-border strips. If in all steps the resulting diagram represents a standard partition then
Cn,k,⫽共⫺兲l 共30兲
with l⫽(number of lines in the k-border strips)⫺n. If in some step the resulting diagram does not represent a standard partition, then Cn,k,⫽0. As example, from the figures above one has
C2,3,兵412其⫽共⫺兲4⫺2⫽1, C2,3,兵321其⫽共⫺兲5⫺2⫽⫺1, C2,3,兵2212其⫽0.
Equation共28兲 allows one to relate the plethysm of two symmetric Schur functions with the
plethysms of symmetric Schur functions of smaller degrees. In this way, using兵n其丢兵1其⬅兵n其 as starting point one computes all the plethysms of type兵n其丢兵m其. This equation, together with
兵n其丢兵1m其⫽共⫺兲m⫹1兵n其丢兵m其⫹
兺
k⫽1 m⫺1 共⫺兲k⫹1共兵n其丢兵k其兲共兵n其丢兵1m⫺k其兲, 共31兲 兵1n其丢兵1m其⫽共⫺兲m⫹1兵1n其丢兵m其⫹兺
k⫽1 m⫺1 共⫺兲k⫹1共兵1n其 丢兵k其兲共兵1n其丢兵1m⫺k其兲, 共32兲 兵1n其丢兵m其⫽再
关兵 n其丢兵m其兴T for n even, 关兵n其丢兵1m其兴T for n odd 共33兲allows us to compute plethysms with both Schur functions symmetric and/or antisymmetric. A very common situation which arises in applications is when one needs to compute
plethysms of a same Schur function by many共sometimes all兲 Schur functions of a given degree to
the right.共This is the case of the applications that we will make in Secs. III–VII.兲 For such cases we proposed in Ref. 5 the following algorithm that allows to compute, in a build up way, all plethysms兵其丢兵其r with兵其 a fixed Schur function and 兵其r all Schur functions of degree r, once the plethysms兵其丢兵r其 and兵其丢兵其r⬘, with r
⬘
⬍r have already been computed.共1兲 Find all partitions of r and order them in descending order of all their parts read from left to right.
共2兲 For each partition 兵其⫽兵1,2, . . . ,t,0, . . . ,0其 perform the outer product
兵1,2, . . . ,t⫺1其兵t其, order the irreps in the reduction as in item共1兲, then use Eqs. 共11兲 to obtain the equation
兵其丢兵其⫽共兵其丢兵1,2, . . . ,t⫺1其兲共兵其丢兵t其兲⫺
兺
兵⬘其Ɱ兵其␣共兵1
,2, . . . ,t⫺1其兵t其
→兵
⬘
其兲兵其丢兵⬘
其, 共34兲where the symbolⱮ means preceding, following the ordering in item 共1兲.
Since 兵1,2, . . . ,t⫺1其 and 兵t其 have smaller degree than 兵其, the plethysms 兵其 丢兵1,2, . . . ,t⫺1其and兵其丢兵t其have already been computed in the induction process. On the other hand, the plethysms兵其丢兵
⬘
其 also have been computed since 兵⬘
其 precedes兵其.The formulas here given and the above algorithm suffice for calculating all plethysms needed in this work.
B. Special branching rules
The use of plethysms to compute branching rules is based in the theorem.4
关– 1 兴– ⫽ 共–␣兲– ⫹ 共–兲– ⫹¯ ⫹ 共–兲–, 共35兲 then the character 关– 兴– of G decomposes into the characters (–)– of H according to the char-acters contained in the plethysm
关 共–␣兲– ⫹ 共–兲– ⫹ ¯ ⫹ 共–兲– 兴丢关– 兴–. 共36兲
This plethysm can be obtained expressing the characters of G and H in terms of characters of GL(n), computing the resulting plethysms of GL(n) characters and re-expressing the result in terms of characters of H in order to obtain the final result.
Using the association irrep↔ character this theorem gives us the coefficients of the reduction of the irrep 关– 兴– of G in the direct sum of irreps (–)– of H.
To illustrate the use of this theorem, let us consider some general cases that will be used later on. The first step toward the use of Eq.共36兲 is to find the decomposition 共35兲. One way of finding it is by constructing a realization of basis states of irreps and generators of groups G and H.
One such realization is provided by the boson calculus18in which a set of boson operators bi† 共creation兲 and bi 共annihilation兲 is introduced and the generators and basis states of irreps are written in terms of them.
The boson operators satisfy the usual commutation relations 关bi,bj †兴⫽␦ i j, 关bi,bj兴⫽关bi †,b j †兴⫽0, i, j⫽1,2, . . . ,n, 共37兲
and the bi’s annihilate the vacuum state兩0
典
. For U(n) the generators are realized byCi j⫽b
i †b
j, i, j⫽1,2, . . . ,n, 共38兲
while the maximum weight basis states of symmetric irreps兵N,0, . . . ,0其⬅兵N其 are realized by 兩兵N其m.w.
典
⫽ 1冑
N!共b1†兲N兩0
典
, 共39兲from which it follows that the basis states of irrep 兵1其 of U(n) are realized by
兩兵1其i
典
⫽bi†兩0典
, i⫽1,2, . . . ,n. 共40兲The generators of U(n⫺1) are the Cijgiven in Eq.共38兲 for i, j⫽1,2, . . . ,n⫺1. Acting then in 共40兲 one sees that the U(n) irrep兵1其 splits into two U(n⫺1) irreps 兵1其 and 兵0其 with basis states
兩兵1其i
典
⫽bi†兩0典
, i⫽1,2, . . . ,n⫺1 and 兩兵0其典
⫽bn†兩0典
. 共41兲 Therefore one obtains兵1其⫽兵1其⫹兵0其 for U共n兲傻U共n⫺1兲. 共42兲
For O(n) the generators are Lij⫽Cij⫺Cji and reduction共35兲 read as
兵1其⫽共1兲. 共43兲
关We denote the irreps of unitary 共U兲 and orthogonal 共O兲 groups as quantities inside braces and parentheses, respectively.兴
Consider the case in which U(n) acts on a vector spaceE⫽E
⬘
⫹E⬙
with dimensions n⬘
andn
⬙
such that n⫽n⬘
⫹n⬙
. We then split n into two terms n⬘
and n⬙
and consider U(n⬘
) as the group with generators Cij⬘⬘ for i⬘
, j⬘
⫽1,2, . . . ,n⬘
and U(n⬙
) that with generators Cij⬙⬙ with i⬙
, j⬙
⫽n⬘
⫹1,n⬘
⫹2, . . . ,n⬘
⫹n⬙
⫽n. The basis 共40兲 splits into two兩兵1其i
⬘典
⫽bi ⬘ †兩0
典
, i⬘
⫽1,2, . . . ,n⬘
,兩兵1其i
⬙典
⫽bi†⬙兩0典
, i⬙
⫽n⬘
⫹1,n⬘
⫹2, . . . ,n⬘
⫹n⬙
⫽n, 共44兲 realizing the basis states of irreps兵1其⬘
兵0其⬙
and兵0其⬘
兵1其⬙
of U(n⬘
)U(n
⬙
), respectively. We then have兵1其⫽兵1其
⬘
兵0其⬙
⫹兵0其⬘
兵1其⬙
for U共n⬘
⫹n⬙
兲傻U共n⬘
兲U共n
⬙
兲. 共45兲 For the case in which U(n) acts on a vector spaceE⫽E⬘
丢E⬙
with dimensions n⬘
and n⬙
one uses boson operators with two indices, each one associated to transformations in each subspace,关bis,bjt †兴⫽␦
i j␦st, 关bis,bjt兴⫽关bis †
,b†jt兴⫽0, i, j⫽1,2, . . . ,n
⬘
, s,t⫽1,2, . . . ,n⬙
. 共46兲 The basis states of irrep兵1其 are realized by兩兵1其is
典
⫽bis†兩0典
, i⫽1,2, . . . ,n⬘
, s⫽1,2, . . . ,n⬙
. 共47兲 Since the U(n⬘
) generators Cij⫽兺sbis†bjs act on the first index and those Cst⫽兺ibis†bit of U(n⬙
) on the second, one concludes that兵1其⫽兵1其
⬘
兵1其⬙
for U共n⬘
n⬙
兲傻U共n⬘
兲⫻U共n⬙
兲. 共48兲Using Eq.共42兲 in Eq. 共36兲, the branching rule for the reduction U(n)傻U(n⫺1) is given by
computing the plethysm
共兵1其⫹兵0其兲丢兵其⫽
兺
⬘⬙␣共兵
⬘
其兵⬙
其→兵其兲共兵1其丢兵⬘
其兲(兵0其丢兵⬙
其)⫽
兺
⬘n⬙
␣共兵
⬘
其兵n⬙
其→兵其兲兵⬘
其, 共49兲 where use was made of共12兲, 共18兲, and 共19兲. By Littlewood rules, one sees that the Schur functions that contain兵其 in the expansion of its outer product by a symmetric Schur function are those兵⬘
其 satisfyingi⭓i
⬘
⭓i⫹1, i⫽1,2, . . . ,i⫺1. 共50兲Then one concludes that under restriction U(n)傻U(n⫺1) the U(n) irrep 兵其 reduces as
兵其⫽
兺
⬘ 兵
⬘
其, 共51兲where兵
⬘
其 are the U(n⫺1) irreps satisfying Eq. 共50兲. These are the well known in-betweenessconditions introduced by Gelfand19in the labeling of basis states of U(n) irreps.
To compute the branching of irrep兵其 of U(n
⬘
⫹n⬙
) into irreps of U(n⬘
)U(n
⬙
), accordingto Eqs.共36兲 and 共45兲 we need to compute the plethysm
共兵1其
⬘
兵0其⬙
⫹兵0其⬘
兵1其⬙
兲丢兵其⫽兺
␣共兵其兵其→兵其兲共共兵1其
⬘
兵0其⬙
兲丢兵其兲共兵0其⬘
兵1其⬙
丢兵其兲,共52兲
共共兵1其
⬘
兵0其⬙
兲丢兵其兲⫽兺
␥ ␣共兵␥其⫻兵其→兵其兲共兵1其
⬘
丢兵␥其兲共兵0其⬙
丢兵其兲⫽兵其兵0其⫽兵其 共53兲using Eqs. 共18兲 and 共19兲 and the result 兵其⫻兵n其⫽兵其 on inner product of Schur functions.
Analogously (兵0其
⬘
兵1其⬙
)丢兵其 gives兵其 and one concludes that兵其⫽
兺
␣共兵其兵其→兵其兲兵其
⬘
兵其⬙
for U共n⬘
⫹n⬙
兲傻U共n⬘
兲U共n
⬙
兲, 共54兲兵其
⬘
,兵其⬙
being irreps of U(n⬘
) and U(n⬙
), respectively.To compute the branching of irrep兵其 of U(n
⬘
n⬙
) into irreps of U(n⬘
)⫻U(n⬙
), according to Eqs.共36兲 and 共48兲 we need to compute the plethysm,共兵1其
⬘
兵1其⬙
兲丢兵其⫽兺
␣共兵其⫻兵其→兵其兲共兵1其
⬘
丢兵其兲共兵1其⬙
丢兵其兲, 共55兲where use was made of Eq.共15兲. Using Eq. 共18兲 one concludes that
兵其⫽
兺
, ␣共兵其⫻兵其→兵其兲兵其
⬘
兵其⬙
for U共n⬘
兲⫻U共n⬙
兲, 共56兲兵其
⬘
,兵其⬙
being irreps of U(n⬘
) and U(n⬙
), respectively.When兵其 is a symmetric representation the inner product in Eq. 共56兲 requires that the irreps of U(n
⬘
) and U(n⬙
) be the same.Equation共43兲 is of no use for producing branching rules since it gives a trivial result. For this case one uses the known result.4,6
The character兵其 of U(n) decomposes into O(n) characters (
⬙
) by the relation兵其⫽
兺
⬙
冋
兺
⬘ ␣共兵⬘
其兵⬙
其→兵其兲册
共⬙
兲, 共57兲 where the sum is made in the irreps兵⬘
其 with even parts.When兵其 is a symmetric representation both Schur functions兵
⬘
其 and兵⬙
其 are symmetricand Eq.共57兲 gives
兵N其⫽共N兲⫹共N⫺2兲⫹¯⫹共0兲 or 共1兲 for U共n兲傻O共n兲. 共58兲
We give in Table I the branching rules in the reduction U(n)傻O(n) for the lowest degree
U(n) irreps with no more than three rows.
For small values of n some O(n) characters in Eq. 共57兲 may have more than the allowed
number关n/2兴 of rows. In this case, they are worked out using modified rules.20For U(3)傻O⫹(3), Eq. 共57兲 and the corresponding modification rules are equivalent to the Elliott rules7 for the branching of U共3兲 irrep兵f1, f2, f3其 into O⫹(3) irreps (L).
共1兲 Define
⫽ f1⫺ f2, ⫽ f2⫺ f3, ¯⫽max共,兲, ¯⫽min共,兲. 共59兲
共2兲 Introduce an extra label K that can assume the values
K⫽¯ ,¯⫺2, . . . ,0 or 1. 共60兲
共3兲 To each K corresponds a set of L values
L⫽¯,¯⫺2, . . . ,0 or 1 for K⫽0. 共61兲 The inverse result, that is, the expression of O(n) characters in terms of those of U(n) is also needed. It can be obtained by subtractions using tables of U(n)傻O(n) reductions or by use of the result4,21
共兲⫽兵其⫹
兺
冋
兺
␥ (⫺)r/2␣(兵␥其兵其→兵其
册
兵其, 共62兲where r is the degree of 兵␥其 and these are taken among the set of Schur functions that in
Frobenius’ notation4assume the form
冉
a⫹1 a冊
,冉
a⫹1 b⫹1 a b冊
,冉
a⫹1 b⫹1 c⫹1 a b c冊
, . . . . 共63兲When共兲 is symmetric one obtains from Eq. 共58兲 and also from Eq. 共62兲,
共N兲⫽兵N其⫺兵N⫺2其 for N⭓2. 共64兲
Table B-4 in Ref. 4 gives a list of reductions共62兲 for irreps 兵其 of degree up to 16 and parts not greater than 4.
TABLE I. U(n)傻O(n) branching rules for U(n) irreps with no more than three rows and the lowest degrees. 兵0其⫽(0) 兵1其⫽(1) 兵2其⫽(2)⫹(0) 兵12其⫽(12) 兵3其⫽(3)⫹(1) 兵21其⫽(21)⫹(1) 兵13其⫽(13) 兵4其⫽(4)⫹(2)⫹(0) 兵31其⫽(31)⫹(2)⫹(12) 兵22其⫽(22)⫹(2)⫹(0) 兵212其⫽(212)⫹(12) 兵5其⫽(5)⫹(3)⫹(1) 兵41其⫽(41)⫹(3)⫹(21)⫹(1) 兵32其⫽(32)⫹(3)⫹(21)⫹(1) 兵312其⫽(312)⫹(21)⫹(13) 兵221其⫽(221)⫹(21)⫹(1) 兵6其⫽(6)⫹(4)⫹(2)⫹(0) 兵51其⫽(51)⫹(4)⫹(31)⫹(2)⫹(12) 兵42其⫽(42)⫹(4)⫹(31)⫹(22)⫹2(2)⫹(0) 兵32其⫽(32)⫹(31)⫹(12) 兵412其⫽(412)⫹(31)⫹(212)⫹(12) 兵321其⫽(321)⫹(31)⫹(22)⫹(212)⫹(2)⫹(12) 兵23其⫽(23)⫹(22)⫹(2)⫹(0) 兵7其⫽(7)⫹(5)⫹(3)⫹(1) 兵61其⫽(61)⫹(5)⫹(41)⫹(3)⫹(21)⫹(1) 兵52其⫽(52)⫹(5)⫹(41)⫹(32)⫹2(3)⫹(21)⫹(1) 兵43其⫽(43)⫹(41)⫹(32)⫹(3)⫹(21)⫹(1) 兵512其⫽(512)⫹(41)⫹(312)⫹(21)⫹(13) 兵421其⫽(421)⫹(41)⫹(32)⫹(312)⫹(221)⫹(3)⫹2(21) ⫹(1) 兵321其⫽(321)⫹(32)⫹(312)⫹(21)⫹(13) 兵322其⫽(322)⫹(32)⫹(221)⫹(3)⫹(21)⫹(1) 兵8其⫽(8)⫹(6)⫹(4)⫹(2)⫹(0) 兵71其⫽(71)⫹(6)⫹(51)⫹(4)⫹(31)⫹(2)⫹(12) 兵62其⫽(62)⫹(6)⫹(51)⫹(42)⫹2(4)⫹(31)⫹(22)⫹2(2) ⫹(0)
III. IBM-1
In the original IBM, now named IBM-1, the valence nucleons of even–even nuclei are joined in pairs to form a s- or d-boson, without distinguishing protons from neutrons. Then the building blocks are creation (s†,d†) and annihilation (s,d) boson operators satisfying the commutation relations
关s,s†兴⫽1, 关d
,d†⬘兴⫽␦⬘, ,
⬘
⫽0,⫾1,⫾2, 共65兲all other commutators vanishing. In a compact notation one can define, say,
b† with b†⫽d⫺3† for ⫽1,2,3,4,5 and b6†⫽s† 共66兲
and analogously for b, recovering Eq.共37兲. Using linear combinations of creation and annihila-tion operators that preserve the number of bosons, it is possible to construct O⫹(3) Racah tensors of ranksᐉ⫽0,1,2,3,4. Linear combinations of these tensors realize15,22the infinitesimal generators of U共6兲 subgroups in the three chains ending with O⫹(3)傻O⫹(2),
% U共5兲 傻 O⫹共5兲 傻 O⫹共3兲 傻 O⫹共2兲 共I兲,
U共6兲 → SU共3兲 傻 O⫹共3兲 傻 O⫹共2兲 共II兲,
& O⫹共6兲 傻 O⫹共5兲 傻 O⫹共3兲 傻 O⫹共2兲 共III兲.
共67兲 With one-index boson operators only symmetrical irreps can be realized. Then the U共6兲 irrep
is兵N其 where N denotes the number of bosons.
Let us examine the branching rules in chain共I兲 of Eq. 共67兲. The U共5兲 labels are given by the general result共51兲. Then the U共5兲 irrep is symmetrical兵Nd其, where Ndis the number of d-bosons and can assume the values
Nd⫽N,N⫺1, . . . ,0. 共68兲
Each U共5兲 irrep兵Nd其, being symmetric, branches as共58兲 into O⫹(5) irreps
兵Nd其⫽共Nd兲⫹共Nd⫺2兲⫹¯⫹共0兲 or 共1兲. 共69兲
To find the branching in O⫹(5)傻O⫹(3) one observes that the generators of U共5兲 were
con-structed only with operators d† and d so one has
共1兲⫽共2兲 for O⫹共5兲傻O⫹共3兲. 共70兲
According to Eq.共36兲 the branching of a general O⫹(5) irrep兵其 into O⫹(3) irreps is found computing the plethysm (2)丢(). The character 共2兲 of O⫹(3) is given by (2)⫽兵2其⫺兵0其. Since 共兲 is an O⫹(5) irrep it has at most two lines, then we expand it using Eq.共62兲 in terms of Schur
functions with up to two rows:
共兲⫽
兺
k ␣k兵
k其⫹
兺
1,2 ␣1,2兵1
,2其. 共71兲
The plethysm (2)丢() is then 共2兲丢共兲⫽共兵2其⫺兵0其兲丢
冋
兺
k ␣k兵 k其⫹兺
1,2 兵1,2其册
⫽兺
k ␣k关兵 2其丢兵k其⫺兵2其丢兵k⫺1其兴⫹兺
12 ␣12关兵2其丢兵1,2其⫺兵2其丢兵1⫺1,2其 ⫺兵2其丢兵1,2⫺1其⫹兵2其丢兵1⫺1,2⫺1其兴, 共72兲where plethysms with Schur functions associated to nonstandard partitions are disregarded. The final result is obtained by expressing the Schur functions resulting from plethysms in terms of O⫹(3) irreps (L) using Eqs.共59兲–共61兲.
In IBM-1 the O⫹(5) irrep 共兲 is symmetric, then one uses Eqs. 共71兲 and 共72兲 with ␣
1,2
⫽0. The terms with ␣1,2⫽0 will be used in IBM-2 and 3. In Table II the O
⫹(5)傻O⫹(3)
branching rules for O⫹(5) irreps with the lowest degrees are given.
Now let us find the branching rules in chain共II兲 of Eq. 共67兲. To find the decomposition 共35兲
for U(6)傻SU(3) in chain 共II兲 one observes that the U共3兲 irreps must have the (L) multiplets 共2兲
and共0兲 contained in irrep 兵1其 of U共6兲 关the reduction U(3)傻SU(3) has only one SU共3兲 irrep with labels given by Eq.共59兲兴. Using Elliott’s rules 共59兲–共61兲 one sees that the U共3兲 irrep must be 兵2其. We then have
兵1其⫽兵2其⬅共2,0兲 for U共6兲傻U共3兲 关or SU共3兲兴. 共73兲
Using Eq.共73兲 and Eq. 共36兲 one has that the U共3兲关SU共3兲兴 irreps contained in the irrep 兵其 of U共6兲 are
兵其⫽
兺
␣共兵2其丢兵其→兵其兲兵其, 共74兲
where in the plethysms only irreps with no more than three rows are considered and these produce SU共3兲 irreps (1⫺2,2⫺3) in Elliott’s notation. Table III presents the branching U(6)傻SU(3) for U共6兲 irreps with no more than three rows and the lowest degrees.
The branching rule in SU(3)傻O⫹(3)傻O⫹(2) is given by Elliott’s rules共59兲–共61兲.
Since in IBM-1 the U共6兲 irrep 兵其 is a symmetric irrep兵N其, the plethysm in Eq.共74兲 is given
by Eq.共22兲 and one obtains
TABLE II. O⫹(5)傻O⫹(3) branching rules for O⫹(5) irreps of degrees up to 8. (0)⫽(0) (1)⫽(2) (2)⫽(2)⫹(4) (12)⫽(1)⫹(3) (3)⫽(0)⫹(3)⫹(4)⫹(6) (21)⫽(1)⫹(2)⫹(3)⫹(4)⫹(5) (4)⫽(2)⫹(4)⫹(5)⫹(6)⫹(8) (31)⫽(1)⫹(2)⫹2(3)⫹(4)⫹2(5)⫹(6)⫹(7) (22)⫽(0)⫹(2)⫹(3)⫹(4)⫹(6) (5)⫽(2)⫹(4)⫹(5)⫹(6)⫹(7)⫹(8)⫹(10) (41)⫽(1)⫹(2)⫹2(3)⫹2(4)⫹2(5)⫹2(6)⫹2(7)⫹(8)⫹(9) (32)⫽(1)⫹2(2)⫹(3)⫹2(4)⫹2(5)⫹(6)⫹(7)⫹(8) (6)⫽(0)⫹(3)⫹(4)⫹2(6)⫹(7)⫹(8)⫹(9)⫹(10)⫹(12) (51)⫽(1)⫹(2)⫹2(3)⫹2(4)⫹3(5)⫹2(6)⫹3(7)⫹2(8)⫹2(9)⫹(10)⫹(11) (42)⫽(0)⫹(1)⫹2(2)⫹2(3)⫹3(4)⫹2(5)⫹3(6)⫹2(7)⫹2(8)⫹(9)⫹(10) (32)⫽(1)⫹2(3)⫹(4)⫹(5)⫹(6)⫹(7)⫹(9) (7)⫽(2)⫹(4)⫹(5)⫹(6)⫹(7)⫹2(8)⫹(9)⫹(10)⫹(11)⫹(12)⫹(14) (61)⫽(1)⫹(2)⫹2(3)⫹2(4)⫹3(5)⫹3(6)⫹3(7)⫹3(8)⫹3(9)⫹2(10)⫹2(11)⫹(12)⫹(13) (52)⫽(0)⫹(1)⫹2(2)⫹3(3)⫹3(4)⫹3(5)⫹4(6)⫹3(7)⫹3(8)⫹3(9)⫹2(10)⫹⫹(11)⫹(12) (43)⫽(1)⫹2(2)⫹2(3)⫹2(4)⫹3(5)⫹2(6)⫹2(7)⫹2(8)⫹(9)⫹(10)⫹(11) (8)⫽(2)⫹(4)⫹(5)⫹(6)⫹(7)⫹2(8)⫹(9)⫹2(10)⫹(11)⫹(12)⫹(13)⫹(14)⫹(16) (71)⫽(1)⫹(2)⫹2(3)⫹2(4)⫹3(5)⫹3(6)⫹4(7)⫹3(8)⫹4(9)⫹3(10)⫹3(11)⫹2(12)⫹2(13)⫹(14) ⫹(15) (62)⫽(1)⫹3(2)⫹2(3)⫹4(4)⫹4(5)⫹4(6)⫹4(7)⫹5(8)⫹3(9)⫹4(10)⫹3(11)⫹2(12)⫹(13)⫹(14) (53)⫽2(1)⫹2(2)⫹3(3)⫹3(4)⫹4(5)⫹3(6)⫹4(7)⫹3(8)⫹3(9)⫹2(10)⫹2(11)⫹(12)⫹(13) (42)⫽(0)⫹(2)⫹(3)⫹2(4)⫹(5)⫹2(6)⫹(7)⫹(8)⫹(9)⫹(10)⫹(12)
兵N其⫽
兺
1,2,3
兵21,22,23其⬅
兺
123共2共1⫺2兲,2共2⫺3兲兲, 共75兲
where (1,2,3) are共standard兲partitions of N into three parts.
The branching for the first link U(6)傻O⫹(6) in chain共III兲 is found using Eq. 共57兲. Note that in共57兲 the branching is for U(n)傻O(n) and we need a further reduction O(n)傻O⫹(n). For the cases treated here the O共6兲 and O⫹(6) irreps are the same.
In IBM-1 the U共6兲 irrep 兵其 being symmetric implies that Eq. 共57兲 has a simple expression:
兵N其⫽共N兲⫹共N⫺2兲⫹¯⫹共0兲 or 共1兲 for U共6兲傻O⫹共6兲. 共76兲
To find the branching rule in the link O⫹(6)傻O⫹(5) one first observes that Eq.共42兲 gives
共1兲⫽共1兲⫹共0兲 for O⫹共6兲傻O⫹共5兲. 共77兲
TABLE III. Branching rules for U(6)傻SU(3) for U共6兲 irreps with no more than three rows and lowest degrees. 兵0其⫽(0,0) 兵1其⫽(2,0) 兵2其⫽(4,0)⫹(0,2) 兵12其⫽(2,1) 兵3其⫽(6,0)⫹(2,2)⫹(0,0) 兵21其⫽(4,1)⫹(2,2)⫹(1,1) 兵13其⫽(3,0)⫹(0,3) 兵4其⫽(8,0)⫹(4,2)⫹(0,4)⫹(2,0) 兵31其⫽(6,1)⫹(4,2)⫹(2,3)⫹(1,2)⫹(2,0)⫹(3,1) 兵22其⫽(4,2)⫹(0,4)⫹(2,0)⫹(3,1) 兵212其⫽(5,0)⫹(2,3)⫹(1,2)⫹(0,1)⫹(3,1) 兵5其⫽(10,0)⫹(6,2)⫹(2,4)⫹(4,0)⫹(0,2) 兵41其⫽(8,1)⫹(6,2)⫹(4,3)⫹(5,1)⫹(2,4)⫹(3,2)⫹(4,0)⫹(1,3)⫹(2,1)⫹(0,2) 兵32其⫽(6,2)⫹(4,3)⫹(5,1)⫹(2,4)⫹(3,2)⫹2(4,0)⫹(1,3)⫹(2,1)⫹(0,2) 兵312其⫽(7,0)⫹(4,3)⫹(5,1)⫹2(3,2)⫹(0,5)⫹(1,3)⫹2(2,1)⫹(1,0) 兵221其⫽(5,1)⫹(2,4)⫹(3,2)⫹(4,0)⫹(1,3)⫹(2,1)⫹(0,2) 兵6其⫽(12,0)⫹(8,2)⫹(4,4)⫹(6,0)⫹(0,6)⫹(2,2)⫹(0,0) 兵51其⫽(10,1)⫹(8,2)⫹(6,3)⫹(7,1)⫹(4,4)⫹(5,2)⫹(6,0)⫹(2,5)⫹(3,3)⫹(4,1)⫹(1,4)⫹2(2,2)⫹(1,1) 兵42其⫽(8,2)⫹(6,3)⫹(7,1)⫹2(4,4)⫹(5,2)⫹2(6,0)⫹2(3,3)⫹2(4,1)⫹(0,6)⫹(1,4)⫹3(2,2)⫹(1,1)⫹(0,0) 兵412其⫽(9,0)⫹(6,3)⫹(7,1)⫹2(5,2)⫹(2,5)⫹2(3,3)⫹2(4,1)⫹(1,4)⫹(2,2)⫹2(3,0)⫹2(0,3)⫹(1,1) 兵32其⫽(6,3)⫹(5,2)⫹(6,0)⫹(2,5)⫹(3,3)⫹(4,1)⫹(2,2)⫹(3,0)⫹(0,3) 兵321其⫽(7,1)⫹(4,4)⫹2(5,2)⫹(6,0)⫹(2,5)⫹2(3,3)⫹3(4,1)⫹2(1,4)⫹3(2,2)⫹(3,0)⫹(0,3)⫹2(1,1) 兵23其⫽(6,0)⫹(3,3)⫹(0,6)⫹2(2,2)⫹(0,0) 兵7其⫽(14,0)⫹(10,2)⫹(6,4)⫹(8,0)⫹(2,6)⫹(4,2)⫹(0,4)⫹(2,0) 兵61其⫽(12,1)⫹(10,2)⫹(8,3)⫹(9,1)⫹(6,4)⫹(7,2)⫹(8,0)⫹(4,5)⫹(5,3)⫹(6,1)⫹(2,6)⫹(3,4)⫹2(4,2) ⫹(1,5)⫹(2,3)⫹(3,1)⫹(0,4)⫹(1,2)⫹(2,0) 兵52其⫽(10,2)⫹(8,3)⫹(9,1)⫹2(6,4)⫹(7,2)⫹2(8,0)⫹(4,5)⫹2(5,3)⫹2(6,1)⫹(2,6)⫹2(3,4)⫹4(4,2)⫹(1,5) ⫹2(2,3)⫹2(3,1)⫹2(0,4)⫹(1,2)⫹2(2,0) 兵512其⫽(11,0)⫹(8,3)⫹(9,1)⫹2(7,2)⫹(4,5)⫹2(5,3)⫹2(6,1)⫹2(3,4)⫹(4,2)⫹2(5,0)⫹(0,7)⫹(1,5) ⫹3(2,3)⫹2(3,1)⫹2(1,2)⫹(0,1)
Next one writes the O⫹(6) irrep 共兲 in terms of U共6兲 irreps 兵其:
共兲⫽
兺
␣兵其 共78兲
and computes the plethysm
共共1兲⫹共0兲兲丢共兲⫽
兺
␣共共1兲⫹共0兲兲丢兵其⫽
兺
␣冋
兺
⬘,k␣共兵
⬘
其兵k其→兵其兲兵⬘
其册
. 共79兲 The resulting U共5兲 irreps are then converted into O⫹(5) irreps by use of Eq.共57兲.In IBM-1 the O⫹(6) irrep is symmetric and, in this case, Eq.共78兲 becomes
共0兲⫽兵0其, 共1兲⫽兵1其 and 共k兲⫽兵k其⫺兵k⫺2其 for k⭓2 共80兲
and Eq.共79兲 gives
共共1兲⫹共0兲兲丢共k兲⫽
兺
p,q 关␣共兵
p其兵q其→兵k其兲⫺␣共兵p其兵q其→兵k⫺2其兲兴兵p其. 共81兲 Expressing兵p其 in terms of O⫹(5) irreps by means of Eq.共58兲 one has the final result
共k兲⫽共k兲⫹共k⫺1兲⫹¯ ⫹共0兲 for O⫹共6兲傻O⫹共5兲. 共82兲
For general O⫹(6) irreps ()⫽(1,2,3), computer calculations using Eq. 共79兲 shows the branching rule 共兲⫽
兺
1⫽2 1兺
2⫽3 2 共1,2兲 for O⫹共6兲傻O⫹共5兲, 共83兲the usual inbetweeness conditions for Gelfand labels.
IV. IBM-2
In IBM-2 two kinds of bosons are considered, one formed by proton pairs and other by neutron pairs, denoted by
s†, d,† , s†, d,† , ⫽0,⫾1,⫾2, for protons, for neutrons 共84兲
and similarly for annihilation operators. The commutation relations are the same as Eq. 共65兲
concerning angular momentum labels and neutron operators commute with proton operators. Using the compact notation
b,␣† , b,␣, ⫽,, ␣⫽1,2, . . . ,6, 共85兲
the commutation relations become 关b␣,b⬘␣⬘
†
兴⫽␦⬘␦␣␣⬘, 关b␣,b⬘␣⬘兴⫽关b␣† ,b⬘␣⬘ †
兴⫽0. 共86兲
With these operators one constructs operators C␣⬘␣⬘⫽b␣†
that under commutation close the Lie algebra of U共12兲. The operators C␣␣⬘⫽C␣␣⬘ and C”␣␣⬘ ⫽C␣␣⬘generate the Lie algebras of U(6) and U(6), respectively. We have then a particular case of the reduction U(n1⫹n2)傻U(n1)
U(n2) studied in Sec. II B. Using the results there obtained one has the branching rule
兵其⫽
兺
⬘,⬙
␣共兵
⬘
其兵⬙
其→兵其兲兵⬘
其兵⬙
其 for U共12兲傻U共6兲U共6兲. 共88兲
With operators共84兲 关or 共85兲兴 one can construct only symmetrical irreps兵N其 of U共12兲 and Eq. 共88兲 reduces to
兵N其⫽
兺
k⫽0 N
兵N⫺k其兵k其 for U共12兲傻U共6兲
U共6兲. 共89兲
The basis states of irrep兵N其 must be also basis states for an irrep of O⫹⫹ (3), the group of simultaneous rotations of protons and neutrons. This can be achieved by use of lattice of algebras, in contrast with chains of algebras in IBM-1.
The simplest lattice is obtained when we use chains共I兲, 共II兲, and 共III兲 separately for protons and for neutrons and only in the last step one couples O⫹(3) with O⫹(3) to obtain O⫹⫹ (3):
%U共5兲傻 O⫹共5兲& U共6兲 →SU共3兲 → O⫹共3兲 % &O⫹共6兲傻 O⫹共5兲% & U共12兲 O⫹⫹ 共3兲傻 O⫹⫹ 共2兲. & %U共5兲傻 O⫹共5兲& % U共6兲 →SU共3兲 → O⫹共3兲 &O⫹共6兲傻 O⫹共5兲% 共90兲 This is a trivial extension of IBM-1 and L and L are coupled to give
L⫹⫽L⫹L, L⫹L⫺1, . . . ,兩L⫺L兩. 共91兲 Another lattice is U共6兲 U⫹共5兲傻O⫹⫹ 共5兲 共I1兲 & % & U⫹共6兲 → SU⫹共3兲 → O⫹⫹ 共3兲傻O⫹⫹ 共2兲 共II1兲 % & % U共6兲 O⫹⫹ 共6兲傻O⫹⫹ 共5兲 共III1兲 共92兲 in which the algebras of U(6) and U(6) are joined in the first step. In the first link one has
U(6)⫻U(6)→U⫹(6) and the branching rules are given by the Kronecker product of U共6兲
irreps. In this case, the irreps of U(6) and U(6) are both symmetric by Eq.共89兲 and the irreps of U⫹(6) can have one or two rows. Chains (I1), (II1), and (III1) are the same as共I兲, 共II兲, and
共III兲 but now the U共6兲, U共5兲, O⫹(6), and O⫹(6) irreps can be two-rowed.
Another type of lattice of algebras is obtained by joining the neutron and proton algebras at the second step:
U共6兲傻U共5兲 &
U⫹共5兲傻O⫹⫹ 共5兲傻O⫹⫹ 共3兲傻O⫹⫹ 共2兲 共I2兲, %
U共6兲傻U共5兲 U共6兲傻SU共3兲
&
SU⫹共3兲傻O⫹⫹ 共3傻O⫹⫹ 共2兲, 共II2兲, %
U共6兲傻SU共3兲 U共6兲傻O⫹共6兲
&
O⫹⫹ 共6兲傻O⫹⫹ 共5兲傻O⫹⫹ 共3兲傻O⫹⫹ 共2兲 共III2兲. %
U共6兲傻O⫹共6兲
共93兲
The branching rules for the irreps of the joined algebras are obtained by Kronecker products and the resulting irreps can be one- and two-rowed. The Kronecker product expansion of irreps of unitary groups are given by the outer product of Schur functions:
兵其兵其⫽
兺
␣共兵其兵其→兵其兲兵其⫹. 共94兲
The Kronecker product of O共6兲 irreps is done by expressing the O共6兲 characters in terms of Schur functions, making the outer products and re-expressing the result in terms of O共6兲 irreps. In Table
IV we give the Kronecker product of O共6兲 irreps with the lowest product degrees.
V. IBM-3
This model was proposed by Elliott and White23 in order to take into account the isospin
degree of freedom. It differs from IBM-2 by the inclusion of a third kind of boson, the␦-boson, formed by a proton–neutron pair. There are 18 creation operators
s†, d,† , s†, d,† , s␦†, d␦,† 共⫽0,⫾1,⫾2兲 共95兲 and the corresponding annihilation operators. Operators of different pairs of bosons commute
among themselves while each set , and␦ satisfies bose commutation relations.
One has again lattices of algebras now starting with
U共18兲傻U共6兲
U共6兲
U␦共6兲 共96兲
and ending with O⫹⫹␦⫹ (3)傻O⫹⫹␦⫹ (2).
By an extension of the calculation done to obtain Eq.共54兲 one obtains
兵其⫽
兺
,, ␣共兵其兵其兵其→兵其兲兵其兵其兵其␦ for U共6兲
U共6兲
U␦共6兲. 共97兲
兵N其⫽
兺
p⫽0 N兺
q⫽0 N⫺p 兵p其兵q其兵N⫺p⫺q其␦ for U共6兲U共6兲
U␦共6兲. 共98兲
As in IBM-2, a trivial lattice is obtained joining the three algebras in the first step by the link
U共6兲
U共6兲
U␦共6兲傻U⫹⫹␦共6兲. 共99兲
In this case we will have a triple Kronecker product of U共6兲 irreps and the resulting
U⫹⫹␦(6) irreps can be three-rowed. From this point on one follows chains共I兲, 共II兲, and 共III兲 in
which the irreps of U共5兲 and O共6兲 can be three-rowed and those of O共5兲 two-rowed.
A more interesting lattice, from the physical point of view, is the one that works separately with space and isospin degrees of freedom and joins then at the end. To this end let us denote creation and annihilation operators by
bᐉm† , bᐉm 共⫽,,␦, m⫽⫺ᐉ,⫺ᐉ⫹1, . . . ,ᐉ兲, or
TABLE IV. Kronecker product of O共6兲 irreps with lowest total degrees. (1)(1)⫽(0)⫹(12)⫹(2) (2)(1)⫽(1)⫹(21)⫹(3) (12)(1)⫽(1)⫹(13)⫹(21) (3)(1)⫽(2)⫹(31)⫹(4) (21)(1)⫽(12)⫹(2)⫹(212)⫹(22)⫹(31) (13)(1)⫽(12)⫹(212) (2)(2)⫽(0)⫹(12)⫹(2)⫹(22)⫹(31)⫹(4) (12)(2)⫽(12)⫹(2)⫹(212)⫹(31) (12)(12)⫽(0)⫹(12)⫹(2)⫹(212)⫹(22) (4)(1)⫽(3)⫹(41)⫹(5) (31)(1)⫽(21)⫹(3)⫹(312)⫹(32)⫹(41) (22)(1)⫽(21)⫹(221)⫹(32) (212)(1)⫽(13)⫹(21)⫹(221)⫹(312) (3)(2)⫽(1)⫹(21)⫹(3)⫹(32)⫹(41)⫹(5) (21)(2)⫽(1)⫹(13)⫹2(21)⫹(221)⫹(3)⫹(312)⫹(32) ⫹(41) (13)(2)⫽(13)⫹(21)⫹(312) (3)(12)⫽(21)⫹(3)⫹(312)⫹(41) (21)(12)⫽(1)⫹(13)⫹2(21)⫹(221)⫹(3)⫹(312)⫹(32) (13)(12)⫽(1)⫹(13)⫹(21)⫹(221) (5)(1)⫽(4)⫹(51)⫹(6) (41)(1)⫽(31)⫹(4)⫹(412)⫹(42)⫹(51) (32)(1)⫽(22)⫹(31)⫹(321)⫹(32)⫹(42) (312)(1)⫽(212)⫹(31)⫹(321)⫹(412) (221)(1)⫽(212)⫹(22)⫹(23)⫹(321) (4)(2)⫽(2)⫹(31)⫹(4)⫹(42)⫹(51)⫹(6) (31)(2)⫽(12)⫹(2)⫹(212)⫹(22)⫹2(31)⫹(321)⫹(32) ⫹(4)⫹(412)⫹(42)⫹(51) (22)(2)⫽(2)⫹(212)⫹(22)⫹(23)⫹(31)⫹(321)⫹(42) (212)(2)⫽(12)⫹2(212)⫹(22)⫹(31)⫹(321)⫹(412) (4)(12)⫽(31)⫹(4)⫹(412)⫹(51) (31)(12)⫽(2)⫹(212)⫹(22)⫹2(31)⫹(321)⫹(4)⫹(412) ⫹(42) (22)(12)⫽(12)⫹(212)⫹(22)⫹(31)⫹(321)⫹(32) (212)(12)⫽(12)⫹(2)⫹2(212)⫹(22)⫹(23)⫹(31) ⫹(321) (3)(3)⫽(0)⫹(12)⫹(2)⫹(22)⫹(31)⫹(32)⫹(4)⫹(42) ⫹(51)⫹(6)
b␣† , b␣ 共⫽,,␦, ␣⫽1,2, . . . ,6兲 共100兲 so that the commutation relations read as
关b␣,b⬘␣⬘ †
兴⫽␦⬘␦␣␣⬘, 关b␣,b⬘␣⬘兴⫽关b␣† ,b⬘␣⬘ †
兴⫽0. 共101兲
The U共18兲 infinitesimal generators will then be realized by C␣⬘␣⬘⫽b ␣ † b ⬘␣⬘, 共102兲 while C⬘⫽
兺
␣⫽1 6 C␣⬘␣ and C” ␣ ␣⬘⫽兺
⫽1 3 C␣␣⬘ 共103兲are generator of the Lie algebras of US(6) 共space兲 and UT(3) 共isospin兲, respectively. We then have as first link in this lattice,
U共18兲傻US共6兲⫻UT共3兲. 共104兲
The U共18兲 irrep will be symmetric and, according to Eq. 共56兲, the branching law in Eq. 共104兲 will be
兵N其⫽
兺
n1,n2,n3 兵
n1,n2,n3其S兵n1,n2,n3其T, 共105兲 where (n1,n2,n3) is a共standard兲 partition of N into three parts.
For UT(3) one uses the chain UT(3)傻OT⫹(3)傻OT⫹(2) and the branching rule is given by Elliott’s rules, Eqs. 共59兲–共61兲.
From US(6) one can follow each of chains共I兲, 共II兲, and 共III兲 and use the results of Sec. III for three-rowed U共6兲 irreps.
For US(6)傻US(5), Eqs.共50兲 and 共51兲 give
兵f1, f2, f3其⫽
兺
f1⬘⫽ f2 f1兺
f2⬘⫽ f3 f2兺
f3⬘⫽0 f3 兵f1⬘
, f2⬘
, f3⬘
其. 共106兲 For U(5)傻O(5) one uses Eq. 共57兲 and Table I. The three-rowed O⫹(5) irreps (1,2,3) in Eq. 共57兲 must be interpreted using the modification rules共1,2,1兲⬅共1,2兲, 共1,2,3⬎1兲 disregarded. 共107兲 For O⫹(6)傻O⫹(5) one uses Eq.共83兲.
For US(6)傻SUS(3) one uses Eq.共74兲 where now Schur functions 兵其 with up to three rows must be considered.
VI. IBM-4
In IBM-4, proposed by Elliott and Evans,24 the bosonic pairs, besides the spatial degree of
freedom, have also spin–isospin degrees of freedom in the combination S⫽0, T⫽1 and S⫽1,
b(ᐉm ᐉ)(SmS)(TmT) † with ᐉ⫽0,2, ⫺ᐉ⭐m ᐉ⭐ᐉ, S⫽mS⫽0, T⫽1, mT⫽0,⫾1, S⫽1, mS⫽0,⫾1, T⫽mT⫽0 共108兲
and corresponding annihilation operators. The operators C(ᐉm ᐉ)(SmS)(TmT) (ᐉ⬘mᐉ⬘)(S⬘mS⬘)(T⬘mT⬘) ⫽b(ᐉmᐉ)(SmS)(TmT) † b(ᐉ⬘m ᐉ ⬘)(S⬘m S ⬘)(T⬘m T ⬘) 共109兲
generate the Lie algebra of U共36兲 while
C(Sm S)(TmT) (S⬘mS⬘)(T⬘mT⬘) ⫽
兺
ᐉmᐉ C(ᐉm ᐉ)(SmS)(TmT) (ᐉmᐉ)(S⬘mS⬘)(T⬘mT)⬘ and C” ᐉm ᐉ ᐉ⬘mᐉ⬘ ⫽兺
SmSTmTC(ᐉmᐉ)(SmS)(TmT) (ᐉ⬘mᐉ⬘)(SmS)(TmT) 共110兲 generate the Lie algebras of UST(6) and UL(6) in the chainU共36兲傻UL共6兲⫻UST共6兲. 共111兲
An arbitrary irrep of U共36兲 branches into irreps of UL(6)⫻UST(6) according to Eq. 共56兲. Since the U共36兲 irreps that one can realize with 共108兲 and 共109兲 are only symmetric ones, Eq. 共56兲 gives
兵N其⫽
兺
兵其L兵其ST, 共112兲
where共兲 are 共standard兲 partitions of N into six parts. For UL(6) one follows chains共I兲, 共II兲, and
共III兲, now with all irreps in their greatest generality. To treat UST(6) one observes that
C(00)(1m)(00)(1m⬘) and C(1m)(00)(1m)(00) generate the Lie algebras of US(6) and UT(3) in the link
UST共6兲傻US共3兲
UT共3兲, 共113兲
which allows us to treat spin and isospin separately. The branching rules in this link are given by Eq. 共54兲:
兵其⫽
兺
␣共兵其兵其→兵其兲兵其S兵其T for UST共6兲傻US共3兲
UT共3兲. 共114兲
The simplest case is when兵其 is symmetric,
兵N其ST⫽
兺
k⫽0N
兵N⫺k其S兵k其T. 共115兲
The next one is
兵N⫺1,1其⫽
兺
k⫽1 N⫺1 兵k其S兵N⫺k其T⫹兺
k⫽2 N 关兵N⫺k其S兵k⫺1,1其T⫹兵k⫺1,1其S兵N⫺k其T兴. 共116兲 Another chain of interest isUST共6兲傻SUST共4兲傻SUS共2兲⫻SUT共2兲. 共117兲 To find the reduction 共35兲 for this chain one observes that the basis states for irrep 兵1其 of UST(6) are
兺
ᐉmᐉ b(ᐉmᐉ)(SmS)(TmT) † 兩0典
with S⫽MS⫽0, T⫽1, MT⫽0,⫾1, S⫽1, mS⫽0,⫾1, T⫽mT⫽0. 共118兲The states with the first and second sets of labels are basis states for irreps 兵0其S兵1其T and
兵1其S兵0其T of SUS(2)⫻SUT(2), respectively. The U共4兲 irrep 兵11其 has exactly this SUS(2)
⫻SUT(2) reduction, so one has
兵1其⫽兵11其 for UST共6兲傻UST共4兲关SUST共4兲兴 共119兲
and according to Eq.共36兲 one has the branching rule
兵其⫽
兺
␣共兵11其丢兵其→兵其兲兵其, for UST共6兲傻UST共4兲关SUST共4兲兴, 共120兲
where only the Schur functions with up to four rows are considered in the plethysm. This plethysm
can be computed by use of Eq.共23兲 as input in the algorithm given in Sec. II A. In Table V one
gives the branching rules for the reduction UST(6)傻UST(4) for UST(6) irreps with the lowest degrees.
The branching rules for the reduction U(4)傻SU(2)⫻SU(2) are given by Eq. 共56兲. Table
11-18 in Ref. 25 gives the branching rules for this reduction for U共3兲 irreps of degrees up to 10.
VII. IBM-1 G AND F
IBM-1 can be extended by introducing bosons with angular momenta 3,4, . . . . In order to deal with states of positive parity only bosons with even angular momenta are introduced. Bosons with odd angular momenta are used to deal with spectra with even and odd parity levels.
The inclusion of boson pairs of angular momentaᐉ⫽4 in IBM-1 gave birth to IBM-1G. In
this model one has the boson creation operators
bᐉm with ᐉ⫽0,2,4, ⫺ᐉ⭐m⭐ᐉ 共121兲
and the corresponding annihilation operators and the group involved will be U共15兲. One then has
to search for chains ending with O⫹(3)傻O⫹(2). One of such chains,
U共15兲傻SU共3兲傻O⫹共3兲傻O⫹共2兲 共122兲
was studied in Ref. 26. There, the generators of the Lie algebra of SU共3兲 in chain 共122兲 are
realized as X(1)⫽
冑
1/7 关d†⫻d˜兴(1) ⫹冑
6/7 关g†⫻g˜兴(1), 共123兲 X(2)⫽冑
1/70 兵4冑
7/15 关d†⫻s˜⫹s†⫻d˜ 兴(2)⫺11冑
2/21 关d†⫻d˜兴(2) ⫹36冑
1/105 关d†⫻g˜⫹g†⫻d˜兴(2)⫺2冑
33/7 关g†⫻g˜兴(2)兴其, where 关T(k1)⫻T(k2)兴 m(k) denotes coupling of O⫹(3) Racah tensors via Clebsch–Gordan
coeffi-cients to produce m components of O⫹(3) tensors of rank k and b˜ , as in Ref. 15, is defined by b
˜
