Plethysms and interacting boson models

Abstract

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.

On the calculation of inner products of Schur functions

“…Similar to Eqs. (12), (13) and (14), it is possible to obtain for a two rowed irrep { f 1 f 2 }, the plethysm {1} {1} ⊗ { f 1 f 2 } by using the relation (15) and the basic result (used often for boson systems [5,11]),…”

“…(10) as it is possible to expand any Schur function {λ} in terms of totally antisymmetric or symmetric Schur functions. This can be implemented in a recursive procedure by assuming that the results for all irreps of the integers 1 to n − 1 are known and then obtain the inner products for the irreps of n. Alternatively one can use a procedure given in [11] to compute plethysms adapted for computing the LHS of Eq. (10) for all {λ} of S n starting with…”

“…Similarly the inner product gives the reduction of the Kronecker product of S n irreps labeled by partitions (λ) and (µ) into S n irreps (ν) and g λµν is the multiplicity of the irrep (ν). The third product is called plethysm or symmetrized power and it is explained in [5][6][7][9][10][11]. For example say the states of a single particle correspond to the irrep {λ} of U(n).…”

“…Then with m identical particles occupying these states, the m particle states with permutation symmetry (µ) of S m transform as the irreps {ν} of U(n) and the {ν} are given by the plethysm {λ} ⊗ {µ}. It should be stressed that, although the subject of Schur function multiplications is old, there is recent interest in developing algorithms for these [6,[11][12][13][14] and this is mainly due to the developments in group theoretical models for atomic nuclei.…”

“…In addition, recently computer codes are also developed for evaluating plethysms [11][12][13] and one of the authors [16] has created a bank of plethysms. On the other hand inner products are more complicated.…”