The plethysm technique applied to the classification of nuclear states

Plethysms and interacting boson models

“…III-VII.͒ For such cases we proposed in Ref. 5 the following algorithm that allows to compute, in a build up way, all plethysms ͕ ͖ ͕ ͖ r with ͕ ͖ a fixed Schur function and ͕ ͖ r all Schur functions of degree r, once the plethysms ͕ ͖ ͕r͖ and ͕ ͖ ͕ ͖ r Ј , with rЈϽr have already been computed.…”

“…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 Littlewood 6 as a third way of combining two Schur function to obtain a linear combination of Schur functions of a same degree. With few exceptions, 4,7,8 it 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][10][11][12][13] In most applications to physical problems such as in nuclear structure 5,7 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.…”

“…In that case, using an induction formula for computing plethysms with both factors being symmetric Schur functions given in Ref. 13, we developed an algorithm 5 to compute plethysms with a symmetric Schur function in the left and all Schur functions of a given degree at right.…”

On the calculation of inner products of Schur functions

“…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).…”