Coupling and Recoupling Coefficients for Wigner's U(4) Supermultiplet Symmetry

TL;DR

This work presents a streamlined, null-space–based method to compute all $U(4)$ coupling and Racah recoupling coefficients for both the canonical chain $U(4)\supset U(3)\supset U(2)\supset U(1)$ and the physical chain $U(4)\supset SU_S(2)\otimes SU_T(2)$. It provides a consistent framework for handling inner and outer multiplicities, constructs highest- and lower-weight CGCs, and derives $6$- and $9$-U(4) recoupling coefficients by solving linear systems built from CGCs. The approach yields a practical projection-based transformation between physical and canonical bases, enabling the extraction of reduced spin-isospin Wigner coefficients necessary for nuclear-structure calculations. The results pave the way for a modern C++ library to support symmetry-adapted shell-model calculations and broader applications of Wigner’s supermultiplet symmetry in low-energy nuclear physics.

Abstract

A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group-subgroup chains is presented. The canonical U(4)->U(3)->U(2)->U(1) coupling and recoupling coefficients are applicable to any system that possesses U(4) symmetry, while the physical U(4)->SU_S(2)xSU_T(2) coupling coefficients are more specific to nuclear structure studies that utilize Wigner's Supermultiplet Symmetry concept. The procedure that is proposed sidesteps the use of binomial coefficients and alternating sum series, and consequently enables fast and accurate computation of any and all U(4)-underpinned features. The inner multiplicity of a (S,T) pair within a single U(4) irreducible representation is obtained from the dimension of the null space of the SU(2) raising generators; while the resolution for the outer multiplicity follows from the work of Alex et al. on U(N). It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the \textit{canonical} and the \textit{physical} group-subgroup chains of U(4).

Figures (8)

• Figure 1: Young tableau representation of U(4) [on the left] and SU(4) [on the right] irreps.
• Figure 2: A schematic depiction of the nine generators of the canonical $\rm U(4) \supset U(3) \supset U(2) \supset U(1)$ group-subgroup chain.
• Figure 3: Dimension of some U(4) irreps of the type $[4k,2k,k,0]$ (with even $k$), where $A=\sum_i n_{i4} = 7k$ is the total number of particles that are distributed into the irreps.
• Figure 4: The z-weight diagram for the U(4) irrep $[2,1,1,0]$, with a demonstration of the action -- using colored arrows -- on all of the Gelfand states of the U(4) raising generators that push upward towards the so-called highest weight state (sometimes referred to as the extremal state) at the top right corner of the plot. The conjugate of the highest weight state is the lowest weight state located at the origin at the bottom left corner. Since the irrep is specified, the top row of the Gelfand patterns are omitted.
• Figure 5: Three different orders of coupling three U(4) irreps $f_1$, $f_2$ and $f_3$.