{'fraction_non_alphanumeric': 0.03183877891724302, 'fraction_numerical': 0.012640114476508467, 'mean_word_length': 5.7096, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Method of N-Body Hyperspherical Basis Symmetrization

Parentage Scheme of Summarization (PSS) to the N-body symmetrized basis construction necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles is introduced. Proposed method allows to construct N-particle symmetrized hyperspherical functions on the bases of N-particle hyperspherical functions symmetrized with respect to N–1-particles by the use of the Kinematic Rotation Coefficients (KRC) related with the (N-1)-th and N-th particle permutations. The main problem that arises when investigating dynamics of few-body systems in physics is the problem of kinematic rotations under particle permutations. When number of particles increases, kinematic rotations include not only particle permutation but also transitions between different possible configurations, and mathematical calculations using complex general formula become impossible. Moreover, no general formula exists for systems with more than four particles. In order to solve kinematic rotation problem for N-particle systems, the recurrence method of determination of the KRC is applied. According to this method, the initial coefficients with the lowest quantum numbers are calculated by solving the overlap integral analytically, wave functions with arbitrary quantum numbers are expanded in terms of the basic hyperspherical functions, and the kinematic rotations of the obtained expansion are performed with the use of already known coefficients with the lowest quantum numbers. Significant advantage of the recurrence method is that no principal difficulties arise when increasing number of particles. Furthermore, recurrence relations contain numerical coefficients that are easy to evaluate by substituting appropriate quantum numbers. The problem of symmetrization of N-Body hyperspherical functions is solved by the use of the PSS. A construction scheme is given for the symmetrized hyperspherical basis in (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the basic hyperspherical functions are obtained. Parentage coefficients for N-Body systems are introduced. It is demonstrated that no principal difficulties arise when increasing number of particles.


Introduction
While investigating few-body systems with the use Hyperspherical Function Method (HFM) the problem of the hyperspherical basis transformation between different sets of Jacobi coordinates arises (the kinematic rotation problem) [1][2][3][4][5][6]. Coefficients of unitary transformations of three-particle hyperspherical functions under particle permutations were introduced in [1] (Raynal-Revai coefficients). For four-and more particle systems kinematic rotations include both particle permutations and transitions from one configuration to another. Four-particle transformations were discussed in [2][3][7][8][9][10]17]. When number of particles increases, mathematical calculations using complex general formula become impossible. Moreover, no general formula exists for systems with more than four particles.
In order to solve kinematic rotation problem for N>3 particle systems, the recurrence method of determination of the kinematic rotation coefficients (KRC) was introduced [2,11,18]. The symmetrized basis may be constructed easily if the KRC are available [2, 3, 8, 9, 10, 16, 17, and 18]. However, the transformations of Hyperspherical Functions (HF) become sufficiently complex when the number of particles increases. The recurrence method allows avoiding difficulties related with the direct calculations of the KRC. A construction scheme for recurrence relations, proposed in [11], can be easily generalized and applied to the determination of the transformation coefficients for hyperspherical basis with any number of particles. According to this method the coefficients with general formula for the coefficients is not needed at all. Transformation coefficients for N=3 and N=4 were obtained in [10]. This paper discusses the problem of symmetrization of N-body hyperspherical functions. In the next section, parentage scheme of symmetrization is applied to the construction of the symmetrized five-and six-particle hyperspherical basis in the (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the corresponding hyperspherical functions under particle permutations are obtained. It is demonstrated that the KRC contain only permutations related with the last two particles.
In the third section parentage scheme of symmetrization is generalized to construct N-particle symmetrized hyperspherical functions.

Construction of a Symmetrized Hyperspherical Basis for (4+1) and (5+1) Configurations
Four body Hyperspherical Functions symmetrized with respect to three identical particles (nuclei) were introduced in [ corresponding to x, y, z four-body systems' Jacobi coordinates for either (2+2) or (3+1) configurations [10], μ and K are four and three particle hypermomenta, L is a total four-particle angular momentum, Transformation coefficients of the functions under particle permutations were introduced in [10] (2) Transformations of four-body hyperspherical functions under particle permutations were obtained with the use of four-body coefficients [3], similar to three body coefficients introduced by Raynal and Revai ] where i=1,2,3 and 4 i P includes only permutations involving the fourth particle. From Equations (1) and (2) we obtain KRC for four-particle hyperspherical functions symmetrized with respect to three identical particles [10] (4) where Four-body symmetrized hyperspherical basis in (3+1) configuration was constructed by the use of the parentage scheme of symmetrization [10]. Complete set of Young operators acting on the four-body functions symmetrized with respect to three particles were obtained and all relations between parentage coefficients and the coefficients (4) were found [10].
Parentage Scheme of Symmetrization can be easily generalized for N-body systems by introducing Young operators acting on N-particle functions corresponding to the irreducible representation of the N-1 particle permutation group For the five-particle systems we have the following representations [5] where α represents set of parameters defining 5-particle functionα Ψ by removing the square corresponding to the fifth particle. The method of obtaining relations between four-body parentage coefficients and the KRC (4) was described in [10] The symmetrized hyperspherical basis for five-particle systems can be easily constructed by substituting parentage coefficients (12) into formula (9).

Construction of a Symmetrized Hyperspherical Basis for N-Body Systems
Construction scheme for the symmetrized hyperspherical basis for the (4+1) and (5+1) configurations presented in section 2 can be easily generalized and applied to the systems containing any number of particles.
According to the PSS, N-body hyperspherical functions corresponding to the representation of the N-particle permutation group . N S can be obtained by finding parentage coefficients and constructing linear combinations of the N-particle functions corresponding to the irreducible representations of N-1 particle permutation group N-body hyperspherical functions and the complete sets of the recurrence relations for the KRC of these functions with arbitrary quantum numbers were obtained in [18]. According to the PSS we need to construct N-body functions symmetrized with respect to N-1 particles first and then calculate parentage coefficients acting on these functions. Formula (2) can be easily generalized for N-body systems  Relationships similar to (10) and (14)

Conclusion
The use of the parentage scheme of symmetrization allows developing rather simple construction scheme for N-body symmetrized hyperspherical basis in the ((N-1)+1) configuration. This kind of basis may be used when considering different problems in molecular, atomic, nuclear and particle physics. The symmetrized N-body hyperspherical functions corresponding to the representations of the N particle permutation group . N S are constructed by the use of the parentage coefficients acting on the N-particle functions corresponding to the irreducible representation of the N-1 particle permutation group