1. Constructing Wave Functions for Few-Body Systems in a Hyperspherical Basis Using Parentage Scheme of Symmetrization
Lia Leon Margolin, Ph.D.
Associate Professor of Mathematics
Marymount Manhattan Collage, New York. NY

2. Problem statement
Investigating few-body systems with identical particles in a hyperspherical basis yields the problem of obtaining symmetrized hyperspherical functions from functions with arbitrary quantum numbers.
This article solves the problem of hyperspherical basis symmetrization for four-,five- ,and six- body systems using Parentage Scheme of Symmetrization.
Jibuti R.I., Krupennikova N.B., Chachanidze-Margolin L.L.
Few Body Systems, 4, 151, (1988)
Margolin L.L. Journal of Physics, 343, 1 (2012)
Margolin L.L. EPJ Web of Conferences 66, (2014)

3. Presentation Outline
Hyperspherical Functions (HF) for the systems with different particles
Different Configurations for Four-Five and Six particle systems
Transformation Coefficients for N=3,4,5,6 Body Systems
Transformations of N-Particle Systems with N-1 Identical Particles when N=4,5,6
Parentage Scheme of Symmetrization for N=4,5,and 6 Body Systems
Constructing Young operators acting on N=4,5,and 6 Body HF symmetrized with respect to N-1 particles
Finding Parentage coefficients for N=3,4,5,and 6 Body Systems
Constructing Fully symmetrized HF with specific quantum numbers for N=3,4,5,and 6 Body Systems
Conclusion

4. Abstract
Hyperspherical basis symmetrization for four-,five- and six- body systems using Parentage Scheme of Symmetrization.
Parentage coefficients corresponding to the [4], [31], [22], [211], representations of S_4 groups, [5], [311], [221], [2111], [11111] representations of S_5 groups, and [42] and [51] representations of S_6 groups are obtained,
Young operators, acting on N = 4,5,6 body hyperspherical functions symmetrized with respect to (N-1) particles, are derived. The connection between the transformation coefficients for the identical particle systems and the parentage coefficients is demonstrated and the corresponding formulas are introduced.

5. Four Particle System Configurations
(3+1) Configuration (2+2) Configuration
Jibuti R.I., Krupennikova N.B., Chachanidze-Margolin L.L.
Few Body Systems, 4, 151, (1988)
Margolin L.L. Journal of Physics, 343, 1 (2012)
10/12/2019

6. (3+1) Configuration for
10/12/2019

7. Transformation from Jacobi to Four Body Hyperspherical Coordinates

8. Four Particle HF in Nine Dimensional Space of Jacobi Vectors
4 particle HF forms complete set of the basic orthonormal functions and presents aigenvalues of the square of 9 dimensional angular momentum where k3 is operator of six dimentionla angular momentum, and capital omega isset of five angular momentum alfa, x and y, l(x), l(y) l(z) are orbital momentums corresponsing to jacobi coordinates, k4 is generalizes angular momentum in 9 dimensional space, k3 is 6 dimensional space,

9. Four Body HF with L=l12 +l3

10. Transformations of four-body hyperspherical functions

11. Importance of Recurrence Method
The transformations of Hyperspherical functions become sufficiently complex when number of particles in the system increases
For four and more particles kinematic rotations (KR) include both particle permutations and transitions from one configuration to another
Finding (KR) coefficients for four particle systems using general formula is extremely difficult and is practically impossible for the systems with five and more particles
Recurrent method allows to obtain KR coefficients for the systems with any number of particles

12. Five Particle System Configurations
(4+1) Configuration (2+2+1) Configuration
(3+2) Configuration
Krupennikova N.B., Chachanidze-Margolin L.L.
Proc. X –Europ. Conference on Few-Body Physics (1990)
Margolin, L.L. EPJ Web of Conferences 66, (2014)
10/12/2019

13. Five Body Hyperspherical Basis

14. Five Body Kinematic Rotations

15. (5+1) Configuration for Six Body Systems𝑥
𝑥 𝑦 𝑧 𝑤 𝑢 = 𝑎 11 𝑎 12 𝑎 21 𝑎 𝑎 13 𝑎 14 𝑎 23 𝑎 𝑎 31 𝑎 32 𝑎 41 𝑎 𝑎 33 𝑎 34 𝑎 43 𝑎 𝑎 51 𝑎 𝑎 53 𝑎 𝑎 15 𝑎 𝑎 35 𝑎 𝑎 𝑥 ′ 𝑦 ′ 𝑧 ′ 𝑤 ′ 𝑢 ′ 𝑦 𝑧 𝑤 𝑢

16. Six Body Hyperspherical Functions
Ψ 𝑘 3 𝑘 4 𝑘 5 𝑘 6 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑙 123 𝑙 4 𝑙 𝑙 5 𝐿 Ω = 𝒩 𝑘 6 +9/2 𝑘 5 +9/2, 𝑙 sin 𝛿 𝑘 cos 𝛿 𝑙 5 𝑃 𝑠 𝑘 5 +5, 𝑙 5 +1/2 cos 2𝛿 ⋅
⋅ Ψ 𝑘 3 𝑘 4 𝑘 5 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑙 123 𝑙 4 𝜔 𝒴 𝑙 𝑢 𝑀 𝐿
𝑥 , 𝑦 , 𝑧 , 𝑤 , 𝑢 → 𝜌 2 = 𝑥 2 + 𝑦 2 + 𝑧 2 + 𝑤 2 + 𝑢 2 Ω= 𝛼,𝛽, 𝑥 , 𝑦 , 𝑧 , 𝑤 , 𝑢
𝑥=𝜌 cos 𝛼 sin 𝛽 sin 𝛾 sin⁡𝛿
𝑦=𝜌 sin 𝛼 sin 𝛽 sin 𝛾 𝑠𝑖𝑛𝛿
𝑧=𝜌 cos 𝛽 sin 𝛾 𝑠𝑖𝑛𝛿
w=𝜌 cos 𝛾 𝑠𝑖𝑛𝛿
𝑢=𝜌𝑐𝑜𝑠𝛿

17. Four body Hyperspherical Functions symmetrized with respect to three identical particles
Where are three body symmetrization coefficients, are angular momenta corresponding to x, y, z four body systems Jacobi coordinates for either (2+2) or (3+1) configurations [10], and K are four and three particle hypermomenta, L is a total angular momentum, , and are Young diagrams for the four and three particle systems correspondingly ( is obtained from by means of the removal of a cell, corresponding to the fourth particle), denotes the rows of the representation and is the representation number with given K and L.
Transformation coefficients of the functions under particle permutations 1 2 4 3 1 2 3
Margolin L.L. Journal of Physics, 343, 1 (2012)

18. Transformation matrix for four particle
systems with three identical particles

19. Transformations of Four Body HF with three identical particles
Margolin L.L. Journal of Physics, 343, 1 (2012)

20. Transformation Coefficients of four-body HF with three identical particles
Margolin, L.L. EPJ Web of Conferences 66, (2014

23. Parentage scheme of Symmetrization.
N=3 we have [3],[21] and [111]
N=4 we have [4],[22],[31],[211]
N=5 we have [5],[41],[221],[311],[2111],[11111]
N=6 we have [6],[51],[42],[411],[2211],[3111],[111111]
Young operators acting on N-particle functions corresponding to the irreducible representation of the N-1 particle permutation group . Where T are the elements of permutation matrix related with the representation of Sn-1 . These operators identify contributions of the in N-particle symmetrized functions.
Where are the elements of permutation matrix related with the representation of . These operators identify contributions of the in N-particle symmetrized functions. For the five particle systems we have the following representations [5], [11111], [2111], [311], and [221]. Each of these representations contain representations of the subgroup For example, [2111] contains two representations [1111] and [211] of the subgroup . Below are presented some examples of the actions of the Young operators (5) on the five body hyperspherical functions symmetrized with respect to four particles:

24. Young Operators of Group S3 1  2  3  1  2  3  1  3  2  1  2  3
Whrn constructing Yeung operators its better to use simplest representation of Sn-1 inside group Sn  1  3  2  1  2  3

25. Young Operators of Group S4 1  2  3 4  1  2  3  4  1  3  2  4
These operators identify contributions of the in N-particle symmetrized functions. For the five particle systems we have the following representations [5], [11111], [2111], [311], and [221]. Each of these representations contain representations of the subgroup For example, [2111] contains two representations [1111] and [211] of the subgroup . Below are presented some examples of the actions of the Young operators (5) on the five body hyperspherical functions symmetrized with respect to four particles:
It is remarkable that formula (6) contains only permutations involving the fifth particle. As the functions under consideration are symmetrized relatively to four particles, the matrices , and may be in turn expressed by the element of the matrix:
(7)  1  2  3  4  1  3  2  4  1  4  2  3

26. Young Operators of Group S5 1 2 3 4 5  1  2  3 5  1  3  2  4 5  1  4  2  3 5  1  5  2  3 4

27. Young Operators of Group S51 2 3 4 5 1 2 3 5 4  1  3  2  4  5  1  3  2  5  4  1  4  2  5  3  1 2 3 4 5  1  2  3 4 5

28. Young Operators of Group S5 (cont.) 1  2  3  4 5  1  3  2  4 5  1  3  2  4 5  1  5  2  3 4

29. Young Operators of Group S5 (cont.) 1  2  3  4  5
𝜔 Ψ = 𝑃 15 + 𝑃 25 −2 𝑃 35 Ψ 𝑃 15 − 𝑃 25 Ψ
𝜔 Ψ = 𝑃 45 − 𝑃 15 − 𝑃 25 − 𝑃 35 Ψ 𝑃 15 − 𝑃 25 Ψ 𝜔 Ψ = 𝑃 15 + 𝑃 𝑃 35 + 𝑃 Ψ − 𝑃 15 − 𝑃 25 Ψ − 𝑃 25 − 𝑃 15 Ψ  1  2  4  3  5  1  2  5  3  4
𝜔 Ψ = 𝑃 45 − 𝑃 15 − 𝑃 25 − 𝑃 35 Ψ 𝑃 35 − 𝑃 15 − 𝑃 25 Ψ
𝜔 Ψ = 𝑃 15 + 𝑃 𝑃 35 + 𝑃 45 Ψ 𝑃 25 − 𝑃 15 Ψ − 𝑃 15 + 𝑃 25 −2 𝑃 35 Ψ 𝑃 15 + 𝑃 𝑃 35 + 𝑃 45 Ψ 𝑃 25 − 𝑃 15 Ψ − 𝑃 15 + 𝑃 25 −2 𝑃 35 Ψ
𝜔 Ψ = 𝑃 15 + 𝑃 25 + 𝑃 𝑃 Ψ 𝑃 15 − 𝑃 25 Ψ − 𝑃 15 + 𝑃 25 −2 𝑃 35 Ψ 𝑃 15 + 𝑃 25 + 𝑃 𝑃 Ψ 𝑃 15 − 𝑃 25 Ψ − 𝑃 15 + 𝑃 25 −2 𝑃 35 Ψ  1  2  4  5  1  4  5  2  3

30. Young Operators of Group S5 (cont.)
It is remarkable that formula (6) contains only permutations involving the fifth particle. As the functions under consideration are symmetrized relatively to four particles, the matrices , and may be in turn expressed by the element of the matrix:

31. Young Operators Acting on Six body Functions
𝜔 5 𝑚 Ψ 5 =𝒜 𝑖 Γ 𝑚5 𝑃 𝑖6 𝑃 𝑖6 Ψ^ 5 , 𝑖=1,…,5
𝜔 Ψ 5 =− Ψ │ Ψ Ψ
𝜔 Ψ 5 =− Ψ │ Ψ Ψ
𝜔 Ψ 5 =− Ψ │ Ψ Ψ
𝜔 Ψ 5 =− Ψ │ Ψ Ψ
𝜔 Ψ 5 = − Ψ │ Ψ Ψ

32. Young Operators of Six Body Systems:
Young Operators of Six Body Systems
𝜔 Ψ 41 =− Ψ │ Ψ Ψ
𝜔 Ψ 41 =− Ψ │ Ψ Ψ
𝜔 Ψ 41 = Ψ − Ψ │ Ψ Ψ │ Ψ
𝜔 Ψ 41 =− Ψ │ Ψ Ψ
𝜔 Ψ 41 =− Ψ │ Ψ Ψ
𝜔 Ψ 41 = Ψ − Ψ │ Ψ Ψ │ Ψ
𝜔 Ψ 41 =− Ψ │ Ψ Ψ
𝜔 Ψ 41 = Ψ − Ψ │ Ψ Ψ │ Ψ
𝜔 Ψ 41 = Ψ − Ψ │ Ψ Ψ │ Ψ

33. Parentage Scheme of Symmentrization
B are 4 particle parentage coefficients alfa=l12,l3,K3, nm numbers [f] that corresponds to [f\bar] that can be obtained by removal of cell corresponding to 4th particle

34. Parentage Coefficients for Three Body Systems

35. Parentage Coefficients for Four Body Systems

36. Parentage Coefficients for Five Body Systems

37. Parentage Coefficients for six body HF
Φ 𝑘 6 𝐿 𝑓 𝑚 𝜈 𝜔 = 𝛼𝜈 𝐵 𝑘 6 𝐿 𝑓 𝜈 𝑓 𝑚 𝛼 Φ 𝑘 6 𝐿 𝑓 𝑚 𝑛 𝑚 𝛼 (𝜔)
where 𝐵 𝑘 6 𝐿 𝑓 𝜈 𝑓 𝛼 𝑃𝑎𝑟𝑒𝑛𝑡𝑎𝑔𝑒 coefficients
Φ 𝑘 6 𝐿 𝑓 𝑛 𝛼 𝜔 = 𝑓 𝜈 𝐵 𝑘 6 𝐿 𝑓 𝜈 𝑓 𝛼 Φ 𝑘 6 𝐿 𝑓 𝑚 𝛼 𝜔
𝜈 𝐵 𝑘 6 𝐿 42 𝜈 41 𝛼 ⋅ 𝐵 𝑘 6 𝐿 𝛼 ′ = 3 8 𝛿 𝛼𝛼 ′ − 𝛼 𝛼 ′
𝛼 𝛼 ′ 56 ;
𝜈 𝐵 𝑘 6 𝐿 42 𝜈 41 𝛼 ⋅ 𝐵 𝑘 6 𝐿 𝛼 ′ =− 𝛼 𝛼 ′ 56 ;
𝜈 𝐵 𝑘 6 𝐿 51 𝜈 5 𝛼 ⋅ 𝐵 𝑘 6 𝐿 51 𝜈 5 𝛼 ′ = 5 6 𝛿 𝛼𝛼 ′ − 𝛼 5 𝛼 ′ 56 ;
𝜈 𝐵 𝑘 6 𝐿 51 𝜈 5 𝛼 ⋅ 𝐵 𝑘 6 𝐿 51 𝜈 𝛼 ′ =− 𝛼 41 𝛼 ′ 𝛼= 𝜈 , 𝑙 1234 , 𝑙 5 , 𝑘 5 .

38. Constructing Four Body Symmetrized Hyperspherical Basis
Ψ 𝑘 4 𝐿 𝑓 𝑚 𝜈 = 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑘 3 𝐵 𝑘 4 𝐿 𝑓 𝑚 𝜈 𝑓 𝑚 𝜈 𝑙 12 𝑙 3 𝑘 3 𝐶 𝑘 3 𝑙 𝑓 𝑚 𝑙 1 𝑙 2 Ψ 𝑘 3 𝑘 4 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝐿 Ω
= 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑘 3 𝐶 𝑘 4 𝐿 𝑓 𝑚𝜈 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑘 3 Ψ 𝑘 4 𝑘 3 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝐿 Ω
𝐶 𝑘 4 𝐿 𝑓 𝑚 𝜈 𝑙 1 𝑙 2 𝑙 12 𝑙 3 𝑘 3 = 𝜈 𝐵 𝑘 4 𝐿 𝑓 𝑚𝜈 𝑓 𝑚 𝜈 𝑙 12 𝑙 3 𝑘 3 𝐶 𝑘 4 𝐿 𝑓 𝑚 𝜈 𝑙 1 𝑙 2

39. Four Body Symmetrization Coefficients

40. 𝑃𝑎𝑟𝑒𝑛𝑡𝑎𝑔𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝐵 𝑘 4 𝐿 𝑓 𝑚𝜈 𝑓 𝑚 𝜈 𝑙 12 𝑙 3 𝑘 3 𝐰𝐢𝐭𝐡 𝐊𝟒=𝟒, 𝐋=𝟎
𝑃𝑎𝑟𝑒𝑛𝑡𝑎𝑔𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝐵 𝑘 4 𝐿 𝑓 𝑚𝜈 𝑓 𝑚 𝜈 𝑙 12 𝑙 3 𝑘 3 𝐰𝐢𝐭𝐡 𝐊𝟒=𝟒, 𝐋=𝟎
𝑙 12 𝑙 3 𝑘 3
000
002
222
111
113
004
𝑓 𝜈 𝑓
4 1 [3]
26 /9 /2
− /3 /6
1/ 2
1/ 3
1/ 6
55 /9
− /5
/15
− /15
7 9 5 /5
− /5
− /15
/15
2/5
−1/ 15
−4/ 30
− /10
4 3 5 /5
− /5
2/15 1
211 [21] /6 −
13 /6
1 6
− 2 /3
22 1 [21]
− /3 /5
− /5
4/ 30
1/ 15

41. Symmetrized three and four Body HF N=3:
𝑘 3 =𝐿=1
Ψ 𝑘 3 = = Ψ 𝑘 3 =1 01 ; Ψ 𝑘 3 = = Ψ 𝑘 3 =1 10 ; Ψ 𝑘 3 =1 10 ; Ψ 𝑘 3 = = Ψ 𝑘 3 =1 10 ; Ψ 𝑘 3 = =− Ψ 𝑘 3 =1 01
𝑘 3 =2;𝐿=0
Ψ 𝑘 3 = = Ψ 𝑘 3 =2 00 ; Ψ 𝑘 3 = =− Ψ 𝑘 3 =2 11 ; Ψ 𝑘 3 = =− Ψ 𝑘 3 =2 11 ; Ψ 𝑘 3 = =− Ψ 𝑘 3 =2 ∞
𝑘 4 =2;𝐿=0
Ψ 𝑘 4 = 𝑖 = Ψ 𝑘 3 = 𝑘 4 = 𝑖 𝑙 12 = 𝑙 3 = Ψ 𝑘 3 =1 𝑘 4 = 𝑖 𝑙 12 = 𝑙 3 =1 𝑖=1, 2
Ψ 𝑘 4 = = Ψ 𝑘 3 = 𝑘 4 =2 𝑙 𝑖 = Ψ 𝑘 3 =1 𝑘 4 =2 𝑙 2 = 𝑙 3 =1
Ψ 𝑘 4 = =− Ψ 𝑘 3 = 𝑘 4 =2 𝑙 1 = 𝑙 2 = Ψ 𝑘 3 =1 𝑘 4 =2 𝑙 1 = 𝑙 3 =1
Ψ 𝑘 4 = 𝑖 =− Ψ 𝑘 3 = 𝑘 4 = 𝑖 Ψ 𝑘 3 =1 𝑘 4 = 𝑖 𝑖=1, 2
Ψ 𝑘 4 = = Ψ 𝑘 3 = 𝑘 4 =2 𝑙 1 = 𝑙 2 = Ψ 𝑘 3 =1 𝑘 4 =2 𝑙 1 = 𝑙 3 =1
Ψ 𝑘 4 = = Ψ 𝑘 3 = 𝑘 4 =2 𝑙 𝑖 =0 − Ψ 𝑘 3 =1 𝑘 4 =2 𝑙 2 = 𝑙 3 =1
Ψ 𝑘 4 = = Ψ 𝑘 3 =0 𝑘 4 =2 3 𝑙 12 = 𝑙 3 =0 = Ψ 𝑘 3 =0 𝑘 4 =2 𝑙 𝑖 =0
N=4:
𝑘 4 =𝐿=1
Ψ 𝑘 4 = = Ψ 𝑘 3 = 𝑘 4 = 𝑙 12 =1 𝑙 3 =0 = Ψ 𝑘 3 = 𝑘 4 =1 𝑙 1 =1
Ψ 𝑘 4 = = Ψ 𝑘 3 = 𝑘 4 = 𝑙 12 =1 𝑙 3 =0 =− Ψ 𝑘 3 = 𝑘 4 =1 𝑙 2 =1
Ψ 𝑘 4 = = Ψ 𝑘 3 =0 𝑘 4 = 𝑙 12 =0 𝑙 3 =1 = Ψ 𝑘 3 =0 𝑘 4 =1 𝑙 3 =1

42. Symmetrized Five Body HF
𝑘 5 =𝐿=1
Φ = Φ 𝑘 5 =1 𝑙 1 =1 ; Φ =− Φ 𝑘 5 =1 𝑙 2 =1 ; Φ = Φ 𝑘 5 =1 𝑙 3 =1 ; Φ =− Φ 𝑘 5 =1 𝑙 4 =1
𝑘 5 =2, 𝐿=0
Φ =− Φ 𝑘 2 =2 𝑙 1 = 𝑙 2 =1 − Φ 𝑘 5 =2 𝑙 1 = 𝑙 3 =1 − Φ 𝑘 5 =2 𝑙 1 = 𝑙 4 =1 ;
Φ =− Φ 𝑘 3 = 𝑘 4 = 𝑘 5 =2 𝑙 𝑖 = Φ 𝑘 5 =2 𝑙 2 = 𝑙 3 = Φ 𝑘 5 =2 𝑙 2 = 𝑙 4 =1 ;
Φ =− Φ 𝑘 4 = 𝑘 5 =2 𝑙 𝑖 =0 − Φ 𝑘 5 =2 𝑙 3 = 𝑙 4 =1 ;
Φ = Φ 𝑘 5 =2 𝑙 𝑖 =0 ;

43. Symmetrized Six Body HF
𝑘 5 =𝐿=1,
Φ =− Φ 𝑘 6 =2 𝑙 1 = 𝑙 2 = Φ 𝑘 6 =2 𝑙 1 = 𝑙 3 =1 ;
Φ = 1 3 Φ 𝑘 6 =2 𝑙 1 = 𝑙 2 = Φ 𝑘 6 =2 𝑙 1 = 𝑙 3 =1 − Φ 𝑘 6 =2 𝑙 1 = 𝑙 4 =1 ;
Φ = Φ 𝑘 6 =2 𝑙 1 = 𝑙 2 = Φ 𝑘 6 =2 𝑙 1 = 𝑙 3 =1 − Φ 𝑘 6 =2 𝑙 1 = 𝑙 4 =1
Φ 𝑘 6 =2 𝑙 1 = 𝑙 5 =1 ;
Φ =− Φ 𝑘 3 = 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 =0 − Φ 𝑘 6 =2 𝑙 2 = 𝑙 3 =1 ;
Φ = 1 3 Φ 𝑘 3 = 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 =0 − Φ 𝑘 6 =2 𝑙 2 = 𝑙 3 = Φ 𝑘 6 =2 𝑙 2 = 𝑙 4 =1 ;
Φ =− Φ 𝑘 3 = 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 = Φ 𝑘 6 =2 𝑙 2 = 𝑙 3 =1
Φ 𝑘 6 =2 𝑙 2 = 𝑙 4 =1 − Φ 𝑘 6 =2 𝑙 2 = 𝑙 5 =1 ;
Φ = Φ 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 =0 − Φ 𝑘 6 =2 𝑙 3 = 𝑙 4 =1 ;
Φ = Φ 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 = Φ 𝑘 6 =2 𝑙 3 = 𝑙 4 = Φ 𝑘 6 =2 𝑙 3 = 𝑙 5 =1 ;
Φ = Φ 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 =0 − Φ 𝑘 6 =2 𝑙 4 = 𝑙 5 =1 ;
Symmetrized Six Body HF
Φ =− Φ 𝑘 6 =2 𝑙 1 = 𝑙 2 =1 − 1 2 Φ 𝑘 6 =2 𝑙 1 = 𝑙 3 =1 − Φ 𝑘 6 =2 𝑙 1 = 𝑙 4 =1
− Φ 𝑘 6 =2 𝑙 1 = 𝑙 5 =1 ;
Φ =− Φ 𝑘 3 = 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 = Φ 𝑘 6 =2 𝑙 2 = 𝑙 3 = Φ 𝑘 6 =2 𝑙 2 = 𝑙 4 =1
Φ 𝑘 6 =2 𝑙 2 = 𝑙 5 =1 ;
Φ =− Φ 𝑘 4 = 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 =0 − Φ 𝑘 6 =2 𝑙 3 = 𝑙 4 =1 − Φ 𝑘 6 =2 𝑙 3 = 𝑙 5 =1 ;
Φ = Φ 𝑘 5 = 𝑘 6 =2 𝑙 𝑖 = Φ 𝑘 6 =2 𝑙 4 = 𝑙 5 =1 ;
Φ = Φ 𝑘 6 =2 𝑙 𝑖 =0 .
𝑘 6 =𝐿=1:
𝐵 =1; 𝐵 =−1;
Φ = Φ 𝑘 6 =1 𝑙 1 =1 ; Φ =− Φ 𝑘 6 =1 𝑙 2 =1 ; Φ = Φ 𝑘 6 =1 𝑙 3 =1 ;
Φ =− Φ 𝑘 6 =1 𝑙 4 =1 ; Φ = Φ 𝑘 6 =1 𝑙 5 =1

44. Conclusions
Recurrence method allows to find unitary transformation coefficients for the N particles systems of various types.
According to the PSS, N-body hyperspherical functions corresponding to the representation of the N-particle permutation group Sn 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 Sn-1.
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.

45. Talks and Publications
L Margolin EPJ Web of Conferences 66, (2014)
L.L. Margolin, Sh. Tsiklauri, “Kinematic rotations of N-particle hyperspherical basis,” August 2006, 8pp, e-Print Archive:nucl-th/
ia Leon Margolin 2012 J. Phys.: Conf. Ser
Chachanidze-Margolin L.L, “Development of the Hyperspherical Function Method in Impulse Representation applicable for the Microscopic Investigations of Various Nuclear and Hypernuclear Systems”, Proc. Of the VI-th Joint International AMS-SMM Conference, ”, Houston, TX, May 13-15, 2004
Chachanidze-Margolin L.L., “The Kinematical Rotations of N-particle Hyperspherical basis. Construction of Symmetrized Basic Hyperspherical Functions”, 2003 Spring Eastern Sectional Meeting of American Mathematical Society (AMS), New York, New York, April 12-14

46. Kinematic Rotations of N-particle Hyperspherical Basis at
Kinematic Rotations of N-particle Hyperspherical Basis at
K_{N}=2, L_{N}=0.
(1)

47. Three and Four Body Systems

48. Orthonormal Hyperspherical basis