Geodesic
Classical $6j$-symbols of finite-dimensional representations of the algebra $\mathfrak{gl}_3$
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 3-19 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We find anЁexplicit formula for anЁarbitrary $6j$-symbol of finite-dimensional irreducible representations of the Lie algebra $\mathfrak{gl}_3$. It is given by the result of substituting $\pm 1$s in a hypergeometric-type series similar to the $\Gamma$-series, which is the simplest several-variate hypergeometric series. We present necessary conditions for the $6j$-symbol to be nonzero.
Keywords: $6j$-symbols, hypergeometric functions. 
@article{TMF_2023_216_1_a0,
     author = {D. V. Artamonov},
     title = {Classical $6j$-symbols of finite-dimensional representations of the~algebra $\mathfrak{gl}_3$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--19},
     year = {2023},
     volume = {216},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a0/}
}
TY  - JOUR
AU  - D. V. Artamonov
TI  - Classical $6j$-symbols of finite-dimensional representations of the algebra $\mathfrak{gl}_3$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 3
EP  - 19
VL  - 216
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a0/
LA  - ru
ID  - TMF_2023_216_1_a0
ER  - 
%0 Journal Article
%A D. V. Artamonov
%T Classical $6j$-symbols of finite-dimensional representations of the algebra $\mathfrak{gl}_3$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 3-19
%V 216
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a0/
%G ru
%F TMF_2023_216_1_a0
D. V. Artamonov. Classical $6j$-symbols of finite-dimensional representations of the algebra $\mathfrak{gl}_3$. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/TMF_2023_216_1_a0/

[1] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs, 205, AMS, Providence, RI, 2015 | DOI | MR

[2] G. Racah, “Theory of complex spectra. II”, Phys. Rev., 62:9–10 (1942), 438–462 | DOI

[3] E. Vigner, Teoriya grupp i ee prilozheniya k kvantovomekhanicheskoi teorii atomnykh spektrov, IL, M., 1961 | MR | Zbl

[4] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 3, Kvantovaya mekhanika (nerelyativistskaya teoriya), Fizmatlit, M., 2004 | MR

[5] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Kvantovaya teoriya uglovogo momenta, Nauka, L., 1975 | MR | Zbl

[6] L. C. Biedenharn, J. D. Louck, Angular momentum in quantum mechanics, Encyclopedia of Mathematics and its Applications, 8, ed. G.-C. Rota, Addison–Wesley, Reading, MA, 1981 | MR

[7] N. Ja. Vilenkin, A. U. Klimyk, Representation of Lie Groups and Special Functions, v. 1, Mathematics and its Applications (Soviet Series), 72, Simplest Lie Groups, Special Functions and Integral Transforms, Kluwer, Dordrecht, 1991 | DOI | MR

[8] S. E. Derkachev, V. P. Spiridonov, “O $6j$-simvolakh dlya gruppy $SL(2,\mathbb{C})$”, TMF, 198:1 (2019), 32–53 | DOI | DOI | MR

[9] S. E. Derkachev, A. V. Ivanov, “Koeffitsienty Raka dlya gruppy $\mathrm{SL}(2,\mathbb{R})$”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 28, Zap. nauchn. sem. POMI, 509, POMI, SPb, 2021, 99–112 | MR

[10] C. Rebbi, R. Slansky, “Crossing matrices for $SU(2)$ and $SU(3)$”, Rev. Mod. Phys., 42:1 (1970), 68–86 | DOI | MR

[11] P. Arnold, “Landau–Pomeranchuk–Migdal effect in sequential bremsstrahlung: large-$N$ QCD to $N=3$ via the $SU(N)$ analog of Wigner $6$-$j$ symbols”, Phys. Rev. D, 100:3 (2019), 034030, 17 pp. | DOI | MR

[12] A. V. Sleptsov, Simmetrii kvantovykh invariantov uzlov i kvantovykh $6j$-simvolov, Dis. $\ldots$ doktora fiz.-matem. nauk, ITEF, M., 2022

[13] P. H. Butler, B. G. Wybourne, “Calculation of $j$ and $jm$ symbols for arbitrary compact groups. I. Methodology”, Int. J. Quantum Chem., 10:4 (1976), 581–598 | DOI

[14] K. T. Hecht, “A simple class of $U(N)$ Racah coefficients and their application”, Comm. Math. Phys., 41:2 (1975), 135–156 | DOI | MR

[15] R. A. Gustafson, “A Whipple's transformation for hypergeometric series in $U(N)$ and multivariable hypergeometric orthogonal polynomials”, SIAM J. Math. Anal., 18:2 (1987), 495–530 | DOI | MR

[16] M. K. F. Wong, “On the multiplicity-free Wigner and Racah coefficients of $U(n)$”, J. Math. Phys., 20:12 (1979), 2391–2397 | DOI | MR

[17] J. D. Louck, L. C. Biedenharn, “Canonical adjoit tensor operators in $U(n)$”, J. Math. Phys., 11:8 (1970), 2368–2411 | DOI | MR

[18] L. C. Biedenharn, J. D. Louck, E. Chacón, M. Ciftan, “On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in $U(3)$”, J. Math. Phys., 13:12 (1972), 1957–1984 | DOI | MR

[19] A. Mironov, A. Morozov, A. Sleptsov, “On $6j$-symbols for symmetric representations of $U_q(\mathfrak{su}_N)$”, Pisma v ZhETF, 106:10 (2017), 607–608 | DOI | DOI

[20] V. Alekseev, A. Morozov, A. Sleptsov, “Multiplicity-free $U_q(SU(n))$ $6$-$j$ symbols: Relations, asymptotics, symmetries”, Nucl. Phys. B., 960 (2020), 115164, 33 pp. | DOI | MR

[21] D. V. Artamonov, “Formuly vychisleniya $3j$-simvolov dlya predstavlenii algebry Li $\mathfrak{gl}_3$ v bazise Gelfanda–Tsetlina”, Sib. matem. zhurn., 63:4 (2022), 717–735 | DOI | MR

[22] G. E. Baid, L. C. Biedenharn, “On the representations of semisimple Lie groups. II”, J. Math. Phys., 4:12 (1963), 1449–1466 | DOI | MR

[23] D. V. Artamonov, “Koeffitsienty Klebsha–Gordana dlya $\mathfrak{gl}_3$ i gipergeometricheskie funktsii”, Algebra i analiz, 33:1 (2021), 1–29, arXiv: 2101.01049 | DOI | MR

[24] I. M. Gelfand, M. I. Graev, V. S. Retakh, “Obschie gipergeometricheskie sistemy uravnenii i ryady gipergeometricheskogo tipa”, UMN, 47:4 (1992), 3–82 | DOI | MR | Zbl

[25] D. V. Artamonov, “Antisymmetrization of the Gel'fand–Kapranov–Zelevinskij systems”, J. Math. Sci., 255:5 (2021), 535–542 | DOI | MR

[26] D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, MTsNMO, M., 2007 | MR | Zbl