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Racah Problems for the Oscillator Algebra, the Lie Algebra \(\mathfrak {sl}_n\), and Multivariate Krawtchouk Polynomials


The oscillator Racah algebra \(\mathcal {R}_n(\mathfrak {h})\) is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra \(\mathfrak {h}\). An embedding of the Lie algebra \(\mathfrak {sl}_{n-1}\) into \(\mathcal {R}_n(\mathfrak {h})\) is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for \(\mathfrak {h}\) and matrix elements of \(\mathfrak {sl}_n\)-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.

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  1. Aguirre, L., Felder, G., Veselov, A.P.: Gaudin subalgebras and stable rational curves. Compos. Math. (N.S.) 22(3), 1057–1071 (2016)

    MATH  Google Scholar 

  2. Baseilhac, P.: Deformed Dolan-Grady relations in quantum integrable models. Nucl. Phys. B 709, 491–521 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Baseilhac, P., Crampe, N.: FRT presentation of classical Askey-Wilson algebras. Lett. Math. Phys. 109, 2187–2207 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Bullock, D., Przytycki, J.H.: Multiplicative structure of Kauffman bracket skein module quantization. Proc. Am. Math. Soc. 128, 923–931 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Cooke, J.: Kaufman Skein Algebras and Quantum Teichmüller Spaces via Factorisation Homology. arXiv:1811.09293 (2018)

  6. Crampe, N., Frappat, L., Vinet, L.: Centralizers of the superalgebra \(osp(1|2)\): the Brauer algebra as a quotient of the Bannai-Ito algebra. J. Phys. A. 52(42), 424001 (2019)

    ADS  MathSciNet  Google Scholar 

  7. Crampe, N., Gaboriaud, J., Vinet, L., Zaimi, M.: Revisiting the Askey–Wilson algebra with the universal R-matrix of \(U_q(\mathfrak{su}(2))\). arXiv:1908.04806 (2019)

  8. Crampe, N., Poulain d’Andecy, L., Vinet, L.: Temperley-Lieb, Brauer and Racah algebras and other centralizers of \(su(2)\). arXiv:1905.06346

  9. Crampe, N., Ragoucy, E., Vinet, L., Zhedanov, A.S.: Truncation of the reflection algebra and the Hahn algebra. J. Phys. A Math. Theor. 52, 35LT01 (2019)

    MathSciNet  Google Scholar 

  10. Crampe, N., Vinet, L., Zaimi, M.: Bannai-Ito algebras and the universal \(R\)-matrix of \(\mathfrak{osp} (1|2)\). J. Phys. A 53(5), 05LT01 (2020)

  11. De Bie, H., De Clerq, H., van de Vijver, W.: The higher rank q-deformed Bannai-Ito and Askey-Wilson algebra. Commun. Math. Phys. 374, 277–316 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  12. De Bie, H., Genest, V.X., Tsujimoto, S., Vinet, L., Zhedanov, A.: The Bannai-Ito algebra and some applications. J. Phys. Conf. Ser. 597, 012001 (2015)

    Google Scholar 

  13. De Bie, H., Genest, V.X., van de Vijver, W., Vinet, L.: A higher rank Racah algebra and the \((\mathbb{Z}_2)^n\) Laplace-Dunkl operator. J. Phys. A Math. Theor. 51, 025203 (2018)

    ADS  MATH  Google Scholar 

  14. De Bie, H., Genest, V.X., Vinet, L.: The \(\mathbb{Z}_n^2\) Dirac-Dunkl operator and a higher rank Bannai-Ito algebra. Adv. Math. 303, 390–414 (2016)

    MathSciNet  MATH  Google Scholar 

  15. De Bie, H., van de Vijver, W.: A discrete realization of the higher rank Racah algebra. Constr. Approx. 52, 1–29 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Diaconis, P., Griffiths, R.: An introduction to multivariate Krawtchouk polynomials and their applications. J. Stat. Plan. Inference 154, 39–53 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Dolan, L., Grady, M.: Conserved charges from self-duality. Phys. Rev. D 25, 1587–1604 (1982)

    ADS  MathSciNet  Google Scholar 

  18. Frappat, L., Gaboriaud, J., Ragoucy, E., Vinet, L.: The dual pair \(( U_q (\mathfrak{su}(1,1), \mathfrak{o}_q^{{1/2}}(2n))\), q-oscillators and Askey-Wilson algebras. arXiv:1908.04277 (2019)

  19. Gaboriaud, J., Vinet, L., Vinet, S., Zhedanov, A.: The Racah algebra as a commutant and Howe duality. J. Phys. A Math. Theor. 51, 50LT01 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Gaboriaud, J., Vinet, L., Vinet, S., Zhedanov, A.: The dual pair \(Pin(2n) \times \mathfrak{osp}(1|2)\), the Dirac equation and the Bannai-Ito algebra. Nucl. Phys. B 937, 226–239 (2018)

    ADS  MATH  Google Scholar 

  21. Genest, V.X., Vinet, L., Zhedanov, A.: Superintegrability in two dimensions and the Racah-Wilson algebra. Lett. Math. Phys. 104, 931–952 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Genest, V.X., Vinet, L., Zhedanov, A.: The Bannai-Ito polynomials as Racah coefficients of the \(sl_{-1}(2)\) algebra. Proc. Am. Math. Soc 142, 1545–1560 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Genest, V.X., Vinet, L., Zhedanov, A.: The equitable Racah algebra from three \(\mathfrak{su}(1,1)\) algebras. J. Phys. A Math. Theor. 47, 025203 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  24. Genest, V.X., Vinet, L., Zhedanov, A.: The Racah algebra and superintegrable models. J. Phys. Conf. Ser. 512, 012011 (2014)

    Google Scholar 

  25. Genest, V.X., Vinet, L., Zhedanov, A.: The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Am. Math. Soc. 144, 5217–5226 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Genest, V.X., Vinet, L., Zhedanov, A.: The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states. J. Phys. A Math. Theor. 46, 505203 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Geronimo, J., Iliev, P.: Bispectrality of multivariable Racah-Wilson polynomials. Constr. Approx. 31, 417–457 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Granovskii, Ya A., Zhedanov, A.S.: Nature of the symmetry group of the \(6j\)-symbol. Sov. Phys. JETP 67, 1982–1985 (1988)

    MathSciNet  Google Scholar 

  29. Granovskii, Ya.A., Zhedanov, A.S.: Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra \(sl_q(2)\). arXiv:hep-th/9304138 (1993)

  30. Griffiths, R.: Orthogonal polynomials on the multinomial distribution. Aust. J. Stat. 13(1), 27–35 (1971)

    MathSciNet  MATH  Google Scholar 

  31. Hoare, M.R., Rahman, M.: A probabilistic origin for a new class of bivariate polynomials. SIGMA 4, 89–106 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Huang, H.-W.: An embedding of the universal Askey-Wilson algebra into \(U_q(\mathfrak{sl}_2)) \otimes U_q(\mathfrak{sl}_2)) \otimes U_q(\mathfrak{sl}_2))\). Nucl. Phys. B 922, 401–434 (2017)

    ADS  MATH  Google Scholar 

  33. Iliev, P.: A Lie theoretic interpretation of multivariate hypergeometric polynomials. Compos. Math. 148, 991–1002 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Iliev, P., Terwilliger, P.: The Rahman polynomials and the Lie algebra \(\mathfrak{sl}_3(\mathbb{C})\). Trans. Am. Math. Soc. 364(8), 4225–4328 (2012)

    MATH  Google Scholar 

  35. Iliev, P., Xu, Y.: Hahn polynomials on polyhedra and quantum integrability. Adv. Math. 364, 107032 (2020)

    MathSciNet  MATH  Google Scholar 

  36. Klimyk, A.U., Vilenkin, N.J.: Representation of Lie Groups and Special Functions. Kluwer Academic, Dordrecht (1991–1993)

  37. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Technical Report 98-17. Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft University of Technology (1998)

  38. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer, Berlin (2010)

    MATH  Google Scholar 

  39. Koornwinder, T.: Krawtchouk polynomials, a unification of two different group theoretic interpretations. SIAM J. Math. Anal. 13(6), 1011–1023 (1982)

    MathSciNet  MATH  Google Scholar 

  40. Koornwinder, T.: The relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. SIGMA 3, 063 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Koornwinder, T.: Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra. SIGMA 4, 052 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Mizukawa, H., Tanaka, H.: \((n + 1, m + 1)\)-hypergeometric functions associated to character algebras. Proc. Am. Math. Soc. 132, 2613–2618 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Post, S., Walter, A.: A higher rank extension of the Askey-Wilson Algebra. arXiv:1705.01860

  44. Rosenblyum, A.V.: Spectral analysis of generators of representations of the group \(U(3)\). Theor. Math. Phys. 73, 1352–1356 (1987)

    MathSciNet  Google Scholar 

  45. Streater, R.F.: The representations of the oscillator group. Commun. Math. Phys. 4, 217 (1967)

    ADS  MathSciNet  MATH  Google Scholar 

  46. Terwilliger, P.: The universal Askey-Wilson algebra and DAHA of type \((\cal{C}^{\vee }_1, \cal{C}_1)\). SIGMA 9, 047 (2013)

    MATH  Google Scholar 

  47. Terwilliger, P., Vidunas, R.: Leonard pairs and the Askey-Wilson relations. J. Algebra Appl. 03, 411–426 (2004)

    MathSciNet  MATH  Google Scholar 

  48. Terwilliger, P., Zitnik, A.: Distance-regular graphs of q-Racah type and the universal Askey-Wilson algebra. J. Comb. Theor. Ser. A 125, 98–112 (2014)

    MathSciNet  MATH  Google Scholar 

  49. Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 32, 2337–2342 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  50. Van der Jeugt, J.: Coupling coefficients for Lie algebra representations and addition formulas for special functions. J. Math. Phys. 38, 2728 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  51. Van der Jeugt, J.: \(3nj\)-coefficients and orthogonal polynomials of hypergeometric type. In: Orthogonal Polynomials and Special Functions, vol. 1817, pp. 25–92 (2003)

  52. Zhedanov, A.S.: “Hidden symmetry” of the Askey-Wilson polynomials. Theor. Math. Phys. 89, 1146–1157 (1991)

    MathSciNet  MATH  Google Scholar 

  53. Zhedanov, A.: \(9j\)-symbols for the oscillator algebra and the Krawtchouk polynomials in two variables. J. Phys. A Math. Gen. 30, 8337 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

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NC is partially supported by Agence National de la Recherche Projet AHA ANR-18-CE40-0001 and is gratefully holding a CRM-Simons professorship. The work of WVDV is supported by the Research Foundation Flanders (FWO) under Grant EOS 30889451, as well as the Fonds Professor Frans Wuytack. WVDV is also grateful for the hospitality offered by him at the CRM during his stay. The research of LV is supported in part by a discovery grant of the Natural Science and Engineering Research Council (NSERC) of Canada.

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Correspondence to Wouter van de Vijver.

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Appendix A: Calculation of overlap coefficients

Appendix A: Calculation of overlap coefficients

Let V be a finite-dimensional representation of \(\mathfrak {sl}_2\) and \(\tilde{.}\) an automorphism of \(\mathfrak {sl}_2\). The element h is a Cartan generator of \(\mathfrak {sl}_2\). Let \(\{ \psi _k \}\) be an eigenbasis of h and \(\{ \phi _s \}\) be an eigenbasis for \(\tilde{h}\). The indices k and s run from 0 to N with \(\dim (V)=N+1\). We are interested in the overlap coefficients \(B_{ks}\) between these bases:

$$\begin{aligned} \phi _s= & {} \sum _{k=0}^N B_{sk}\psi _k.\nonumber \\ h\psi _k= & {} \mu _k\psi _k \qquad \tilde{h}\phi _s=\nu _s\phi _s. \end{aligned}$$

The algebra \(\mathfrak {sl}_2\) has algebra relations \([h,e]=2e\) and \([h,f]=-2f\) with e the raising operator and f the lowering operator on \(\{ \psi _k \}\):

$$\begin{aligned} e\psi _k=e_{kk+1}\psi _{k+1} \quad f \psi _k=f_{kk-1} \psi _{k-1} \end{aligned}$$

and \(\mu _k=\mu _0+2k\). From the algebra relation \([e,f]=h\) it follows that

$$\begin{aligned} f_{kk-1}e_{k-1k}- e_{kk+1}f_{k+1k}=\mu _k. \end{aligned}$$

Let \(A_k:=e_{kk-1}f_{k-1k}\). Then, we have

$$\begin{aligned} A_{k}-A_{k+1}=2k+\mu _0. \end{aligned}$$

From this we find

$$\begin{aligned} A_k=-k(k-1)-\mu _0 k-\Omega \end{aligned}$$

with \(\Omega \in \mathbb {R}\). We express \(\tilde{h}\) as a linear combination of h, e and f.

$$\begin{aligned} \tilde{h}=R_h h +R_e e+ R_f f \end{aligned}$$

with \(R_eR_f+R_h^2=1\). We have set up everything we need to find the overlap coefficients. Let the operator \(\tilde{h}\) act on both sides of equality (38).

$$\begin{aligned} \tilde{h}\psi _s=\sum _{k=0}^N B_{sk}(R_h h +R_e e+ R_f f) \psi _k. \end{aligned}$$

This gives

$$\begin{aligned} \nu _s\phi _s=\sum _{k=0}^N B_{sk}(R_h\mu _k\psi _k+R_e e_{kk+1}\psi _{k+1}+R_f f_{kk-1}\psi _{k-1}). \end{aligned}$$

We expand the left-hand side into the basis \(\psi _k\) and we gather the terms on the right-hand side:

$$\begin{aligned} \sum _{k=0}^N \nu _s B_{sk} \psi _k=\sum _{k=0}^N (B_{sk}R_h\mu _k+B_{sk-1}R_e e_{k-1k}+B_{sk+1}R_f f_{k+1k})\psi _k. \end{aligned}$$

From this we find the recurrence relation

$$\begin{aligned} \nu _s B_{sk}=B_{sk+1}R_f f_{k+1k}+ B_{sk}R_h\mu _k+B_{sk-1}R_e e_{k-1k}. \end{aligned}$$

We want to recognize this recurrence relation as one of the family of orthogonal polynomials. Let

$$\begin{aligned} \tilde{B}_{ks}=\left( \prod _{t=2}^k f_{tt-1}R_f\right) B_{sk} \end{aligned}$$

to find

$$\begin{aligned} \nu _s \tilde{B}_{sk}=\tilde{B}_{sk+1}+R_h\mu _k\tilde{B}_{sk}+R_eR_f e_{k-1k}f_{k-1k}\tilde{B}_{sk-1}. \end{aligned}$$

We write the coefficients as polynomials in \(x=\nu _s\):

$$\begin{aligned} x\tilde{B}_k(x)=\tilde{B}_{k+1}(x)+R_h\mu _k\tilde{B}_k(x)+R_eR_fA_k\tilde{B}_{k-1}(x). \end{aligned}$$

We want to compare this with the recurrence relation of the normalized Krawtchouk polynomials as defined in [38]:

$$\begin{aligned} xp_n(x)=p_{n+1}+(n(1-2r)+rN)p_n(x)+r(1-r)n(N+1-n)p_{n-1}(x) \end{aligned}$$

with \(n=0,1, \dots , N\). Let \(x=\alpha y+\beta \) and introduce \(q_n(y)=p_n(\alpha y+\beta )/\alpha ^n\). The polynomial \(q_n(x)\) satisfies the following recurrence relation:

$$\begin{aligned} yq_n(y)=q_{n+1}+\frac{n(1-2r)+rN-\beta }{\alpha }q_n(y)+\frac{r(1-r)n(N+1-n)}{\alpha ^2}q_{n-1}(y). \end{aligned}$$

We retrieve Eq. (39) if we set

$$\begin{aligned} \alpha =\frac{1}{2}, \quad r=\frac{1-R_h}{2}, \quad \beta =\frac{N}{2}, \quad k=n, \quad \Omega =0, \quad \mu _0=-N. \end{aligned}$$

We explicitly write down the polynomials \(\tilde{B}_k(x)\).

$$\begin{aligned} \tilde{B}_k(x)&=2^kp_k\left( \frac{x+N}{2}\right) \\&=(-N)_k (1-R_h)^k K_k\left( \frac{x+N}{2}; \frac{1-R_h}{2},N\right) \\&=(-N)_k (1-R_h)^k {}_2F_1\left( \begin{array}{c} -k,-\frac{x+N}{2}\\ -N \end{array}\big | \frac{2}{1-R_h}\right) . \end{aligned}$$

The overlap coefficients are the Krawtchouk polynomials \(K_k(x)\) (defined in the last line of the equation above) up to a normalization factor.

$$\begin{aligned} B_{sk}=\left( \prod _{t=2}^k f_{tt-1}R_f\right) (-N)_k (1-R_h)^k K_k\left( \frac{\nu _s+N}{2}; \frac{1-R_h}{2},N\right) . \end{aligned}$$

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Crampé, N., van de Vijver, W. & Vinet, L. Racah Problems for the Oscillator Algebra, the Lie Algebra \(\mathfrak {sl}_n\), and Multivariate Krawtchouk Polynomials. Ann. Henri Poincaré 21, 3939–3971 (2020).

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