The matrix elements of unitary SUq(3) corepresentations, which are analogs of the symmetric powers of the natural representation, are shown to be the bivariate q-Krawtchouk orthogonal polynomials, thus, providing an algebraic interpretation of these polynomials in terms of quantum groups.
REFERENCES
1.
T. H.
Koornwinder
, “Orthogonal polynomials in connection with quantum groups
,” Orthogonal Polynomials
294
, 257
–292
(1990
).2.
M. V.
Tratnik
, “Some multivariable orthogonal polynomials of the Askey tableau-discrete families
,” J. Math. Phys.
32
, 2337
–2342
(1991
).3.
V. X.
Genest
, L.
Vinet
, and A.
Zhedanov
, “The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
,” J. Phys. A: Math. Theor.
46
, 505203
(2013
).4.
P.
Iliev
, “A Lie-theoretic interpretation of multivariate hypergeometric polynomials
,” Compositio Math.
148
, 991
–1002
(2012
); e-print arXiv:1101.1683.5.
P.
Iliev
and P.
Terwilliger
, “The Rahman polynomials and the Lie algebra .
,” Trans. Am. Math. Soc.
364
, 4225
–4238
(2012
).6.
T.
Koornwinder
, “Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials
,” Indagationes Math.
92
, 97
–117
(1989
).7.
A. S.
Zhedanov
, “Q rotations and other Q transformations as unitary nonlinear automorphisms of quantum algebras
,” J. Math. Phys.
34
, 2631
–2647
(1993
).8.
V. X.
Genest
, S.
Post
, L.
Vinet
, G.-F.
Yu
, and A.
Zhedanov
, “q-Rotations and Krawtchouk polynomials
,” Ramanujan J.
40
, 335
–357
(2016
).9.
R.
Floreanini
and L.
Vinet
, “On the quantum group and quantum algebra approach to q-special functions
,” Lett. Math. Phys.
27
, 179
–190
(1993
).10.
G.
Gasper
and M.
Rahman
, “Some systems of multivariable orthogonal q-Racah polynomials
,” Ramanujan J.
13
, 389
–405
(2007
).11.
V. X.
Genest
, S.
Post
, and L.
Vinet
, “An algebraic interpretation of the multivariate q-Krawtchouk polynomials
,” Ramanujan J.
43
, 415
–445
(2017
); e-print arXiv:1508.07770.12.
K.
Bragiel
, “The twisted SU(3) group. Irreducible *-representations of the C*-Algebra C(SμU(3))
,” Lett. Math. Phys.
17
, 37
–44
(1989
).13.
Y. S.
Soibelman
, “Algebra of functions on a compact quantum group and its representations
,” Algebra Analiz
2
(1
), 190
–212
(1990
) [Leningrad. Math. J. 2(1), 161–178 (1991)].14.
H. T.
Koelink
, “On *-representations of the Hopf *-algebra associated with the quantum group Uq(n)
,” Compositio Math.
77
(2
), 199
–231
(1991
).15.
L. I.
Korogodski
and Y. S.
Soibelman
, “Algebras of functions on quantum groups: Part I
,” in Mathematical Surveys and Monographs
(American Mathematical Society
, 1998
), Vol. 56, p. 150
.16.
L. L.
Vaksman
and Y. S.
Soibelman
, “Algebra of functions on the quantum group SU(2)
,” Funct. Anal. Appl.
22
, 170
–181
(1988
).17.
S. L.
Woronowicz
, “Compact matrix pseudogroups
,” Commun. Math. Phys.
111
, 613
–665
(1987
).18.
V.
Chari
and A.
Pressley
, A Guide to Quantum Groups
(Cambridge University Press
, 1994
), p. 651
.19.
M.
Noumi
, H.
Yamada
, and K.
Mimachi
, “Finite dimensional representations of the quantum group GLq(n; C) and the zonal spherical functions on Uq(n − 1)Uq(n)
,” Jpn. J. Math.
19
, 31
–80
(1993
).20.
R.
Koekoek
, P. A.
Lesky
, and R. F.
Swarttouw
, , Springer Monographs in Mathematics Series
(Springer
, 2010
).21.
Part of the normalization of Ref. 20 is here included in the polynomial kn(x; p, N, q) themselves.
22.
G.
Gasper
and M.
Rahman
, Encyclopedia of Mathematics and its Application
, 2nd ed. (Cambridge University Press
, 2004
), Vol. 96, p. 428
.23.
W.
Groenevelt
, “A quantum algebra approach to multivariate Askey-Wilson polynomials
,” e-print arXiv:1809.04327 (2018
).24.
N.
Burroughs
, “Relating the approaches to quantised algebras and quantum groups
,” Commun. Math. Phys.
133
, 91
–117
(1990
).25.
T. H.
Koornwinder
, “The addition formula for little q-Legendre polynomials and the SU(2) quantum group
,” SIAM J. Math. Anal.
22
, 295
–301
(1991
).26.
W.
Groenevelt
, “Coupling coefficients for tensor product representations of quantum SU(2)
,” J. Math. Phys.
55
, 101702
(2014
).© 2019 Author(s).
2019
Author(s)
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