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Quantum Grassmannians and $$q$$-hypergeometric series. (English) Zbl 0782.33015
A new point of view relating quantum groups to $$q$$-special functions is presented. It is motivated by a recent work of Horikawa who showed that the contiguity relations for the general $$q$$-hypergeometric series $${}_ 2\varphi_ 1$$ can be described by the quantized universal enveloping algebra $$U_ q(\text{gl}(4))$$, which in turn was found by investigating $$q$$-analogues of Gelfand’s interpretation of Gauss’ hypergeometric functions by Grassmann manifold Grass(2,4). In this article, as an attempt to give an intrinsic explanation of the above relation, the case of $$\text{Grass}(2,n)$$ is discussed and it is shown that a $$q$$-analogue of Lauricella’s hypergeometric series $$F_ D$$ of $$n-3$$ variables is such. As a consequence, the quantized universal enveloping algebra $$U_ q(\text{gl}(n))$$ is obtained naturally as the algebra describing the contiguity relations of the $$q$$-hypergeometric series.
Reviewer: A.Kaneko (Komaba)

##### MSC:
 33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, $$p$$-adic groups, Hecke algebras, and related topics
##### Keywords:
quantum groups; $$q$$-analogues