Language：English   Publishing type：Research paper (scientific journal)   Publisher：INDEPENDENT UNIV MOSCOW-IUM  

We present a fourfold series expansion representing the Askey-Wilson polynomials. To obtain the result, a sequential use is made of several summation and transformation formulas for the basic hypergeometric series, including the Verma's q-extension of the Field and Wimp expansion, Andrews terminating q-analogue of Watson's F-3(2) sum, Singh's quadratic transformation. As an application, we present an explicit formula for the Koornwinder polynomial of type BC,, (n is an element of Z(>0)) with one row diagram. When the parameters are specialized, we recover Lassalle's formula for Macdonald polynomials of type B-n, C-n and D-n with one row diagram, thereby proving his conjectures.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：NATL ACAD SCI UKRAINE, INST MATH  

We present explicit formulas for the Macdonald polynomials of types C-n and D-n in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced W-12(11) series. We remark that the correlation functions of the deformed W algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.

DOI： 10.3842/SIGMA.2015.100

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

We give the explicit forms of the crystal bases B(infinity) for the quantum affine algebras of types A(2n-1)((2)), A(2n)((2)), B-n((1)), C-n((1)), D-n((1)), and D-n+1((2)) by using the method of polyhedral realizations of crystal bases. (C) 2013 AIP Publishing LLC.

DOI： 10.1063/1.4805584

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Language：English   Publisher：Kyoto University  

CiNii Books

Other Link： http://hdl.handle.net/2433/171419

Language：English   Publisher：Kyoto University  

CiNii Books

Other Link： http://hdl.handle.net/2433/141521

Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

We introduce a unital associative algebra A associated with degenerate CP1. We show that A is a commutative algebra and whose Poincare series is given by the number of partitions. Thereby, we can regard A as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by Shiraishi [Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding-Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard-Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t(1),t(2)) of Okounkov-Pandharipande [e-print arXiv:math-ph/0411210]. (c) 2009 American Institute of Physics. [DOI: 10.1063/1.3192773]

DOI： 10.1063/1.3192773

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We give the generalized Littlewood-Richardson rule for exceptional Lie algebras E6 and F4 using polyhedral realizations of crystal bases.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component B-0(lambda) containing the unit element for arbitrary rank 2 cases. We also present the explicit form of B 0 (lambda) containing the highest (or lowest) weight vector by the polyhedral realization method.

DOI： 10.1080/00927870500542978

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

Polyhedral realization of crystal bases is one of the methods for describing the crystal base B(infinity) of a quantized enveloping algebra explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases B(lambda) of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases B(infinity) and B(lambda) are only given in the case of arbitrary rank 2, of A(n) and of A(n)((1)). So, we will give the polyhedral realizations of crystal bases B(infinity) and B(lambda) for all simple Lie algebras in this paper. (c) 2005 American Institute of Physics.

DOI： 10.1063/1.2121308

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

We describe the crystal bases of modified quantum algebras and their connected component containing "zero vector" by the polyhedral realization method for the type A(n) and A(1)((1)). We also present the explicit form of the unique highest-weight vector in the connected component.

DOI： 10.1081/AGB-200063532

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Language：English  

Language：English  

Language：English  

Language：English