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- PHILIPPE CALDERO, ANDREI ZELEVINSKY, Alexander Alexandrovich Kirillov, A. ZELEVINSKY
- 2006

We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.

- Pavel I Etingof, Igor B Frenkel, Alexander A Kirillov
- 1995

Introduction By spherical functions one usually means functions on the double coset space K\G/K, where G is a group and K is a subgroup of G. This is equivalent to considering functions on the homogeneous space G/K left invariant with respect to K. More generally, if V is a fixed irreducible representation of K, for example, finite-dimensional, one can look… (More)

Introduction Recently I.Macdonald deened a family of systems of orthogonal symmetric poly-nomials depending on two parameters q; k which interpolate between Schur's symmetric functions and certain spherical functions on SL(n) over the real and p-adic elds M]. These polynomials are labeled by dominant integral weights of SL(n), and (as was shown by… (More)

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr(Φ 1 (z 1). .. Φ n (z n)q −∂ e h) where Φ i are intertwiners between Verma modules and evaluation modules over an affine Lie algebrâ g, ∂ is the grading operator in… (More)

This is the second part of the paper (the rst part is published in Journal of AMS 9 1135. In the rst part, we deened for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space Hom(L X i X i ; U) is modular invariant. Also, we have shown that in the case of the MTC arising from the… (More)

Introduction. Jack's and Macdonald's polynomials are an important class of symmetric functions associated to root systems. In this paper we define and study an analogue of Jack's and Macdonald's polynomials for affine root systems. Our approach is based on representation theory of affine Lie algebras and quantum affine algebras, and follows the ideas of our… (More)

- Pavel I Etingof, Alexander A Kirillov, M Gelfand, M A Naimark, N Ya Vilenkin
- 1994

Dedicated to I.M.Gel'fand on the occasion of his 80th birthday A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I. and their collaborators (see [V],[VK]). The essence of this approach is the fact that most classical special functions can be obtained as suitable specializations of matrix elements or… (More)

This paper gives a review of Cherednik's results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the q-Jacobi polynomials, and certain… (More)

Classical and quantum family algebras, previously introduced by the author and playing an important role in the theory of semi-smiple Lie algebras and their representations are studied. Basic properties, structure theorems and explicit fomulas are obtained for both types of family algebras in many significant cases. Exact formulas (based on experimental… (More)

In this paper we study some properties of tensor categories that arise in 2-dimensional conformal and 3-dimensional topological quantum field theory – so called modular tensor categories. By definition, these categories are braided tensor categories with duality which are semisimple, have finite number of simple objects and satisfy some non-degeneracy… (More)