Volume 6 (2006), Number 2. Abstracts

Ph. Caldero and A. Zelevinsky. Laurent Expansions in Cluster Algebras via Quiver Representations [PDF]

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.

Keywords. Cluster algebras, Laurent phenomenon, quiver representations, Kronecker quiver

2000 Mathematics Subject Classification. Primary: 16G20; Secondary: 14M15, 16S99.

V. Ginzburg and T. Schedler. Moyal Quantization and Stable Homology of Necklace Lie Algebras [PDF]

We compute the stable homology of necklace Lie algebras associated with quivers and give a construction of stable homology classes from certain A-categories. Our construction is a generalization of the construction of homology classes of moduli spaces of curves due to M. Kontsevich.

In the second part of the paper we produce a Moyal-type quantization of the symmetric algebra of a necklace Lie algebra. The resulting quantized algebra has natural representations in the usual Moyal quantization of polynomial algebras.

Keywords. Graph complex, Moyal product, stable homology, necklace Lie algebra

2000 Mathematics Subject Classification. Primary: 16R30, 14L40

A. Gnedin and G. Olshanski. The Boundary of the Eulerian Number Triangle [PDF]

The Eulerian triangle is a classical array of combinatorial numbers defined by a linear recursion. The associated boundary problem asks one to find all extreme nonnegative solutions to a dual recursion. Exploiting connections with random permutations and Markov chains we show that the boundary is discrete and explicitly identify its elements.

Keywords. Eulerian numbers, extreme boundary, descents

2000 Mathematics Subject Classification. 60J50, 60C05

S. Khoroshkin and M. Nazarov. Yangians and Mickelsson algebras II [PDF]

We study the composition of two functors. The first functor, acting from the category of modules over the Lie algebra glm to the category of modules over the degenerate affine Hecke algebra of GLN, was introduced by Cherednik. The second functor is a skew version of the functor, due to Drinfeld, from the latter category to the category of modules over the Yangian Y(gln). We give a representation-theoretic explanation of a link between intertwining operators on tensor products of Y(gln)-modules and the “extremal cocycle” introduced by Zhelobenko on the Weyl group of glm. We also establish a connection between the composition of two functors and Olshanski's “centralizer construction” of the Yangian Y(gln).

Keywords. Cherednik functor, Drinfeld functor, Zhelobenko cocycle

2000 Mathematics Subject Classification. Primary: 17B35; Secondary: 81R50

A. Kirillov, Jr. McKay Correspondence and Equivariant Sheaves on P1 [PDF]

Let G be a finite subgroup in SU(2), and Q the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of G-equivariant sheaves on P1 and Rep Qh, where h is an orientation of Q, constructing an explicit equivalence of corresponding derived categories.

Keywords. McKay correpondence, equivariant sheaves, quiver representations

2000 Mathematics Subject Classification. 16G20, 14F43, 17B67

A. Molev. Representations of the Twisted Quantized Enveloping Algebra of Type Cn [PDF]

We prove a version of the Poincaré–Birkhoff–Witt theorem for the twisted quantized enveloping algebra U'q(sp2n). This is a subalgebra of Uq(gl2n) and a deformation of the universal enveloping algebra U(sp2n) of the symplectic Lie algebra. We classify finite-dimensional irreducible representations of U'q(sp2n) in terms of their highest weights and show that these representations are deformations of finite-dimensional irreducible representations of sp2n.

Keywords. Quantized enveloping algebra, symplectic Lie algebra, representation

2000 Mathematics Subject Classification. 81R10

A. Okounkov and N. Reshetikhin. The Birth of a Random Matrix [PDF]

We consider the behavior of a random stepped surface near a turning point, that is, a point at which the limit shape is not smooth. When the turning point is a smooth point of the frozen boundary, the resulting point process is identified with the standard Gaussian measure on infinite Hermitian matrices. A different point process appears if the turning point is a cusp of the frozen boundary.

Keywords. Random matrices, random surfaces

2000 Mathematics Subject Classification. 15A52, 60G60

A. Vershik. A New Approach to the Representation Theory of the Symmetric Groups, III: Induced Representations and the Frobenius–Young Correspondence [PDF]

We give a new (inductive) proof of the classical Frobenius–Young correspondence between irreducible complex representations of the symmetric group and Young diagrams, using the new approach, suggested in paper by A. Okounkov and the author, to determining this correspondence. We also give linear relations between Kostka numbers that follow from the decomposition of the restrictions of induced representations to the previous symmetric subgroup. We consider a realization of representations induced from Young subgroups in polylinear forms and describe its relation to Specht modules.

Keywords. Induced represnetations, Young diagram, Frobenius–Young rule, Specht module

2000 Mathematics Subject Classification. 05E05, 81R05

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