195 research outputs found
A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
We identify the Grothendieck group of certain direct sum of singular blocks
of the highest weight category for sl(n) with the n-th tensor power of the
fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by
projective functors and the commuting action of the Temperley-Lieb algebra by
Zuckerman functors. Indecomposable projective functors correspond to Lusztig
canonical basis in U(sl(2)). In the dual realization the n-th tensor power of
the fundamental representation is identified with a direct sum of parabolic
blocks of the highest weight category. Translation across the wall functors act
as generators of the Temperley-Lieb algebra while Zuckerman functors act as
generators of U(sl(2)).Comment: 31 pages, 11 figure
Path representation of maximal parabolic Kazhdan-Lusztig polynomials
We provide simple rules for the computation of Kazhdan--Lusztig polynomials
in the maximal parabolic case. They are obtained by filling regions delimited
by paths with "Dyck strips" obeying certain rules. We compare our results with
those of Lascoux and Sch\"utzenberger.Comment: v3: fixed proof of lemma
Towers of solutions of qKZ equations and their applications to loop models
Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form
a consistent system of linear -difference equations for -valued
meromorphic functions on a complex -torus, with a module over the
GL-type extended affine Hecke algebra . The family
of extended affine Hecke algebras forms a tower of
algebras, with the associated algebra morphisms
the Hecke algebra descends of arc
insertion at the affine braid group level. In this paper we consider qKZ towers
of solutions, which consist of twisted-symmetric
polynomial solutions () of the qKZ equations that are
compatible with the tower structure on . The
compatibility is encoded by so-called braid recursion relations:
is required to coincide up to a quasi-constant
factor with the push-forward of by an intertwiner
of -modules, where
is considered as an -module through the tower
structure on .
We associate to the dense loop model on the half-infinite cylinder with
nonzero loop weights a qKZ tower of solutions. The
solutions are constructed from specialised dual non-symmetric
Macdonald polynomials with specialised parameters using the Cherednik-Matsuo
correspondence. In the special case that the extended affine Hecke algebra
parameter is a third root of unity, coincides with the (suitably
normalized) ground state of the inhomogeneous dense loop model on the
half-infinite cylinder with circumference .Comment: 45 pages. v2: main theorem (Thm. 4.7) strengthened. v3: minor typos
corrected. To appear in Ann. Henri Poincar
A Diagrammatic Temperley-Lieb Categorification
The monoidal category of Soergel bimodules categorifies the Hecke algebra of
a finite Weyl group. In the case of the symmetric group, morphisms in this
category can be drawn as graphs in the plane. We define a quotient category,
also given in terms of planar graphs, which categorifies the Temperley-Lieb
algebra. Certain ideals appearing in this quotient are related both to the
1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We
demonstrate how further subquotients of this category will categorify the cell
modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio
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