50 research outputs found
An observation on highest weight crystals
As shown by Stembridge, crystal graphs can be characterized by their local
behavior. In this paper, we observe that a certain local property on highest
weight crystals forces a more global property. In type , this statement says
that if a node has a single parent and single grandparent, then there is a
unique walk from the highest weight node to it. In other classical types, there
is a similar (but necessarily more technical) statement. This walk is obtained
from the associated level 1 perfect crystal, . (It is unique unless
the Dynkin diagram contains that of as a subdiagram.)
This crystal observation was motivated by representation-theoretic behavior
of the affine Hecke algebra of type , which is known to be captured by
highest weight crystals of type by results of Grojnowski. As
discussed below, the proofs in either setting are straightforward, and so the
theorem linking the two phenomena is not needed. However, the result is
presented here for crystals as one can say something in all types (Grojnowski's
theorem is only in type ), and because the statement seems more surprising
in the language of crystals than it does for affine Hecke algebra modules
Categorifying the tensor product of a level 1 highest weight and perfect crystal in type A
We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism
between a highest weight crystal and the tensor product of a perfect crystal
and another highest weight crystal, all in level 1 type A affine. The nodes of
the perfect crystal correspond to a family of trivial modules and the nodes of
the highest weight crystal correspond to simple modules, which we may also
parameterize by -restricted partitions. In the case is a prime,
one can reinterpret all the results for the symmetric group in characteristic
. The crystal operators correspond to socle of restriction and behave
compatibly with the rule for tensor product of crystal graphs.Comment: 29 pages; to appear in Proc. Sympos. Pure Math. as part of the
Proceedings of the 2012-2014 Southeastern Lie Theory Workshop
Categorifying the tensor product of the Kirillov-Reshetikhin crystal and a fundamental crystal
We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal
isomorphism between a fundamental crystal and the tensor product of a
Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine
type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of
"trivial" modules. The nodes of the fundamental crystal correspond to simple
modules of the corresponding cyclotomic KLR algebra. The crystal operators
correspond to socle of restriction and behave compatibly with the rule for
tensor product of crystal graphs.Comment: 58 pages, 4 figures, 4 table
Rational Dyck Paths in the Non Relatively Prime Case
We study the relationship between rational slope Dyck paths and invariant
subsets of extending the work of the first two authors in the
relatively prime case. We also find a bijection between --Dyck paths
and -tuples of -Dyck paths endowed with certain gluing data. These
are the first steps towards understanding the relationship between rational
slope Catalan combinatorics and the geometry of affine Springer fibers and knot
invariants in the non relatively prime case.Comment: 25 pages, 9 figure
Counting Shi regions with a fixed separating wall
Athanasiadis introduced separating walls for a region in the extended Shi
arrangement and used them to generalize the Narayana numbers. In this paper, we
fix a hyperplane in the extended Shi arrangement for type A and calculate the
number of dominant regions which have the fixed hyperplane as a separating
wall; that is, regions where the hyperplane supports a facet of the region and
separates the region from the origin.Comment: To appear in Annals of Combinatoric
Quadratic transformations of Macdonald and Koornwinder polynomials
When one expands a Schur function in terms of the irreducible characters of
the symplectic (or orthogonal) group, the coefficient of the trivial character
is 0 unless the indexing partition has an appropriate form. A number of
q-analogues of this fact were conjectured in math.QA/0112035; the present paper
proves most of those conjectures, as well as some new identities suggested by
the proof technique. The proof involves showing that a nonsymmetric version of
the relevant integral is annihilated by a suitable ideal of the affine Hecke
algebra, and that any such annihilated functional satisfies the desired
vanishing property. This does not, however, give rise to vanishing identities
for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss
the required modification to these polynomials to support such results.Comment: 32 pages LaTeX, 10 xfig figure
Deformations of permutation representations of Coxeter groups
The permutation representation afforded by a Coxeter group W acting on the
cosets of a standard parabolic subgroup inherits many nice properties from W
such as a shellable Bruhat order and a flat deformation over Z[q] to a
representation of the corresponding Hecke algebra. In this paper we define a
larger class of ``quasiparabolic" subgroups (more generally, quasiparabolic
W-sets), and show that they also inherit these properties. Our motivating
example is the action of the symmetric group on fixed-point-free involutions by
conjugation.Comment: 44 page
The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality
Given a module for the algebra of quantum
differential operators on , and a positive integer , we may equip the
space of invariant tensors in , with an
action of the double affine Hecke algebra of type . Here or
, and is the -dimensional defining representation of .
In this paper we take to be the basic
-module, i.e. the quantized coordinate algebra . We describe a weight basis for
combinatorially in terms of walks in the
type weight lattice, and standard periodic tableaux, and subsequently
identify with the irreducible "rectangular
representation" of height of the double affine Hecke algebra.Comment: 37 pages, 14 figures; Several missing references added with
discussion that includes proper citation to the
A bijection between dominant Shi regions and core partitions
It is well-known that Catalan numbers
count the number of dominant regions in the Shi arrangement of type , and
that they also count partitions which are both -cores as well as
-cores. These concepts have natural extensions, which we call here the
-Catalan numbers and -Shi arrangement. In this paper, we construct a
bijection between dominant regions of the -Shi arrangement and partitions
which are both -cores as well as -cores. The bijection is natural in
the sense that it commutes with the action of the affine symmetric group