648 research outputs found
On the evaluation formula for Jack polynomials with prescribed symmetry
The Jack polynomials with prescribed symmetry are obtained from the
nonsymmetric polynomials via the operations of symmetrization,
antisymmetrization and normalization. After dividing out the corresponding
antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of
interest in applications is the value of the latter polynomial when all the
variables are set equal. Dunkl has obtained this evaluation, making use of a
certain skew symmetric operator. We introduce a simpler operator for this
purpose, thereby obtaining a new derivation of the evaluation formula. An
expansion formula of a certain product in terms of Jack polynomials with
prescribed symmetry implied by the evaluation formula is used to derive a
generalization of a constant term identity due to Macdonald, Kadell and Kaneko.
Although we don't give the details in this work, the operator introduced here
can be defined for any reduced crystallographic root system, and used to
provide an evaluation formula for the corresponding Heckman-Opdam polynomials
with prescribed symmetry.Comment: 18 page
The infinitesimal characters of discrete series for real spherical spaces
Let be the homogeneous space of a real reductive group and a
unimodular real spherical subgroup, and consider the regular representation of
on . It is shown that all representations of the discrete series,
that is, the irreducible subrepresentations of , have infinitesimal
characters which are real and belong to a lattice. Moreover, let be a
maximal compact subgroup of . Then each irreducible representation of
occurs in a finite set of such discrete series representations only. Similar
results are obtained for the twisted discrete series, that is, the discrete
components of the space of square integrable sections of a line bundle, given
by a unitary character on an abelian extension of .Comment: To appear in GAF
On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups
In this paper, we consider the relation between two nonabelian Fourier
transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig
parameters for unipotent elliptic representations of a split p-adic group and
the second is defined in terms of the pseudocoefficients of these
representations and Lusztig's nonabelian Fourier transform for characters of
finite groups of Lie type. We exemplify this relation in the case of the p-adic
group of type G_2.Comment: 17 pages; v2: several minor corrections, references added; v3:
corrections in the table with unipotent discrete series of G
Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials
We realize an extended version of the trigonometric Cherednik algebra as
affine Dunkl operators involving Heaviside functions. We use the quadratic
Casimir element of the extended trigonometric Cherednik algebra to define an
explicit nonstationary Schr\"odinger equation with delta-potential. We use
coordinate Bethe ansatz methods to construct solutions of the nonstationary
Schr\"odinger equation in terms of generalized Bethe wave functions. It is
shown that the generalized Bethe wave functions satisfy affine difference
Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to
the vector valued root system analogs of the quantum Bose gas on the circle
with pairwise delta-function interactions is indicated.Comment: 23 pages; Version 2: expanded introduction and misprints correcte
Extensions of tempered representations
Let be irreducible tempered representations of an affine Hecke
algebra H with positive parameters. We compute the higher extension groups
explicitly in terms of the representations of analytic
R-groups corresponding to and . The result has immediate
applications to the computation of the Euler-Poincar\'e pairing ,
the alternating sum of the dimensions of the Ext-groups. The resulting formula
for is equal to Arthur's formula for the elliptic pairing of
tempered characters in the setting of reductive p-adic groups. Our proof
applies equally well to affine Hecke algebras and to reductive groups over
non-archimedean local fields of arbitrary characteristic. This sheds new light
on the formula of Arthur and gives a new proof of Kazhdan's orthogonality
conjecture for the Euler-Poincar\'e pairing of admissible characters.Comment: This paper grew out of "A formula of Arthur and affine Hecke
algebras" (arXiv:1011.0679). In the second version some minor points were
improve
On the r-matrix structure of the hyperbolic BC(n) Sutherland model
Working in a symplectic reduction framework, we construct a dynamical
r-matrix for the classical hyperbolic BC(n) Sutherland model with three
independent coupling constants. We also examine the Lax representation of the
dynamics and its equivalence with the Hamiltonian equation of motion.Comment: 20 page
Wheat in Pakistan and other Asian Countries
It seems as if in recent years the development literature has
shifted weight towards the agricultural sector, thereby doing more
justice to the relative importance of that sector in developing
economies. The occurrence of the Green Revolution and, subsequently, the
concern for its distribution effects have contributed to this shift.
Another cause may have been the accusation of an urban bias in
development economics and, particularly, in development policies. Or,
more down to earth, the explanation may be simply that in the course of
time it was realized that the neglect of wage goods - among which food
products are prominent - creates a very serious bottleneck which
eventually leads to inflation and balance-of-payments problems, not to
mention social discontent and political tension
Affine cellularity of affine Hecke algebras of rank two
We show that affine Hecke algebras of rank two with generic parameters are
affine cellular in the sense of Koenig-Xi.Comment: 24 pages, 4 figures and 14 tables. New version: added references,
corrected typos. Final versio
Phase Behaviour of Binary Hard-Sphere Mixtures: Free Volume Theory Including Reservoir Hard-Core Interactions
Comprehensive calculations were performed to predict the phase behaviour of
large spherical colloids mixed with small spherical colloids that act as
depletant. To this end, the free volume theory (FVT) of Lekkerkerker et al.
[Europhys. Lett. 20 (1992) 559] is used as a basis and is extended to
explicitly include the hard-sphere character of colloidal depletants into the
expression for the free volume fraction. Taking the excluded volume of the
depletants into account in both the system and the reservoir provides a
relation between the depletant concentration in the reservoir and in the system
that accurately matches with computer simulation results of Dijkstra et al.
[Phys. Rev. E 59 (1999) 5744]. Moreover, the phase diagrams for highly
asymmetric mixtures with size ratios q . 0:2 obtained by using this new
approach corroborates simulation results significantly better than earlier FVT
applications to binary hard-sphere mixtures. The phase diagram of a binary
hard-sphere mixture with a size ratio of q = 0:4, where a binary interstitial
solid solution is formed at high densities, is investigated using a numerical
free volume approach. At this size ratio, the obtained phase diagram is
qualitatively different from previous FVT approaches for hard-sphere and
penetrable depletants, but again compares well with simulation predictions.Comment: The following article has been accepted by The Journal of Chemical
Physics. After it is published, it will be found at
https://doi.org/10.1063/5.003796
Trigonometric Cherednik algebra at critical level and quantum many-body problems
For any module over the affine Weyl group we construct a representation of
the associated trigonometric Cherednik algebra at critical level in
terms of Dunkl type operators. Under this representation the center of
produces quantum conserved integrals for root system generalizations of quantum
spin-particle systems on the circle with delta function interactions. This
enables us to translate the spectral problem of such a quantum spin-particle
system to questions in the representation theory of . We use this
approach to derive the associated Bethe ansatz equations. They are expressed in
terms of the normalized intertwiners of .Comment: 31 page
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