5,802 research outputs found
Poincare series of subsets of affine Weyl groups
In this note, we identify a natural class of subsets of affine Weyl groups
whose Poincare series are rational functions. This class includes the sets of
minimal coset representatives of reflection subgroups. As an application, we
construct a generalization of the classical length-descent generating function,
and prove its rationality.Comment: 7 page
On growth types of quotients of Coxeter groups by parabolic subgroups
The principal objects studied in this note are Coxeter groups that are
neither finite nor affine. A well known result of de la Harpe asserts that such
groups have exponential growth. We consider quotients of by its parabolic
subgroups and by a certain class of reflection subgroups. We show that these
quotients have exponential growth as well. To achieve this, we use a theorem of
Dyer to construct a reflection subgroup of that is isomorphic to the
universal Coxeter group on three generators. The results are all proved under
the restriction that the Coxeter diagram of is simply laced, and some
remarks made on how this restriction may be relaxed.Comment: 10 pages; The exposition has been made more concise and an additional
proposition is proved in the final sectio
A note on exponents vs root heights for complex simple Lie algebras
We give an elementary combinatorial proof of a special case of a result due
to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel.
This can be used to give yet another proof of the classical fact that for a
complex simple Lie algebra, the partition formed by its exponents is dual to
that formed by the numbers of positive roots at each height.Comment: 5 page
Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras
We introduce a generalization of the classical Hall-Littlewood and
Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove
that these Kostka-Foulkes polynomials coincide with the natural generalization
of Lusztig's -analog of weight multiplicities, thereby extending a theorem
of Kato. For an affine Kac-Moody algebra, we define -analogs of string
functions and use Cherednik's constant term identities to derive explicit
product expressions for them.Comment: 19 page
Navier-Stokes solver using Green's functions II: spectral integration of channel flow and plane Couette flow
The Kleiser-Schumann algorithm has been widely used for the direct numerical
simulation of turbulence in rectangular geometries. At the heart of the
algorithm is the solution of linear systems which are tridiagonal except for
one row. This note shows how to solve the Kleiser-Schumann problem using
perfectly triangular matrices. An advantage is the ability to use functions in
the LAPACK library. The method is used to simulate turbulence in channel flow
at (and ) using grid points. An assessment
of the length of time necessary to eliminate transient effects in the initial
state is included
The dynamics of transition to turbulence in plane Couette flow
In plane Couette flow, the incompressible fluid between two plane parallel
walls is driven by the motion of those walls. The laminar solution, in which
the streamwise velocity varies linearly in the wall-normal direction, is known
to be linearly stable at all Reynolds numbers (). Yet, in both experiments
and computations, turbulence is observed for .
In this article, we show that for certain {\it threshold} perturbations of
the laminar flow, the flow approaches either steady or traveling wave
solutions. These solutions exhibit some aspects of turbulence but are not fully
turbulent even at . However, these solutions are linearly unstable and
flows that evolve along their unstable directions become fully turbulent. The
solution approached by a threshold perturbation could depend upon the nature of
the perturbation. Surprisingly, the positive eigenvalue that corresponds to one
family of solutions decreases in magnitude with increasing , with the rate
of decrease given by with
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