4,901 research outputs found
Schur Partial Derivative Operators
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice
cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \|
x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the
study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space
M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The
``shift operators'', which are particular partial symmetric derivative
operators are very useful in the comprehension of the structure of the M_L
spaces. We describe here how a Schur function partial derivative operator acts
on lattice diagrams with distinct cells in the positive quadrant.Comment: 8 pages, LaTe
Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
The aim of this work is to study the quotient ring R_n of the ring
Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous
quasi-symmetric functions. We prove here that the dimension of R_n is given by
C_n, the n-th Catalan number. This is also the dimension of the space SH_n of
super-covariant polynomials, that is defined as the orthogonal complement of
J_n with respect to a given scalar product. We construct a basis for R_n whose
elements are naturally indexed by Dyck paths. This allows us to understand the
Hilbert series of SH_n in terms of number of Dyck paths with a given number of
factors.Comment: LaTeX, 3 figures, 12 page
Combinatorics of Labelled Parallelogram polyominoes
We obtain explicit formulas for the enumeration of labelled parallelogram
polyominoes. These are the polyominoes that are bounded, above and below, by
north-east lattice paths going from the origin to a point (k,n). The numbers
from 1 and n (the labels) are bijectively attached to the north steps of
the above-bounding path, with the condition that they appear in increasing
values along consecutive north steps. We calculate the Frobenius characteristic
of the action of the symmetric group S_n on these labels. All these enumeration
results are refined to take into account the area of these polyominoes. We make
a connection between our enumeration results and the theory of operators for
which the intergral Macdonald polynomials are joint eigenfunctions. We also
explain how these same polyominoes can be used to explicitly construct a linear
basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure
On the importance of local sources of radiation for quasar absorption line systems
A generic assumption of ionization models of quasar absorption systems is
that radiation from local sources is negligible compared with the cosmological
background. We test this assumption and find that it is unlikely to hold for
absorbers as rare as H I Lyman limit systems. Assuming that the absorption
systems are gas clouds centered on sources of radiation, we derive analytic
estimates for the cross-section weighted moments of the flux seen by the
absorbers, of the impact parameter, and of the luminosity of the central
source. In addition, we compute the corresponding medians numerically. For the
one class of absorbers for which the flux has been measured: damped Ly-alpha
systems at z~3, our prediction is in excellent agreement with the observations
if we assume that the absorption arises in clouds centered on Lyman-break
galaxies. Finally, we show that if Lyman-break galaxies dominate the UV
background at redshift 3, then consistency between observations of the UV
background, the UV luminosity density from galaxies, and the number density of
Lyman limit systems requires escape fractions of order 10 percent.Comment: Accepted for publication in the Astrophysical Journal, 11 pages, 1
figure. Version 2: Added alternative method. Decreased fiducial escape
fraction to guarantee consistency between observed luminosity density, mean
free path, and UV background. This increased the column density above which
local radiation is importan
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