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 $n$ 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