A Direct Way to Find the Right Key of a Semistandard Young Tableau

The right and left key of a semistandard Young tableau were introduced by Lascoux and Schutzenberger in 1990. Most prominently, the right key is a tool used to find Demazure characters for sl(n,C). Previous methods used to compute these keys require introducing other types of combinatorial objects. This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau.Comment: 8 pages, 1 figure. Virtually identical to the version submitted in July 2011. To be contained in the author's doctoral thesis written under the supervision of Robert A. Procto

A deep, narrow J-band search for proto-galactic Lyman Alpha emission at redshifts z~9

We present a deep, narrow J-band search for proto--galactic Lyman Alpha emission at redshifts z~9. We combine an exceptionally deep image of the Hubble Deep Field South, obtained using a narrow band filter centred on the wavelength 1.187 microns using the VLT/ISAAC facility, with existing deep, broad band images covering optical to near infrared wavelengths. We search for z~9 Lyman Alpha emitting galaxies displaying a significant narrow band excess relative to the Js-band that are undetected at optical wavelengths. We detect no sources consistent with this criterion to the 90% point source flux limit of the NB image, F_{NB} = 3.28 x 10^{-18} ergs/s/cm2. The survey selection function indicates that we have sampled a volume of approximately 340 h^{-3} Mpc^3 to a Lyman Alpha emission luminosity of 10^{43} h^{-2} ergs/s. We conclude by considering the potential implications for the physics of the high--redshift universe.Comment: 11 pages, accepted MNRAS. Please note that the automatic postscript generation may result in a corrupted postscript file. Please use ftp://astroftp.phys.uvic.ca/pub/jwillis/jwillis_zen1.ps.gz to obtain an uncorrupted versio

Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v