Pre-enriched, not primordial ellipticals

We follow the chemical evolution of a galaxy through star formation and its feedback into the inter-stellar medium, starting from primordial gas and allowing for gas to inflow into the region being modelled. We attempt to reproduce observed spectral line-strengths for early-type galaxies to constrain their star formation histories. The efficiencies and times of star formation are varied as well as the amount and duration of inflow. We evaluate the chemical enrichment and the mass of stars made with time. Single stellar population (SSP) data are then used to predict line-strengths for composite stellar populations. The results are compared with observed line-strengths in ten ellipticals, including some features which help to break the problem of age-metallicity degeneracy in old stellar populations. We find that the elliptical galaxies modelled require high metallicity SSPs (>3 x solar) at later times. In addition the strong lines observed cannot be produced by an initial starburst in primordial gas, even if a large amount of inflow is allowed for during the first few x 10E+8 years. This is because some pre-enrichment is required for lines in the bulk of the stars to approach the observed line-strengths in ellipticals.Comment: 18 pages, 8 figures, Latex, accepted for publication in MNRA

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

Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)

Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is two-thirds of our preprint "Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters", arXiv:1612.06323v

Random Field XY Model in Three Dimensions: The Role of Vortices

We study vortex states in a 3d random-field XY model of up to one billion lattice spins. Starting with random spin orientations, the sample freezes into the vortex-glass state with a stretched-exponential decay of spin correlations, having short correlation length and a low susceptibility, compared to vortex-free states. In a field opposite to the initial magnetization, peculiar topological objects -- walls of spins still opposite to the field -- emerge along the hysteresis curve. On increasing the field strength, the walls develop cracks bounded by vortex loops. The loops then grow in size and eat the walls away. Applications to magnets and superconductors are discussed.Comment: 5 pages, 8 figure

Constraining the Star Formation Histories of Spiral Bulges

Long-slit spectroscopic observations of line-strengths and kinematics made along the minor axes of four spiral bulges are reported. Comparisons are made between central line-strengths in spiral bulges and those in other morphological types. The bulges are found to have central line-strengths comparable with those of single stellar populations (SSPs) of approximately solar abundance or above. Negative radial gradients are observed in line-strengths, similar to those in elliptical galaxies. The bulge data are consistent with correlations between Mg2, and central velocity dispersion observed in elliptical galaxiess. In contrast to elliptical galaxies, central line-strengths lie within the loci defining the range of and Mg2 achieved by Worthey's (1994) solar abundance ratio, SSPs. The implication of solar abundance ratios indicates differences in the star formation histories of spiral bulges and elliptical galaxies. A ``single zone with in- fall'' model of galactic chemical evolution, using Worthey's (1994) SSPs, is used to constrain possible star formation histories in our sample. We show that , Mg2 and Hbeta line-strengths observed in these bulges cannot be reproduced using primordial collapse models of formation but can be reproduced by models with extended in-fall of gas and star formation (2-17 Gyr) in the region modelled. One galaxy (NGC 5689) shows a central population with luminosity weighted average age of ~5 Gyr, supporting the idea of extended star formation. Kinematic substructure, possibly associated with a central spike in metallicity, is observed at the centre of the Sa galaxy NGC 3623.Comment: 14 pages. MNRAS latex file. Accepted for publication in MNRA

The Age, Metallicity and Alpha-Element Abundance of Galactic Globular Clusters from Single Stellar Population Models

Establishing the reliability with which stellar population parameters can be measured is vital to extragalactic astronomy. Galactic GCs provide an excellent medium in which to test the consistency of Single Stellar Population (SSP) models as they should be our best analogue to a homogeneous (single) stellar population. Here we present age, metallicity and $\alpha$-element abundance measurements for 48 Galactic globular clusters (GCs) as determined from integrated spectra using Lick indices and SSP models from Thomas, Maraston & Korn, Lee & Worthey and Vazdekis et al. By comparing our new measurements to independent determinations we are able to assess the ability of these SSPs to derive consistent results -- a key requirement before application to heterogeneous stellar populations like galaxies. We find that metallicity determinations are extremely robust, showing good agreement for all models examined here, including a range of enhancement methods. Ages and $\alpha$-element abundances are accurate for a subset of our models, with the caveat that the range of these parameters in Galactic GCs is limited. We are able to show that the application of published Lick index response functions to models with fixed abundance ratios allows us to measure reasonable $\alpha$-element abundances from a variety of models. We also examine the age-metallicity and [$\alpha$/Fe]-metallicity relations predicted by SSP models, and characterise the possible effects of varied model horizontal branch morphology on our overall results.Comment: 22 pages, 19 figures, accepted for publication in MNRA