We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion
of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formula to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kost
ka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of sl_2(C)-modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.
We are carrying out multi-frequency radio continuum observations, using the Australia Telescope Compact Array, to systematically search for collimated ionized jets towards high-mass young stellar objects (HMYSOs). Here we report observations at 1.4,
2.4, 4.8 and 8.6 GHz, made with angular resolutions of about 7, 4, 2, and 1 arcsec, respectively, towards six objects of a sample of 33 southern HMYSOs thought to be in very early stages of evolution. The objects in the sample were selected from radio and infrared catalogs by having positive radio spectral indices and being luminous (L_bol > 20,000 L_sun), but underluminous in radio emission compared to that expected from its bolometric luminosity. This criteria makes the radio sources good candidates for being ionized jets. As part of this systematic search, two ionized jets have been discovered: one previously published and the other reported here. The rest of the observed candidates correspond to three hypercompact hii regions and two ultracompact hii regions. The two jets discovered are associated with two of the most luminous (70,000 and 100,000 Lsun) HMYSOs known to harbor this type of objects, showing that the phenomena of collimated ionized winds appears in the formation process of stars at least up to masses of ~ 20 M_sun and provides strong evidence for a disk-mediated accretion scenario for the formation of high-mass stars. From the incidence of jets in our sample, we estimate that the jet phase in high-mass protostars lasts for 40,000 yr.
The Carina Nebula (NGC 3372) is our richest nearby laboratory in which to study feedback through UV radiation and stellar winds from very massive stars during the formation of an OB association, at an early phase before SNe have disrupted the environ
ment. This feedback is triggering new generations of star formation around the periphery of the nebula, while simultaneously evaporating the gas and dust reservoirs out of which young stars are trying to accrete. Carina is currently powered by UV radiation from 65 O-type stars and 3 WNH stars, but for most of its lifetime when its most massive star (Eta Car) was on the main-sequence, the Carina Nebula was powered by 70 O-type stars that produced an ionizing luminosity 150 times stronger than in Orion. At a distance of 2.3 kpc, Carina has the most extreme stellar populationwithin a few kpc of the Sun, and suffers little interstellar extinction. It is our best bridge between the detailed star-formation processes that can be studied in nearby regions like Orion, and much more extreme but also more distant regions like 30 Doradus. Existing observations have only begun to tap the tremendous potential of this region for understanding the importance of feedback in star formation; it will provide a reservoir of new discoveries for the next generation of large ground-based telescopes, space telescopes, and large submillimeter and radio arrays.
We present observations of 1.2-mm dust continuum emission, made with the Swedish ESO Submillimeter Telescope, towards eighteen luminous IRAS point sources, all with colors typical of compact HII regions and associated with CS(2-1) emission, thought t
o be representative of young massive star forming regions. Emission was detected toward all the IRAS objects. We find that the 1.2-mm sources associated with them have distinct physical parameters, namely sizes of 0.4 pc, dust temperatures of 30 K, masses of 2x10^3 Msun, column densities of 3x10^23 cm^-2, and densities of 4x10^5 cm^-3. We refer to these dust structures as massive and dense cores. Most of the 1.2-mm sources show single-peaked structures, several of which exhibit a bright compact peak surrounded by a weaker extended envelope. The observed radial intensity profiles of sources with this type of morphology are well fitted with power-law intensity profiles with power-law indices in the range 1.0-1.7. This result indicates that massive and dense cores are centrally condensed, having radial density profiles with power-law indices in the range 1.5-2.2. We also find that the UC HII regions detected with ATCA towards the IRAS sources investigated here (Paper I) are usually projected at the peak position of the 1.2-mm dust continuum emission, suggesting that massive stars are formed at the center of the centrally condensed massive and dense cores.
The first paper in this series took a direct census of energy input from the known OB stars in the Carina Nebula, and in this paper we study the global properties of the surrounding nebulosity. We find that the total IR luminosity of Carina is about
1.2E7 Lsun, accounting for only about 50-60% of the known stellar luminosity from Paper I. Similarly, the ionizing photon luminosity -- (abridged; many important details omitted). Synchronized star formation around the periphery of Carina provides a strong case that star formation here was indeed triggered by stellar winds and UV radiation. This second generation appears to involve a cascade toward preferentially intermediate- and low-mass stars, but this may soon change when eta Car and its siblings explode. If the current reservoir of atomic and molecular gas can be tapped at that time, massive star formation may be rejuvinated around the periphery of Carina much as if it were a young version of Goulds Belt. Also, when these multiple SNe occur, the triggered second generation will be pelted repeatedly with SN ejecta bearing short-lived radioactive nuclides. Carina may therefore represent the most observable analog to the cradle of our own Solar System.
The Australia Telescope Compact Array (ATCA) and the Mopra facility have been used to search for new southern class I methanol masers at 9.9, 25 (J=5) and 104 GHz, which are thought to trace more energetic conditions in the interface regions of molec
ular outflows, than the widespread class I masers at 44 and 95 GHz. One source shows a clear outflow association.
