Involutions and Total Orthogonality in Some Finite Classical Groups

A group $G$ is called \emph{real} if every element is conjugate to its inverse, and $G$ is \emph{strongly real} if each of the conjugating elements may be chosen to be an involution, an element in $G$ which squares to the identity. Real groups are called as such because every irreducible character of a real group is real valued. A group $G$ is called \emph{totally orthogonal} if every irreducible complex representation is realizable over the field of real numbers. Total orthogonality is sufficient, but not necessary for reality. Reality of representations is quantified in the Frobenius-Schur indicator. For an irreducible character $\chi$ of $G$, the Frobenius-Schur indicator is given by $$\varepsilon(\chi) = \frac{1}{|G|}\sum_{g \in G} \chi(g^2).$$ Frobenius and Schur showed that $$\varepsilon(\chi) = \begin{cases} 1 & \text{if } \chi \text{ is the character of a real representation} \\ -1 & \text{if }\chi \text{ is real-valued but is not the character of a real representation} \\ 0 & \text{otherwise} \end{cases}.$$ They also showed that \sum_{\chi \in \Irr(G)} \varepsilon(\chi)\chi(1) = \big | \{g \in G : g^2 =1 \} \big |, where \Irr(G) is the collection of irreducible characters of $G$. Hence a group is totally orthogonal if and only if its character degree sum is equal to the number of involutions in the group. It is conjectured that a finite simple group is strongly real if and only if it is totally orthogonal. In this work, we verify this conjecture for all strongly real simple groups except \PSp(2n,q) and \POmega^\pm(4n,q) when $q$ is even. The methods in this paper reduce this conjecture to showing that \Or^\pm(4n,q) and \Sp(2n,q) are totally orthogonal. Motivated by the Frobenius-Schur count of involutions and Conjecture 7.1 of \cite{vinroot}, we prove an upper bound on the number of involutions in \Or^\pm(2n,q) and \Sp(2n,q) by $q^{(d-r)/2}(q+1)^r$ where $r$ is the rank of the group and $d$ is its dimension. If indeed, \Or^\pm(4n,q) and \Sp(2n,q) are totally orthogonal, this verifies Conjecture 7.1 of \cite{vinroot} for these groups. Finally, we obtain generating functions for the number of involutions in subgroups of the orthogonal groups. We apply these generating functions to compute the limiting behavior of the number of involutions in these groups as the authors did in \cite{fulman} for some other classical groups

New Associations of Gamma-Ray Sources from the Fermi Second Source Catalog

We present the results of an all-sky radio survey between 5 and 9 GHz of the fields surrounding all unassociated gamma-ray objects listed in the Fermi Large Area Telescope Second Source Catalog (2FGL). The goal of these observations is to find all new gamma-ray AGN associations with radio sources >10 mJy at 8 GHz. We observed with the Very Large Array and the Australia Telescope Compact Array the areas around unassociated sources, providing localizations of weak radio point sources found in 2FGL fields at arcmin scales. Then we followed-up a subset of those with the Very Long Baseline and the Long Baseline Arrays to confirm detections of radio emission on parsec-scales. We quantified association probabilities based on known statistics of source counts and assuming a uniform distribution of background sources. In total we found 865 radio sources at arcsec scales as candidates for association and detected 95 of 170 selected for follow-up observations at milliarcsecond resolution. Based on this we obtained firm associations for 76 previously unknown gamma-ray AGN. Comparison of these new AGN associations with the predictions from using the WISE color-color diagram shows that half of the associations are missed. We found that 129 out of 588 observed gamma-ray sources at arcmin scales not a single radio continuum source was detected above our sensitivity limit within the 3-sigma gamma-ray localization. These "empty" fields were found to be particularly concentrated at low Galactic latitudes. The nature of these Galactic gamma-ray emitters is not yet determined.Comment: accepted for publication by ApJS, 18 pages, 10 figures, 12 tables; full electronic versions of tables 2-8 are available as ancillary file

The Jet-Driven Outflow In The Radio Galaxy SDSS J1517+3353: Implications For Double-Peaked Narrow-Line Active Galactic Nucleus

We report on the study of an intriguing active galaxy that was selected as a potential multiple supermassive black hole merger in the early-type host SDSS J151709.20+335324.7 (z = 0.135) from a complete search for double-peaked [O III] lines from the SDSS spectroscopic quasi-stellar object (QSO) database. Ground-based SDSS imaging reveals two blue structures on either side of the photometric center of the host galaxy, separated from each other by about 5.7 kpc. From a combination of SDSS fiber and Keck/HIRES long-slit spectroscopy, it is demonstrated that, in addition to these two features, a third distinct structure surrounds the nucleus of the host galaxy. All three structures exhibit highly ionized line emission with line ratios characteristic of Seyfert II active galactic nuclei. The analysis of spatially resolved emission-line profiles from the HIRES spectrum reveal three distinct kinematic subcomponents, one at rest and the other two moving at -350 km s(-1) and 500 km s(-1) with respect to the systemic velocity of the host galaxy. A comparison of imaging and spectral data confirm a strong association between the kinematic components and the spatial knots, which implies a highly disturbed and complex active region in this object. A comparative analysis of the broadband positions, colors, kinematics, and spectral properties of the knots in this system lead to two plausible explanations: (1) a multiple active galactic nucleus (AGN) produced due to a massive dry merger, or (2) a very powerful radio jet-driven outflow. Subsequent VLA radio imaging reveals a clear jet aligned with the emission-line gas, confirming the latter explanation. We use the broadband radio measurements to examine the impact of the jet on the interstellar medium of the host galaxy, and find that the energy in the radio lobes can heat a significant fraction of the gas to the virial temperature. Finally, we discuss tests that may help future surveys distinguish between jet-driven kinematics and true black-hole binaries. J1517+3353 is a remarkable laboratory for AGN feedback and warrants deeper follow-up study. In the Appendix, we present high-resolution radio imaging of a second AGN with double-peaked [O III] lines, SDSS J112939.78+605742.6, which shows a sub-arcsecond radio jet. If the double-peaked nature of the narrow lines in radio-loud AGNs are generally due to radio jet interactions, we suggest that extended radio structure should be expected in most of such systems.NSF AST-0507483, AST-0808133University of Texas at AustinAlfred P. Sloan FoundationParticipating InstitutionsNational Aeronautics and Space AdministrationU.S. Department of EnergyJapanese MonbukagakushoMax Planck SocietyAstronom

Agribusiness Capstone Courses Design: Objectives and Strategies

This paper discusses the benefits of using strategic management principles as the cornerstone for building the agribusiness capstone experience. The necessity for agribusiness firms to create and implement strategies that build a sustainable competitive advantage in turn necessitates the development of strategic management skills in the leaders/managers of the future. As such, the objectives of a capstone course lean heavily toward the integrative development of strategic decision-making competence. This has a number of implications for the capstone professor in terms of course content, pedagogies, and subsequent measurement of student performance.Agribusiness, Teaching/Communication/Extension/Profession,

A data model of the Climate and Forecast metadata conventions (CF-1.6) with a software implementation (cf-python v2.1)

The CF (Climate and Forecast) metadata conventions are designed to promote the creation, processing, and sharing of climate and forecasting data using Network Common Data Form (netCDF) files and libraries. The CF conventions provide a description of the physical meaning of data and of their spatial and temporal properties, but they depend on the netCDF file encoding which can currently only be fully understood and interpreted by someone familiar with the rules and relationships specified in the conventions documentation. To aid in development of CF-compliant software and to capture with a minimal set of elements all of the information contained in the CF conventions, we propose a formal data model for CF which is independent of netCDF and describes all possible CF-compliant data. Because such data will often be analysed and visualised using software based on other data models, we compare our CF data model with the ISO 19123 coverage model, the Open Geospatial Consortium CF netCDF standard, and the Unidata Common Data Model. To demonstrate that this CF data model can in fact be implemented, we present cf-python, a Python software library that conforms to the model and can manipulate any CF-compliant dataset

An Ice-Core-Based, Late Holocene History for the Transantarctic Mountains, Antarctica

Ice core records (major anions and cations, MSA, oxygen isotopes and particles) developed from two shallow (~200 m depth) sites in the Transantarctic Mountains provide documentation of much of the Holocene paleoenvironmental history of this region. From the more southerly site, Dominion Range, an ~7000-year-long record reveals change in the influence of tropospheric transport to the region. At this site, milder conditions and increased tropospheric inflow prior to ~1500 yr BP are characterized by increased seasalt (ss), terrestrial and marine biogenic inputs. Increased persistence and/or extent of polar stratospheric clouds accompanying generally cooler conditions characterize much of the period since ~1500 yr BP. From the more northerly site, Newall Glacier, the dramatic influence of the retreat of grounded ice from McMurdo Sound dated at[Denton et al., 1989] dominates much of the ice core record. This regional environmental change is documented by massive influxes to the core site of evaporitic salts from areas exposed during low lake level stands. During the past ~150 yr, both Dominion Range and Newall Glacier appear to be experiencing an overall increase in the exposure of ice-free terrain