Dade's Projective Conjecture for p-Solvable Groups

Robinson, Geoffrey R.

doi:10.1006/jabr.2000.8307

Cited by 25 publications

(32 citation statements)

References 7 publications

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“…The distribution of the number counts is presented in Figure 2 and compared to the expectation from the null hypothesis (no clustering), which is a Poisson distribution with average value N = 0.913. This comparison has been suggested as a method to infer the clustering properties of dropout samples in pure-parallel surveys Robertson (2010). Visual inspection immediately identifies the excess at N = 5 (field BoRG58), which will be investigated in detail in the subsequent sections of this paper.…”

Section: Correlation Between Faint and Brightmentioning

confidence: 99%

“…The remainder of the number-count distribution appears to deviate slightly from Poisson, but because of the small number of fields and dropouts per field, a marked difference is neither expected nor statistically significant. At low number counts, Poisson uncertainty dominates the contribution to the standard deviation of the distribution Robertson 2010).…”

Section: Correlation Between Faint and Brightmentioning

confidence: 99%

“…If we assume a mean number density of N = 0.913 z ∼ 8 candidates per WFC3 field, we derive from our cosmic variance calculator a bias 13 b = 8.9 that is a cosmic variance of 48% . This estimate relies on an abundance match of high-redshift galaxies with dark-matter halos, which is the working assumption adopted by several studies (Newman & Davis 2002;Somerville et al 2004;Muñoz et al 2010;Robertson 2010). This modeling has been tested observationally at z 6 by determination of the two-point correlation function of Lyman break galaxies (Overzier et al 2006b;Lee et al 2009), and indirectly at 7 z 10 because conditional luminosity function models correctly predict the observed abundance of dropout galaxies in the HUDF and ERS/CANDELS surveys (Trenti et al 2010b;Oesch et al 2012).…”

Section: Statistical Significance Of Clustering In Borg58mentioning

confidence: 99%

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OVERDENSITIES OFY-DROPOUT GALAXIES FROM THE BRIGHTEST-OF-REIONIZING GALAXIES SURVEY: A CANDIDATE PROTOCLUSTER AT REDSHIFTz≈ 8

Trenti¹,

Bradley

Stiavelli

et al. 2012

ApJ

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Theoretical and numerical modeling of the assembly of dark-matter halos predicts that the most massive and luminous galaxies at high redshift are surrounded by overdensities of fainter companions. We test this prediction with Hubble Space Telescope observations acquired by our Brightest-of-Reionizing Galaxies (BoRG) survey, which identified four very bright z ∼ 8 candidates as Y 098 -dropout sources in four of the 23 non-contiguous Wide Field Camera 3 fields observed. We extend here the search for Y 098 -dropouts to fainter luminosities (M * galaxies with M AB ∼ −20), with detections at 5σ confidence (compared to the 8σ confidence threshold adopted earlier) identifying 17 new candidates. We demonstrate that there is a correlation between number counts of faint and bright Y 098 -dropouts at 99.84% confidence. Field BoRG58, which contains the best bright z ∼ 8 candidate (M AB = −21.3), has the most significant overdensity of faint Y 098 -dropouts. Four new sources are located within 70 (corresponding to 3.1 comoving Mpc at z = 8) from the previously known brighter z ∼ 8 candidate. The overdensity of Y 098 -dropouts in this field has a physical origin to very high confidence (p > 99.975%), independent of completeness and contamination rate of the Y 098 -dropout selection. We modeled the overdensity by means of cosmological simulations and estimate that the principal dark-matter halo has mass M h ≈ (4-7) × 10 11 M (∼5σ density peak) and is surrounded by several M h ≈ 10 11 M halos which could host the fainter dropouts. In this scenario, we predict that all halos will eventually merge into a M h > 2 × 10 14 M galaxy cluster by z = 0. Follow-up observations with ground-and space-based telescopes are required to secure the z ∼ 8 nature of the overdensity, discover new members, and measure their precise redshift.

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Section: Correlation Between Faint and Brightmentioning

confidence: 99%

Section: Correlation Between Faint and Brightmentioning

confidence: 99%

Section: Statistical Significance Of Clustering In Borg58mentioning

confidence: 99%

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OVERDENSITIES OFY-DROPOUT GALAXIES FROM THE BRIGHTEST-OF-REIONIZING GALAXIES SURVEY: A CANDIDATE PROTOCLUSTER AT REDSHIFTz≈ 8

Trenti¹,

Bradley

Stiavelli

et al. 2012

ApJ

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“…A consequence drawn by G. R. Robinson of the well-known conjectures on representation theory of groups by E. C. Dade and himself is that, given χ ∈ Irr(G), there always can be found a radical p-subgroup R of G which is "big" in a defect group of the p-block of χ, and which has an irreducible character η ∈ Irr(R) with d(χ) = d(η). For p-solvable groups this is now a fact (see Theorem 2 of [11], or [1] for a partial result). Our Theorem A, parts (e) and (f), gives a canonical choice for Robinson's predicted R and η: if (Q, δ) is a vertex of χ, it suffices to take R the radical closure of Q in G and η = δ R .…”

Section: (F ) Suppose That (Q δ) Is a Vertex Of χ ∈ Irr(g) And Let mentioning

confidence: 98%

Vertices for characters of $p$-solvable groups

Navarro¹

2002

Trans. Amer. Math. Soc.

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Abstract. Suppose that G is a finite p-solvable group. We associate to every irreducible complex character χ ∈ Irr(G) of G a canonical pair (Q, δ), where Q is a p-subgroup of G and δ ∈ Irr(Q), uniquely determined by χ up to Gconjugacy. This pair behaves as a Green vertex and partitions Irr(G) into "families" of characters. Using the pair (Q, δ), we give a canonical choice of a certain p-radical subgroup R of G and a character η ∈ Irr(R) associated to χ which was predicted by some conjecture of G. R. Robinson.

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“…Although a full reduction has not been attained, G. R. Robinson and C. W. Eaton have shown that in a minimal counterexample, the Fitting subgroup would be central, and there would be a single conjugacy class of components. (See [18] and [6]. )…”

Section: Introductionmentioning

confidence: 99%

A reduction theorem for the McKay conjecture

2007

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The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p -degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL 2 (q) for all prime powers q ≥ 4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q = 2 e or q = 3 e respectively, and e > 1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1 , it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.

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Dade's Projective Conjecture for p-Solvable Groups

Cited by 25 publications

References 7 publications

OVERDENSITIES OFY-DROPOUT GALAXIES FROM THE BRIGHTEST-OF-REIONIZING GALAXIES SURVEY: A CANDIDATE PROTOCLUSTER AT REDSHIFTz≈ 8

OVERDENSITIES OFY-DROPOUT GALAXIES FROM THE BRIGHTEST-OF-REIONIZING GALAXIES SURVEY: A CANDIDATE PROTOCLUSTER AT REDSHIFTz≈ 8

Vertices for characters of $p$-solvable groups

A reduction theorem for the McKay conjecture

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