A note on maximal length elements in conjugacy classes of finite coxeter groups

The maximal lengths of elements in each of the conjugacy classes of Coxeter groups of types $B_n$, $D_n$ and $E_6$ are determined. Additionally, representative elements are given that attain these maximal lengths

Zero excess and minimal length in finite coxeter groups

Let \mathcal{W} be the set of strongly real elements of W, a Coxeter group. Then for $w \in \mathcal{W}$, $e(w)$, the excess of w, is defined by e(w) = \min min \{l(x)+l(y) - l(w)| w = xy; x^2 = y^2 =1}. When $W$ is finite we may also define E(w), the reflection excess of $w$. The main result established here is that if $W$ is finite and $X$ is a $W$-conjugacy class, then there exists $w \in X$ such that $w$ has minimal length in $X$ and $e(w) = 0 = E(w)$

A note on commuting graphs for symmetric groups

The commuting graph C(G;X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G;X) is investigated when G is the symmetric group and X a conjugacy class of G

Corrigendum to "Involution products in Coxeter groups" [J. Group Theory 14 (2011), no. 2, 251–259]

In Involution products in Coxeter groups [J. Group Theory 14 (2011), no. 2, 251–259], Theorem 2.4 states a well-known result on Coxeter groups which gives conditions under which the stabilizer of a nonzero vector is a proper parabolic subgroup. However the hypothesis of this result is incorrectly stated in our paper: it holds for finite Coxeter groups but is not true in general for infinite Coxeter groups. We are grateful to an anonymous referee of a subsequent paper for pointing this out. As a consequence, the proof of Theorem 1.1 in that paper, which uses Theorem 2.4, is incomplete. Here we complete the proof of Theorem 1.1 without recourse to Theorem 2.4

Involution statistics in finite coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial LW,X(t) is defined by LW,X(t) = P x∈X t `(x) , where ` is the length function on W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group W. In particular, these results correct errors in [6] for the involution length polynomials of Coxeter groups of type Bn and Dn. Moreover, we give a counterexample to a unimodality conjecture stated in [6]

X-posets of certain Coxeter groups

Let X be a subgroup of a Coxeter group W. In the paper "On Cosets in Coxeter Groups" Turk. J. Math. 36(2012),77 -- 93, the authors developed the notion of X-posets, which are defined on certain equivalence classes of the (right) cosets of X in W. These posets can be thought of as a generalization of the well-known Bruhat order of W. This article provides a catalogue of all the X-posets for various small Coxeter groups

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The Wide Field Spectrograph (WiFeS)

This paper describes the Wide Field Spectrograph (WiFeS) under construction at the Research School of Astronomy and Astrophysics (RSAA) of the Australian National University (ANU) for the ANU 2.3m telescope at the Siding Spring Observatory. WiFeS is a powerful integral field, double-beam, concentric, image-slicing spectrograph designed to deliver excellent thoughput, wavelength stability, spectrophotometric performance and superb image quality along with wide spectral coverage throughout the 320-950 nm wavelength region. It provides a 25x38 arcsec. field with 0.5 arcsec. sampling along each of twenty five 38X1 arcsec slitlets. The output format is optimized to match the 4096x4096 pixel CCD detectors in each of two cameras individually optimized for the blue and the red ends of the spectrum, respectively. A process of "interleaved nod-and-shuffle" will be applied to permit quantum noise-limited sky subtraction. Using VPH gratings, spectral resolutions of 3000 and 7000 are provided. The full spectral range is covered in a single exposure at R=3000, and in two exposures in the R=7000 mode. The use of transmissive coated optics, VPH gratings and optimized mirror coatings ensures a throughput (including telescope atmosphere and detector) > 30% over a wide spectral range. The concentric image-slicer design ensures an excellent and uniform image quality across the full field. To maximize scientific return, the whole instrument is configured for remote observing, pipeline data reduction, and the accumulation of calibration image libraries.Comment: Accepted for publication in Astrophysics & Space Science, 16 pages, 14 figure

Perturbation theory vs. simulation for tadpole improvement factors in pure gauge theories

We calculate the mean link in Landau gauge for Wilson and improved SU(3) anisotropic gauge actions, using two loop perturbation theory and Monte Carlo simulation employing an accelerated Langevin algorithm. Twisted boundary conditions are employed, with a twist in all four lattice directions considerably improving the (Fourier accelerated) convergence to an improved lattice Landau gauge. Two loop perturbation theory is seen to predict the mean link extremely well even into the region of commonly simulated gauge couplings and so can be used remove the need for numerical tuning of self-consistent tadpole improvement factors. A three loop perturbative coefficient is inferred from the simulations and is found to be small. We show that finite size effects are small and argue likewise for (lattice) Gribov copies and double Dirac sheets.Comment: 13 pages of revtex

Coordinating Unions, Wages and Employment

In this paper we consider a two-sector economy in which individual unions are affiliated into a federation of unions. We analyze the consequences of two different types of wage setting. Firstly, individual unions set wages in their own sector without taking into account the effect of their wages on the employment level in the other sector. There may be positive as well as negative externalities. A positive (negative) externality may exist if a higher (lower) wage in one sector implies a higher level of employment in the other sector. Both cases may occur in our model. Secondly, wages in the two sectors are set by the federation of unions. We show that in this case higher (lower) wages result than in the first case if a positive (negative) externality exists