scholarly journals The absolute order of a permutation representation of a Coxeter group

2013 ◽  
Vol 39 (1) ◽  
pp. 75-98 ◽  
Author(s):  
Christos A. Athanasiadis ◽  
Yuval Roichman
Keyword(s):  
Coxeter Group ◽  
Permutation Representation ◽  
Absolute Order ◽  
The Absolute

Tits Indices

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Coxeter Group ◽  
Coxeter System ◽  
Relative Rank ◽  
Coxeter Diagram ◽  
The Absolute ◽  
Relative Type

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.

scholarly journals The absolute order on the hyperoctahedral group

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Myrto Kallipoliti
Keyword(s):  
Closed Interval ◽  
Order Ideal ◽  
Order Complex ◽  
Hyperoctahedral Group ◽  
Closed Intervals ◽  
Absolute Order ◽  
Nous Montrons ◽  
International Audience ◽  
Strong Constructibility ◽  
The Absolute

International audience The absolute order on the hyperoctahedral group $B_n$ is investigated. It is shown that every closed interval in this order is shellable, those closed intervals which are lattices are characterized and their zeta polynomials are computed. Moreover, using the notion of strong constructibility, it is proved that the order ideal generated by the Coxeter elements of $B_n$ is homotopy Cohen-Macaulay and the Euler characteristic of the order complex of the proper part of this ideal is computed. Finally, an example of a non Cohen-Macaulay closed interval in the absolute order on the group $D_4$ is given and the closed intervals of $D_n$ which are lattices are characterized. Nous étudions l'ordre absolu sur le groupe hyperoctahédral $B_n$. Nous montrons que chaque intervalle fermé de cet ordre est shellable, caractérisons les treillis parmi ces intervalles et calculons les polynômes zêta de ces derniers. De plus, en utilisant la notion de constructibilité forte, nous prouvons que l'idéal engendré par les éléments de Coxeter de $B_n$ est Cohen-Macaulay pour l'homotopie, et nous calculons la caractéristique d'Euler du complexe associé à cet idéal. Pour finir, nous exhibons un exemple d'intervalle fermé non Cohen-Macaulay dans l'ordre absolu du groupe $D_4$, et caractérisons les intervalles fermés de $D_n$ qui sont des treillis.

Bulk standards for low-temperature biological x-ray microanalysis

1984 ◽  
Vol 42 ◽  
pp. 282-283
Author(s):  
P. Echlin ◽  
M. McKoon ◽  
E.S. Taylor ◽  
C.E. Thomas ◽  
K.L. Maloney ◽  
...  
Keyword(s):  
Phase Separation ◽  
Low Temperature ◽  
Analytical Method ◽  
Salt Solutions ◽  
Absolute Concentration ◽  
Cooling Process ◽  
X Ray ◽  
The Absolute ◽  
Analytical Procedures ◽  
Electrical Conductors

Although sections of frozen salt solutions have been used as standards for x-ray microanalysis, such solutions are less useful when analysed in the bulk form. They are poor thermal and electrical conductors and severe phase separation occurs during the cooling process. Following a suggestion by Whitecross et al we have made up a series of salt solutions containing a small amount of graphite to improve the sample conductivity. In addition, we have incorporated a polymer to ensure the formation of microcrystalline ice and a consequent homogenity of salt dispersion within the frozen matrix. The mixtures have been used to standardize the analytical procedures applied to frozen hydrated bulk specimens based on the peak/background analytical method and to measure the absolute concentration of elements in developing roots.

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