Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction

Abstract

This is a contribution to the theory of Hecke algebras. A class of algebras called generic pro-p Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-p Hecke algebras are given by pro-p-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define 'Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra hold true, equivalent to Bernstein's relations in the case of Iwahori-Hecke algebras.For a certain subclass called affine pro-p Hecke algebras, containing Iwahori-Hecke and pro-p-Iwahori Hecke algebras, an explicit canonical and integral basis of the center is constructed and finiteness results are proved about the center and the module-structure of the algebra over its center, recovering results of Bernstein-Zelevinsky-Lusztig and Vignéras. ZusammenfassungEs wird ein Beitrag zur Theorie der Hecke-Algebren geleistet. Speziell wird eine Klasse von Algebren eingeführt, die generischen pro-p Hecke-Algebren, welche die Klasse der generischen Hecke-Algebren erweitert durch Übergang von Coxetergruppen zu Erweiterungen solcher durch abelsche Gruppen. Beispiele sind gegeben durch pro-p-Iwahori Hecke-Algebren und Yokonuma-Hecke Algebren. Es wird der Begriff der Orientierung einer Coxetergruppe eingeführt und benutzt um sogenannte Bernsteinabbildungen definieren, welche eng verwandt sind mit der Bernsteinpräsentierung und dem Cherednik-Kozykel. Sodann wird gezeigt, dass zwischen den Bildern der Bernsteinabbildungen gewisse Relationen herrschen, welche sich im Spezialfall der Iwahori-Hecke Algebra auf die bekannten Bernsteinrelationen reduzieren.Ferner wird für die Unterklasse der affinen pro-p Hecke-Algebren, welche sowohl die Iwahori-Hecke als auch die pro-p-Iwahori Hecke-Algebren umfassen, eine kanonische und ganzzahlige Basis des Zentrums konstruiert und es werden Endlichkeitssätze über das Zentrum, aufgefasst als Algebra, und über die Hecke-Algebra selbst, aufgefasst als Modul über dem Zentrum, bewiesen. Dabei werden bereits bekannte Ergebnisse von Bernstein-Zelevinsky-Lusztig und Vignéras verallgemeinert.I would like to thank my advisor for his support, and I would like to thank my family, without which this thesis would not exist. Generic pro-p Hecke algebras and Bernstein maps 1.Basic definitions and some geometric terminologyWe recall some standard facts and terminology from the theory of Coxeter groups (cf. [Bou07, Ch. IV] or [Bro89, II]). Definition.A Coxeter group W = (W, S) consists of a group W and a set S ⊆ W of generators of order 2 satisfying the action condition. That is, there exists an actionon the set H × {±1}, wheresuch that a generator s ∈ S acts asThere are several other equivalent definitions of the notion of a Coxeter group (see [Bro89, II.4]). In particular, given a group W and a set S of generators of order 2, the action conditio...

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