Generalized reflection groups II

Projective Representations of Reflection Groups Ii

Generalized Nil-coxeter Algebras over Discrete Complex Reflection Groups

We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the ‘usual’ nil-Coxeter algebras: a novel 2-parameter type A family that we call NCA(n, d). We explore several combinatorial properties of NCA(n, d), including its Coxeter word basis, length...

PRECOMPACT TOPOLOGICAL GENERALIZED GROUPS

In this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.

Reflection Groups and Polytopes over Finite Fields, II

When the standard representation of a crystallographic Coxeter group Γ is reduced modulo an odd prime p, a finite representation in some orthogonal space over Zp is obtained. If Γ has a string diagram, the latter group will often be the automorphism group of a finite regular polytope. In Part I we described the basics of this construction and enumerated the polytopes associated with the groups ...

On ground fields of arithmetic hyperbolic reflection groups. II

This paper continues [17] (arXiv.org:math.AG/0609256) and [18] (arXiv:0708.3991). Using authors’s methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtained. This could be important for further classification. Thus, ...