Semiinvariants of Finite Reflection Groups
نویسنده
چکیده
Let G be a finite group of complex n× n unitary matrices generated by reflections acting on C. Let R be the ring of invariant polynomials, and χ be a multiplicative character of G. Let Ω be the R-module of χ-invariant differential forms. We define a multiplication in Ω and show that under this multiplication Ω has an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between χ-invariant forms and logarithmic forms.
منابع مشابه
Nichols-Woronowicz model of coinvariant algebra of complex reflection groups
Let V be a finite dimensional complex vector space. A finite subgroup G ⊂ GL(V ) is called a complex reflection group, if G can be generated by the set of pseudoreflections, i.e., transformations that fix a complex hyperplane in V pointwise. Any real reflection group becomes a complex reflection group if one extends the scalars from R to C. In particular all Coxeter groups give examples of comp...
متن کاملReflection Groups. a Contribution to the Handbook of Algebra
This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included. This chapter is concerned with the theory of finite reflection groups, that is, finite groups generated by reflections in a real or complex vector space. This is a rich theory, both for intrinsic reasons and as far as applications in other mathematical areas or m...
متن کاملLimit Roots of Lorentzian Coxeter Systems
A reflection in a real vector space equipped with a positive definite symmetric bilinear form is any automorphism that sends some nonzero vector v to its negative and pointwise fixes its orthogonal complement, and a finite reflection group is a discrete group generated by such transformations. We note two important classes of groups which occur as finite reflection groups: for a 2-dimensional v...
متن کاملSimple Root Systems and Presentations for Certain Complex Reflection Groups
We find all the inequivalent simple root systems for the complex reflection groups G12, G24, G25 and G26. Then we give all the non-congruent essential presentations of these groups by generators and relations. The methed used in the paper is applicable to any finite (complex) reflection groups. Introduction. Shephard and Todd classified all the finite complex reflection groups in paper [5]. Lat...
متن کاملOn Rank-two Complex Reflection Groups
We describe a class of groups with the property that the finite ones among them are precisely the complex reflection groups of rank two. This situation is reminiscent of Coxeter groups, among which the finite ones are precisely the real reflection groups. We also study braid relations between complex reflections and indicate connections to an axiomatic study of root systems and to the Shephard-...
متن کامل