By B. Apanasov

A revised and considerably enlarged version of the Russian e-book Discrete transformation teams and manifold constructions released via Nauka in 1983, this quantity provides a finished therapy of the geometric thought of discrete teams and the linked tessellations of the underlying area. additionally

Read Online or Download Discrete Groups in Space and Uniformization Problems PDF

Best group theory books

Weyl Transforms

The sensible analytic homes of Weyl transforms as bounded linear operators on \$ L^{2}({\Bbb R}^{n}) \$ are studied when it comes to the symbols of the transforms. The boundedness, the compactness, the spectrum and the practical calculus of the Weyl rework are proved intimately. New effects and methods at the boundedness and compactness of the Weyl transforms when it comes to the symbols in \$ L^{r}({\Bbb R}^{2n}) \$ and when it comes to the Wigner transforms of Hermite capabilities are given.

Discrete Groups and Geometry

This quantity includes a choice of refereed papers offered in honour of A. M. Macbeath, one of many prime researchers within the zone of discrete teams. the topic has been of a lot present curiosity of past due because it comprises the interplay of a couple of different themes akin to team idea, hyperbolic geometry, and intricate research.

Transformations of Manifolds and Application to Differential Equations

The interplay among differential geometry and partial differential equations has been studied because the final century. This dating relies at the indisputable fact that lots of the neighborhood houses of manifolds are expressed by way of partial differential equations. The correspondence among convinced periods of manifolds and the linked differential equations should be priceless in methods.

Additional resources for Discrete Groups in Space and Uniformization Problems

Sample text

25) Exercise: Hironaka decomposition. Let G = α, β ∼ = C2 × C4 be the abelian group of order 8 defined by     1 . −1 . DV : G → GL3 (C) : α →  . 1 .  , β →  . −1 .  . . i . 1 1 a) Show that the Hilbert series of S[V ]G is given as HS[V ]G = (1−T 2 )3 ∈ C(T ). b) Show that there is no set of primary invariants {f1 , . . , f3 } ⊆ S[V ]G such that deg(f1 ) = deg(f2 ) = deg(f2 ) = 2. Invariant Theory of Finite Groups 34 c) Find primary invariants {f1 , . . , f3 } ⊆ S[V ]G such that deg(f1 ) = deg(f2 ) = 2 and deg(f3 ) = 4, and secondary invariants {g1 , .

Fn−1 } ⊆ S[V ]Sn such that S[V ]Sn = Q[f1 , . . , fn−1 ]. 18) Exercise: Modular pseudoreflection groups. Let p be a prime, let    1 . a+b b        . 1 b b + c   G :=  ∈ GL (F ); a, b, c ∈ F ≤ GL4 (Fp ), 4 p p . 1 .        . . 1 and let V := F1×4 be the natural F G-module. p a) Show that G is a pseudoreflection group of order |G| = p3 . b) Show that S[V ]G is not a polynomial ring. Proof. 7]. 19) Exercise: Coefficient growth. (T ) d ±1 ] as well as Let H := r f(1−T di ) = d≥0 hd T ∈ C((T )), where f ∈ Z[T i=1 r ≥ 1 and di ∈ N.

Weyl: The classical groups, their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, 1997.