In this note, we prove that the projective normality of (P(V )/G,L), the celebrated theorem of Erdös-Ginzburg-Ziv and normality of an affine semigroup are all equivalent, where V is a finite dimensional representation of a finite cyclic group G over C and L is the descent of the line bundle O(1)⊗|G|.

For a geometrically finite group Γ of G = SO(n, 1), we survey recent developments on counting and equidistribution problems for orbits of Γ in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In ou...

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

This paper is a continuation of [3] in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hec...

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of Γ×C and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU2, where our construction yields a group theoretic realization of the representation...

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X] in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X] in terms of a so-called colon-operation...