We generalize the construction of affine polar graphs in two different ways to obtain new partial difference sets and amorphic association schemes. The first generalization uses a combination of quadratic forms and uniform cyclotomy. In the second generalization we replace the quadratic form in the affine polar graph construction by higher degree homogeneous functions that are p-ary weakly regu...

We define certain natural finite sums of n’th roots of unity, called GP (n), that are associated to each convex integer polytope P , and which generalize the classical 1-dimensional Gauss sum G(n) defined over Z/nZ, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group W, generated by the reflections with respect to the coordinate hyperplanes, as well as ...

Let Uq(g) be the Drinfeld-Jimbo quantum group, which is a deformation of the universal enveloping algebra of a finite dimensional semisimple Lie algebra g. In the generic case, i.e. the parameter q is not a root of unity, several models have been raised to realize it. For example, the Ringel-Hall algebra approach is one successful model among them, see [10, 5, 11]. The case where q is a root of...

In [7] G.Navarro proposed a refinement of the McKay conjecture involving a special class of automorphisms. In [6] this new conjecture was verified for the alternating groups A(Π) when p = 2. In this paper, it is verified for the p-singular characters of A(Π) when |Π| = wp, w < p and p is odd. 1. McKay and Navarro conjectures Let G be a finite group, where |G| = n. Let p be a prime dividing n, D...

The purpose of this paper is to calculate the restriction of the p-adic polylogarithm sheaf to p-th power torsion points. 2000 Mathematics Subject Classification: 14F30,14G20

There is an Mayer-Vietoris exact sequence 0 → Vn → PicZCpn → Cl Q(ζn−1) × Pic ZCpn−1 → 0 involving the Picard group of the integer group ring ZCpn where Cpn is the cyclic group of order p and ζn−1 is a primitive p -th root of unity. The group Vn splits as Vn ∼= V + n ⊕ V − n and V − n is explicitly known. V + n is a quotient of an in some sense simpler group Vn. In 1977 Kervaire and Murthy conj...

In this survey, we shall be concerned with the category of finite–dimensional representations of the untwisted quantum affine algebra when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and q-characters. We then concentrate on particular families of irreducible represent...