In this paper, we define twinnings for affine R-buildings. We thus extend the theory of simplicial twin buildings of affine type to the non-simplicial case. We show how classical results can be extended to the non-discrete case, and, as an application, we prove that the buildings at infinity of aMoufang twin R-building have the induced structure of aMoufang building. The latter is not true for ...

Let n, r ∈ N. The affine Schur algebra S̃(n, r) (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type Ar−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the U(ĝl n ) action on the tensor space when K is the field of complex numbers. We show that S̃(n, r) can be defined...

Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This was shown by Siebert [Si] and, independently, by Hauser and the third author [HM]. In both papers the assumption of normality is essential. There are nonisomorphic non-normal varieties with isomorphic Lie algebras. The third author [M...

We prove that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and describe a method for computing this number. We use this method to determine the F-signature of Segre products of polynomial rings, and of Veronese subrings of polynomial rings. Our technique involves expressing the F-signature of an affine semigroup ring as the difference of th...

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type axmod b ≤ cx . We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generatin...