Unidirectional Littelmann Paths for Crystals of Type A and Rank 2

Algebras and Representation Theory - For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight Λ, there is a labeling by...     

Hurwitz quaternion order and arithmetic Riemann surfaces

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.     

A non-recursive criterion for weights of a highest-weight module for an affine Lie algebra

Let Λ be a dominant integral weight of level k for the affine Lie algebra g and let α be a non-negative integral combination of simple roots. We address the question of whether the weight η = Λ ? α lies in the set P(Λ) of weights in the irreducible highest-weight module with highest weight Λ. We give a non-recursive criterion in terms of the coefficients of α modulo an integral lattice kM, where M is the lattice parameterizing the abelian normal subgroup T of the Weyl group. The criterion requires the preliminary computation of a set no larger than the fundamental region for kM, and we show how this set can be efficiently calculated.     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.