The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying the emptiness formation probability (EFP) in the domain-wall six-vertex model. Assuming that the limit shape arises in correspondence to the 'condensation' of almost all solutions of the saddle-point equations for certain multiple integral representation for EFP, the limit shape of large ASMs is fo...

For certain vertex operator algebras (e.g., lattice type) and given finite group of automorphisms, we prove existence of a positive definite integral form invariant under the group. Applications include an integral form in the Moonshine VOA which is invariant under the Monster, and examples in other lattice type VOAs.

According to the Ringel-Green Theorem([G],[R1]), the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, the Drinfeld double of the generic composition algebra can be identified with the whole quantum group([X],[XY]), in which the BGP-reflection functors coincide with Lusztig’s symmetries. We first assert the elements co...