Incidence geometry in a Weyl chamber I: GL

Brownian Motion in a Truncated Weyl Chamber

We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. More precisely, we compute, for fixed time t, the probability that the motion does not leave by time t the intersection of a Weyl chamber and a t-dependent centred box, and we identify its asymptotics for t →∞. Different regimes are identified according to the growth speed, ranging f...

Isotropic cosmologies in Weyl geometry

We study homogeneous and isotropic cosmologies in a Weyl spacetime. We show that for homogeneous and isotropic spacetimes, the field equations can be reduced to the Einstein equations with a twofluid source. We write the equations as a two-dimensional dynamical system and analyze the qualitative, asymptotic behavior of the models. We examine the possibility that in certain theories the Weyl 1-f...

Lattice Walks in a Weyl Chamber and Truncated Random Matrices

Let u(d, n) denote the number of permuations in the symmetric group Sn with no increasing subsequence of length greater than d. u(d, n) may alternatively be interpreted as the number of closed Z-lattice walks which begin and end at the origin and take n positive steps followed by n negative steps while remaining confined to the Weyl chamber W = {(t1, t2, . . . , td) ∈ R : t1 ≥ t2 ≥ · · · ≥ td}....

Einstein { Weyl Geometry , the dKP

On the Ext Groups between Weyl Modules for Gl N

This paper studies extension groups between certain Weyl modules for the algebraic group GLn over the integers. Main results include: (1) A complete determination of Ext groups between Weyl modules whose highest weights differ by a single root and (2) Determination of Ext between an exterior power of the defining representation and any Weyl module. The significance of these results for modular ...