Weyl modules and Levi subalgebras

For a simple complex Lie algebra of finite rank and classical type, we fix a triangular decomposition and consider the simple Levi subalgebras associated to closed subsets of roots. We study the restriction of global and local Weyl modules of current algebras to this Levi subalgebra. We identify necessary and sufficient conditions on a pair of a Levi subalgebra and a dominant integral weight, such that the restricted module is a global (resp. a local) Weyl module.Comment: 22 pages, final version, to appear in Jol

Some remarks on generalized roundness

By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank $\geq 2$ equals 1. This answers a question of J-F. Lafont and S. Prassidis.Comment: 3 page

Truths qua Grounds

A number of philosophers have recently found it congenial to talk in terms of grounding. Grounding discourse features grounding sentences that are answers to questions about what grounds what. The goal of this article is to explore and defend a counterpart-theoretic interpretation of grounding discourse. We are familiar with David Lewis's applications of the method of counterpart theory to de re modal discourse. Counterpart-theoretic interpretations of de re modal idioms and grounding sentences share similar motivations, mechanisms, and applications. I shall explain my motivations and describe two applications of a counterpart theory for grounding discourse. But, in this article, my main focus is on counterpart-theoretic mechanisms

On Ozawa kernels

We explicit Ozawa kernels for classical group theoretical constructions, for discrete metric spaces of finite asymptotic dimension, of large enough Hilbert space compression, and for suitable actions of countable groups on metric spaces. We also obtain an alternative proof of stability results concerning Yu's property A.Comment: 17 page

New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules

New graded modules for the current algebra of $\mathfrak{sl}_n$ are introduced. Relating these modules to the fusion product of simple $\mathfrak{sl}_n$-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple $\mathfrak{sl}_n$-modules and use them to provide decomposition formulas for these new modules in important cases.Comment: 23 page

Extended partial order and applications to tensor products

We extend the preorder on k-tuples of dominant weights of a simple complex Lie algebra g of classical type adding up to a fixed weight $\lambda$ defined by V. Chari, D. Sagaki and the author. We show that the induced extended partial order on the equivalence classes has a unique minimal and a unique maximal element. For k=2 we compute its size and determine the cover relation. To each k-tuple we associate a tensor product of simple g-modules and we show that for k=2 the dimension increases also along with the extended partial order, generalizing a theorem proved in the aforementioned paper. We also show that the tensor product associated to the maximal element has the biggest dimension among all tuples for arbitrary k, indicating that this might be a symplectic (resp. orthogonal) analogon of the row shuffle defined by Fomin et al. The extension of the partial order reduces the number of elements in the cover relation and may facilitate the proof of an analogon of Schur positivity along the partial order for symplectic and orthogonal types.Comment: 16 pages, final version, to appear in AJo

Perfect Prediction in Minkowski Spacetime: Perfectly Transparent Equilibrium for Dynamic Games with Imperfect Information

The assumptions of necessary rationality and necessary knowledge of strategies, also known as perfect prediction, lead to at most one surviving outcome, immune to the knowledge that the players have of them. Solutions concepts implementing this approach have been defined on both dynamic games with perfect information and no ties, the Perfect Prediction Equilibrium, and strategic games with no ties, the Perfectly Transparent Equilibrium. In this paper, we generalize the Perfectly Transparent Equilibrium to games in extensive form with imperfect information and no ties. Both the Perfect Prediction Equilibrium and the Perfectly Transparent Equilibrium for strategic games become special cases of this generalized equilibrium concept. The generalized equilibrium, if there are no ties in the payoffs, is at most unique, and is Pareto-optimal. We also contribute a special-relativistic interpretation of a subclass of the games in extensive form with imperfect information as a directed acyclic graph of decisions made by any number of agents, each decision being located at a specific position in Minkowski spacetime, and the information sets and game structure being derived from the causal structure. Strategic games correspond to a setup with only spacelike-separated decisions, and dynamic games to one with only timelike-separated decisions. The generalized Perfectly Transparent Equilibrium thus characterizes the outcome and payoffs reached in a general setup where decisions can be located in any generic positions in Minkowski spacetime, under necessary rationality and necessary knowledge of strategies. We also argue that this provides a directly usable mathematical framework for the design of extension theories of quantum physics with a weakened free choice assumption.Comment: 25 pages, updated technical repor

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for $\mathfrak{sl}_{n+1}$. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.Comment: 17 page