9,488 research outputs found

    Property (T) and rigidity for actions on Banach spaces

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    We study property (T) and the fixed point property for actions on LpL^p and other Banach spaces. We show that property (T) holds when L2L^2 is replaced by LpL^p (and even a subspace/quotient of LpL^p), and that in fact it is independent of 1≤p<∞1\leq p<\infty. We show that the fixed point property for LpL^p follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for LpL^p holds for any 1<p<∞1< p<\infty if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement

    Affine Buildings and Tropical Convexity

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    The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SLd(K)_d(K) and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure

    On p-adic lattices and Grassmannians

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    It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math. Zeitschrift, of the previously published preprint "On p-adic loop groups and Grassmannians

    Brane Realizations of Quantum Hall Solitons and Kac-Moody Lie Algebras

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    Using quiver gauge theories in (1+2)-dimensions, we give brane realizations of a class of Quantum Hall Solitons (QHS) embedded in Type IIA superstring on the ALE spaces with exotic singularities. These systems are obtained by considering two sets of wrapped D4-branes on 2-spheres. The space-time on which the QHS live is identified with the world-volume of D4-branes wrapped on a collection of intersecting 2-spheres arranged as extended Dynkin diagrams of Kac-Moody Lie algebras. The magnetic source is given by an extra orthogonal D4-brane wrapping a generic 2-cycle in the ALE spaces. It is shown as well that data on the representations of Kac-Moody Lie algebras fix the filling factor of the QHS. In case of finite Dynkin diagrams, we recover results on QHS with integer and fractional filling factors known in the literature. In case of hyperbolic bilayer models, we obtain amongst others filling factors describing holes in the graphene.Comment: Lqtex; 15 page

    Demazure resolutions as varieties of lattices with infinitesimal structure

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    Let k be a field of positive characteristic. We construct, for each dominant coweight \lambda of the standard maximal torus in the special linear group, a closed subvariety D(\lambda) of the multigraded Hilbert scheme of an affine space over k, such that the k-valued points of D(\lambda) can be interpreted as lattices in k((z))^n endowed with infinitesimal structure. Moreover, for any \lambda we construct a universal homeomorphism from D(\lambda) to a Demazure resolution of the Schubert variety associated with \lambda in the affine Grassmannian. Lattices in D(\lambda) have non-trivial infinitesimal structure if and only if they lie over the boundary of the big cell.Comment: 24 pages, added the missing bibliograph
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