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- ItemOn reflection subgroups of finite Coxeter groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, GerhardLet W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.
- ItemUnconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Djakov, Plamen; Mityagin, Boris[no abstract available]
- ItemAn inductive approach to Coxeter arrangements and Solomon’s descent algebra(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Douglass, J.Matthew; Pfeiffer, Götz; Röhrle, GerhardIn our recent paper [3], we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for nite Coxeter groups of rank up to 2.
- ItemCrystal energy functions via the charge in types A and C(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Lenart, Cristian; Schilling, AnneThe Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.
- ItemDefinable orthogonality classes in accessible categories are small(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Bagaria, Joan; Casacuberta, Carles; Mathias, A.R.D.; Rosický, JiríWe lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class S of morphisms in an accessible category C, the orthogonal class of objects S⊥ is a small-orthogonality class (hence reflective, if C is cocomplete) is provable in ZFC if S is Σ1, while it follows from the existence of a proper class of supercompact cardinals if S is Σ2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is Σn+2 for n≥1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, follows from the existence of sufficiently large supercompact cadianls, since E∗-equivalences are Σ2-definable for every cohomology theory E∗. On the other hand, E∗-equivalences are Σ1-definable, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.
- ItemStochastic mean payoff games: Smoothed analysis and approximation schemes(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Boros, Endre; Elbassioni, Khaled; Fouz, Mahmoud; Gurvich, Vladimir; Makino, Kazuhisa; Manthey, BodoWe consider two-person zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question whether a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (i.e., in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudo-polynomial algorithm for BWR-games with constant number of random vertices implies smoothed polynomial time complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial time complexity and derive absolute and relative approximation schemes for the above two classes.
- ItemVector bundles on degenerations of elliptic curves and Yang-Baxter equations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Burban, Igor; Kreussler, BerndIn this paper we introduce the notion of a gemetric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.
- ItemUpper tails for intersection local times of random walks in supercritical dimensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Chen, Xia; Mörters, PeterWe determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in Zd under the assumption p(d−2)>d. Our approach allows a direct treatment of the infinite time horizon.
- ItemStratifying modular representations of finite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Benson, Dave; Iyengar, Srikanth B.; Krause, HenningWe classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.
- ItemHölder-differentiability of Gibbs distribution functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Kesseböhmer, Marc; Stratmann, Bernd O.In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil’s staircases) supported on limit sets of finitely generated conformal iterated function systems in R. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not a-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.