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- 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.
- ItemYet another algorithm for the symmetric eigenvalue problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known implementations in terms of both speed and accuracy.
- ItemRepresentation theory of imprimitive of non-commutative association schemes of order(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Hanaki, Akihide; Zieschang, Paul-HermannIn this note, we investigate association schemes of order 6. We prove that non-normal closed subsets of such schemes have order 2 and that normal closed subsets of non-commutative schemes have order 2 or 3. After that, we investigate more closely schemes of order 6 which possess non-normal closed subsets and non-commutative schemes of order 6 which possess a symmetric normal closed subset of order 3. (Non-commutative schemes of order 6 which possess a non-symmetric normal closed subset of order 3 or a normal closed subset of order 2 will be investigated in a forthcoming article.) In both cases, we explicitly give all irreducible (complex) representations of such schemes. Among other structural consequences we obtain that association schemes of order 6 are Coxeter schemes if they have two di erent non-normal closed subsets and that they are semidirect products if they possess a normal and a non-normal closed subset or a thin non-normal closed subset.
- 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.
- ItemAbundance of 3-planes on real projective hypersurfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Finashin, S.; Kharlamov, V.We show that a generic real projective n-dimensional hypersurface of odd degree d , such that 4(n - 2) = (d + 3 3), contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, d3 log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
- ItemPlethysms, replicated Schur functions, and series, with applications to vertex operators(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Fauser, Bertfried; Jarvis, Peter D.; King, Ronald C.Specializations of Schur functions are exploited to define and evaluate the Schur functions sλ[αX] and plethysms sλ[αsν(X))] for any α-integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, Mπ and Lπ, specified by arbitrary partitions π. These are used in turn to define and provide generating functions for formal characters, s(π)λ, of certain groups Hπ, thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the series M=M(0) and various L⊥σ dual to Lσ, and then more explicitly in exponential form. Finally the replicated form of such vertex operators are written down.
- ItemThe algebra of differential operators for a Gegenbauer weight matrix(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Zurrián, Ignacio NahuelIn this work we study in detail the algebra of differential operators D(W) associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representation theory.
- 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.
- ItemThe initial and terminal cluster sets of an analytic curve(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Gauthier, PaulFor an analytic curve γ:(a,b)→C, the set of values approaches by γ(t), as t↘a and as t↗b can be any two continuua of C∪{∞}.
- ItemObservability of systems with delay convoluted observation(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Ivanov, Anatoli F.; Verriest, Erik I.This paper analyzes finite dimensional linear time-invariant systems with observation of a delay, where that delay satisfies a particular implicit relation with the state variables, rendering the entire problem nonlinear. The objective is to retrieve the state variables from the measured delay. The first contribution involves the direct inversion of the delay, the second is the design of a finite dimensional state observer, and the third involves the derivation of certain properties of the delay - state relation. Realistic examples treat vehicles with ultrasonic position sensors