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323
Permutations of strongly selfabsorbing C∗algebras
 TRANS. AM. MATH. SOC
"... Let G be a finite group acting on {1,..., n}. For any C∗algebra A, this defines an action of α of G on A⊗n. We show that if A tensorially absorbs a UHF algebra of infinite type, the JiangSu algebra, or is approximately divisible, then A×α G has the corresponding property as well. ..."
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Cited by 67 (16 self)
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Let G be a finite group acting on {1,..., n}. For any C∗algebra A, this defines an action of α of G on A⊗n. We show that if A tensorially absorbs a UHF algebra of infinite type, the JiangSu algebra, or is approximately divisible, then A×α G has the corresponding property as well.
Finitedimensional representations of rational Cherednik algebras
 MR MR1961261 (2004h:16027
, 2003
"... Abstract. We study lowest weight representations of the rational Cherednik algebra attached to a complex reflection group W. In particular, we generalize a number of previous results due to Berest, Etingof, and Ginzburg. 1. ..."
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Cited by 66 (7 self)
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Abstract. We study lowest weight representations of the rational Cherednik algebra attached to a complex reflection group W. In particular, we generalize a number of previous results due to Berest, Etingof, and Ginzburg. 1.
Heterotic compactification, an algorithmic approach
 arXiv:hepth/0702210. – “Complete Intersections, Monads and Heterotic Compactification
"... We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection CalabiYau manifolds in a s ..."
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Cited by 44 (26 self)
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We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection CalabiYau manifolds in a single projective space where we classify positive monad bundles. Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets. In particular, we find that the number of antigenerations vanishes
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 35 (5 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
FImodules: a new approach to stability for Snrepresentations
, 2012
"... In this paper we introduce and develop the theory of FImodules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of ordered ntuples on an arbitrary manifold • the diagonal coinvariant algebra on r sets of n variables • the cohomology and tautological r ..."
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Cited by 28 (4 self)
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In this paper we introduce and develop the theory of FImodules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of ordered ntuples on an arbitrary manifold • the diagonal coinvariant algebra on r sets of n variables • the cohomology and tautological ring of the moduli space of npointed curves • the space of polynomials on rank varieties of n × n matrices • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 and more. The symmetric group Sn acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cyclecounting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. FImodules are a refinement of Church–Farb’s theory of representation stability for Snrepresentations. In this framework, a complicated sequence of Snrepresentations becomes a
EFFECTIVE ALGEBRAIC DEGENERACY
, 2009
"... We show that for every generic smooth projective hypersurface X ⊂ P n+1, n � 2, there exists a proper algebraic subvariety Y � X such that every nonconstant entire holomorphic curve f: C → X has image f(C) which lies in Y, provided deg X � 2 n5 ..."
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Cited by 26 (7 self)
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We show that for every generic smooth projective hypersurface X ⊂ P n+1, n � 2, there exists a proper algebraic subvariety Y � X such that every nonconstant entire holomorphic curve f: C → X has image f(C) which lies in Y, provided deg X � 2 n5
Preprojective algebras and cluster algebras
, 2008
"... We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups. ..."
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Cited by 24 (0 self)
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We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
Representation theory and homological stability
, 2010
"... We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. One main goal is to expand the important and wellstudied concept of homological stability so that it applies to a much broader variety of examples. Representation stability ..."
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Cited by 23 (3 self)
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We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. One main goal is to expand the important and wellstudied concept of homological stability so that it applies to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1) n−1 conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in