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197
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 22 (3 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.
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 16 (10 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
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 16 (5 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.
THE MODULARITY CONJECTURE FOR RIGID CALABI–YAU Threefolds Over Q
, 2000
"... We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A3. Our proof is based on geometric analysis. ..."
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Cited by 15 (1 self)
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We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A3. Our proof is based on geometric analysis.
Vertices of GelfandTsetlin polytopes
 DISCRETE COMPUT. GEOM
, 2008
"... This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowestdimensional face containing a given ..."
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Cited by 14 (5 self)
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This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowestdimensional face containing a given GelfandTsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov [1] about the integrality of all vertices of the GelfandTsetlin polytopes. We can construct for each n ≥ 5 a counterexample, with arbitrarily increasing denominators as n grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when n is fixed.
Extraspecial 2groups and images of braid group representations
 J. Knot Theory Ramifications
"... Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompo ..."
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Cited by 13 (6 self)
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Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the YangBaxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the wellknown Jones representations of Bn factoring over TemperleyLieb algebras and the corresponding link invariants. 1.
The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type.
, 2003
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Casimir Operators and Monodromy Representations of Generalised Braid Groups
, 2003
"... Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group ..."
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Cited by 13 (6 self)
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Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group Bg of type g is a deformation of the action of (a finite extension of) W on V. The residues of ∇κ are the Casimirs κα of the subalgebras sl α 2 ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊆ V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g–modules if g = sl3 but that this is not the case if g ≇ sl2, sl3. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group Bn on the zero weight spaces of all simple U�sln–modules for n ≥ 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self–dual g–modules and obtain complete classification results for g = sln and g2 and conjecturally complete results for g orthogonal or symplectic.
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 12 (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.
Combinatorial representation theory
 in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38
, 1999
"... Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when ..."
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Cited by 12 (0 self)
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Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.