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280
Symmetry Groups, Semidefinite Programs, and Sums of Squares
, 2002
"... We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete symmetries in semidefinite programs. It is shown that symmetr ..."
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Cited by 62 (7 self)
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We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete symmetries in semidefinite programs. It is shown that symmetry exploitation allows a significant reduction in both matrix size and number of decision variables. This result is applied to semidefinite programs arising from the computation of sum of squares decompositions for multivariate polynomials. The results, reinterpreted from an invarianttheoretic viewpoint, provide a novel representation of a class of nonnegative symmetric polynomials. The main theorem states that an invariant sum of squares polynomial is a sum of inner products of pairs of matrices, whose entries are invariant polynomials. In these pairs, one of the matrices is computed based on the real irreducible representations of the group, and the other is a sum of squares matrix. The reduction techniques enable the numerical solution of largescale instances, otherwise computationally infeasible to solve.
On the toric algebra of graphical models
, 2006
"... We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of con ..."
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Cited by 36 (6 self)
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We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley–Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
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Cited by 32 (0 self)
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We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
Counting gauge invariants: The Plethystic program
 JHEP 0703 (2007) 090 hepth/0701063
"... We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singulari ..."
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Cited by 30 (11 self)
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We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of HardyRamanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the
Multigraded CastelnuovoMumford Regularity
 J. REINE ANGEW. MATH
, 2003
"... We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric ..."
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Cited by 29 (8 self)
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We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric
Algebraic invariants for rightangled Artin groups
 Math. Ann., posted on Dec
"... Abstract. A finite simplicial graph Γ determines a rightangled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation vw = wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of ..."
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Cited by 27 (17 self)
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Abstract. A finite simplicial graph Γ determines a rightangled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation vw = wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of GΓ, directly from the graph Γ. 1.
Permutation branes and linear matrix factorisations
, 2005
"... All the known rational boundary states for Gepner models can be regarded as permutation branes. On general grounds, one expects that topological branes in Gepner models can be encoded as matrix factorisations of the corresponding LandauGinzburg potentials. In this paper we identify the matrix facto ..."
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Cited by 24 (2 self)
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All the known rational boundary states for Gepner models can be regarded as permutation branes. On general grounds, one expects that topological branes in Gepner models can be encoded as matrix factorisations of the corresponding LandauGinzburg potentials. In this paper we identify the matrix factorisations associated to arbitrary Btype permutation branes.
On the rank of a tropical matrix
"... Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise ..."
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Cited by 24 (5 self)
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Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed. 1.
Momentangle complexes, monomial ideals, and Massey products
 Pure and Applied Math. Quarterly
"... Abstract. Associated to every finite simplicial complex K there is a “momentangle” finite CWcomplex, ZK; if K is a triangulation of a sphere, ZK is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Mas ..."
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Cited by 21 (6 self)
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Abstract. Associated to every finite simplicial complex K there is a “momentangle” finite CWcomplex, ZK; if K is a triangulation of a sphere, ZK is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a momentangle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of nonformal manifolds and subspace arrangements are given. Contents
Toric ideals, real toric varieties, and the moment map
 In Topics in algebraic geometry and geometric modeling, volume 334 of Contemp. Math
, 2003
"... This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the moment map. In particular, we explain the relation between linear precision and the moment ma ..."
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Cited by 21 (7 self)
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This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the moment map. In particular, we explain the relation between linear precision and the moment map.