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Pavel Etingof - One of the best experts on this subject based on the ideXlab platform.
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Finite Symmetric Tensor categories with the Chevalley property in characteristic 2
Journal of Algebra and Its Applications, 2020Co-Authors: Pavel Etingof, Shlomo GelakiAbstract:We prove an analog of Deligne’s theorem for finite Symmetric Tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a Symmetric fiber functor to the Symmetric Tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is Symmetric Tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].
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on the frobenius functor for Symmetric Tensor categories in positive characteristic
arXiv: Representation Theory, 2019Co-Authors: Pavel Etingof, Victor OstrikAbstract:We develop a theory of Frobenius functors for Symmetric Tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear Symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ is finite and semisimple then it admits a fiber functor to ${\rm Ver}_p$. The main new feature is that when $\mathcal{C}$ is not semisimple, $F$ need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor $\mathcal{C}\to {\rm Ver}_p$. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of $F$, and use it to show that for categories with finitely many simple objects $F$ does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which $F$ is exact, and define the canonical maximal Frobenius exact subcategory $\mathcal{C}_{\rm ex}$ inside any STC $\mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by $F$. We prove that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to ${\rm Ver}_p$. We also show that a sufficiently large power of $F$ lands in $\mathcal{C}_{\rm ex}$. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category and show that a STC with Chevalley property is (almost) Frobenius exact.
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Symmetric Tensor categories in characteristic 2
Advances in Mathematics, 2019Co-Authors: Dave Benson, Pavel EtingofAbstract:Abstract We construct and study a nested sequence of finite Symmetric Tensor categories Vec = C 0 ⊂ C 1 ⊂ ⋯ ⊂ C n ⊂ ⋯ over a field of characteristic 2 such that C 2 n are incompressible, i.e., do not admit Tensor functors into Tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C 1 described by Venkatesh [28] and the category C 2 defined by Ostrik. The Grothendieck rings of the categories C 2 n and C 2 n + 1 are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2 n + 2 -th root of unity, O n = Z [ 2 cos ( π / 2 n + 1 ) ] .
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Finite Symmetric Tensor categories with the Chevalley property in characteristic $2$
arXiv: Quantum Algebra, 2019Co-Authors: Pavel Etingof, Shlomo GelakiAbstract:We prove an analog of Deligne's theorem for finite Symmetric Tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$ admits a Symmetric fiber functor to the Symmetric Tensor category $\mathcal{D}$ of representations of the triangular Hopf algebra $(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. Equivalently, we prove that there exists a unique finite group scheme $G$ in $\mathcal{D}$ such that $\mathcal{C}$ is Symmetric Tensor equivalent to $\Rep_{\mathcal{D}}(G)$. Finally, we compute the group $H^2_{\rm inv}(A,K)$ of equivalence classes of twists for the group algebra $K[A]$ of a finite abelian $p$-group $A$ over an arbitrary field $K$ of characteristic $p>0$, and the Sweedler cohomology groups $H^i_{\rm{Sw}}(\mathcal{O}(A),K)$, $i\ge 1$, of the function algebra $\mathcal{O}(A)$ of $A$.
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finite Symmetric Tensor categories with the chevalley property in characteristic 2
arXiv: Quantum Algebra, 2019Co-Authors: Pavel Etingof, Shlomo GelakiAbstract:We prove an analog of Deligne's theorem for finite Symmetric Tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$ admits a Symmetric fiber functor to the Symmetric Tensor category $\mathcal{D}$ of representations of the triangular Hopf algebra $(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. Equivalently, we prove that there exists a unique finite group scheme $G$ in $\mathcal{D}$ such that $\mathcal{C}$ is Symmetric Tensor equivalent to $\Rep_{\mathcal{D}}(G)$. Finally, we compute the group $H^2_{\rm inv}(A,K)$ of equivalence classes of twists for the group algebra $K[A]$ of a finite abelian $p$-group $A$ over an arbitrary field $K$ of characteristic $p>0$, and the Sweedler cohomology groups $H^i_{\rm{Sw}}(\mathcal{O}(A),K)$, $i\ge 1$, of the function algebra $\mathcal{O}(A)$ of $A$.
Eugene Zhang - One of the best experts on this subject based on the ideXlab platform.
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robust and fast extraction of 3d Symmetric Tensor field topology
IEEE Transactions on Visualization and Computer Graphics, 2019Co-Authors: Lawrence Roy, Prashant Kumar, Yue Zhang, Eugene ZhangAbstract:3D Symmetric Tensor fields appear in many science and engineering fields, and topology-driven analysis is important in many of these application domains, such as solid mechanics and fluid dynamics. Degenerate curves and neutral surfaces are important topological features in 3D Symmetric Tensor fields. Existing methods to extract degenerate curves and neutral surfaces often miss parts of the curves and surfaces, respectively. Moreover, these methods are computationally expensive due to the lack of knowledge of structures of degenerate curves and neutral surfaces. In this paper, we provide theoretical analysis on the geometric and topological structures of degenerate curves and neutral surfaces of 3D linear Tensor fields. These structures lead to parameterizations for degenerate curves and neutral surfaces that can not only provide more robust extraction of these features but also incur less computational cost. We demonstrate the benefits of our approach by applying our degenerate curve and neutral surface detection techniques to solid mechanics simulation data sets.
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feature surfaces in Symmetric Tensor fields based on eigenvalue manifold
IEEE Transactions on Visualization and Computer Graphics, 2016Co-Authors: Jonathan Palacios, Yue Zhang, Harry Yeh, Wenping Wang, Robert S Laramee, Ritesh Sharma, Thomas Schultz, Eugene ZhangAbstract:Three-dimensional Symmetric Tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D Symmetric Tensor fields focus on degenerate Tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces , into Tensor field analysis, based on the notion of eigenvalue manifold . Neutral surfaces are the boundary between linear Tensors and planar Tensors, and the traceless surfaces are the boundary between Tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete Tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of Tensor modes, Tensor isotropy, and Tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches , to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.
Q.-s. Zheng - One of the best experts on this subject based on the ideXlab platform.
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On the representations for isotropic vector-valued, Symmetric Tensor-valued and skew-Symmetric Tensor-valued functions
International Journal of Engineering Science, 1993Co-Authors: Q.-s. ZhengAbstract:Abstract In this paper, we give a new derivation procedure for determining the representations for 3- and 2-dimensional isotropic vector-valued, Symmetric Tensor-valued and skew-Symmetric Tensorvalued functions of vectors, Symmetric Tensors and skew-Symmetric Tensors. Simplified representations for isotropic scalar-valued and vector-valued functions are suggested.
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On transversely isotropic, orthotropic and relative isotropic functions of Symmetric Tensors, skew-Symmetric Tensors and vectors. Part V: The irreducibility of the representations for three dimensional orthotropic functions and the summary
International Journal of Engineering Science, 1993Co-Authors: Q.-s. ZhengAbstract:Abstract In this part, the irreducibility of the representations derived in Part IV for three dimensional orthotropic scalar-valued, vector-valued, Symmetric Tensor-valued and skew-Symmetric Tensor-valued functions of Symmetric Tensors, skew-Symmetric Tensors and vectors is proved. Finally, a summary for all of this five-part paper is also given.
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On transversely isotropic, orthotropic and relative isotropic functions of Symmetric Tensors, skew-Symmetric Tensors and vectors. Part I: Two dimensional orthotropic and relative isotropic functions and three dimensional relative isotropic functions
International Journal of Engineering Science, 1993Co-Authors: Q.-s. ZhengAbstract:Abstract In this part, we derive the complete and irreducible representations for two dimensional orthotropic functions and two and three dimensional relative isotropic (i.e. the symmetry group is the proper orthogonal Tensor group) functions of Symmetric Tensors, skew-Symmetric Tensors and vectors. The functions may be any general (not only polynomial) scalar-valued, vector-valued, Symmetric Tensor-valued and skew-Symmetric Tensor-valued ones. The complete representations for three dimensional transversely isotropic and orthotropic functions are derived in Parts II and IV respectively and their irreducibility is proved in Parts III and V respectively.
Jiawang Nie - One of the best experts on this subject based on the ideXlab platform.
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Symmetric Tensor Decompositions On Varieties.
arXiv: Numerical Analysis, 2020Co-Authors: Jiawang Nie, Lihong ZhiAbstract:This paper discusses the problem of Symmetric Tensor decomposition on a given variety $X$: decomposing a Symmetric Tensor into the sum of Tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable Tensors, which are crucial to the practical computations of such decompositions. For a given Tensor, we also develop a criterion for the existence of a Symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing Symmetric Tensor decompositions on an arbitrary $X$. As a specific application, Vandermonde decompositions for nonSymmetric Tensors can be computed by the proposed algorithm.
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Symmetric Tensor Nuclear Norms
SIAM Journal on Applied Algebra and Geometry, 2017Co-Authors: Jiawang NieAbstract:This paper studies Symmetric Tensor nuclear norms. As shown by Friedland and Lim, the nuclear norm of a Symmetric Tensor can be achieved at a Symmetric decomposition. We discuss how to compute Symmetric Tensor nuclear norms, depending on the order and ground field. Lasserre relaxations are proposed for the computation. Theoretical properties of the relaxations are studied. For Symmetric Tensors, we can compute their nuclear norms, as well as the Symmetric nuclear decompositions. The methods can be extended to nonSymmetric Tensors.
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Low Rank Symmetric Tensor Approximations
SIAM Journal on Matrix Analysis and Applications, 2017Co-Authors: Jiawang NieAbstract:For a given Symmetric Tensor, we aim at finding a new one whose Symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank Symmetric Tensors. Such linear relations can be expressed by polynomials, which are called generating polynomials. We propose a new approach for computing low rank approximations by using generating polynomials. First, we estimate a set of generating polynomials that are approximately satisfied by the given Tensor. Second, we find approximate common zeros of these polynomials. Third, we use these zeros to construct low rank Tensor approximations. If the Symmetric Tensor to be approximated is sufficiently close to a low rank one, we show that the computed low rank approximations are quasi-optimal.
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Symmetric Tensor Nuclear Norms
arXiv: Optimization and Control, 2016Co-Authors: Jiawang NieAbstract:This paper studies nuclear norms of Symmetric Tensors. As recently shown by Friedland and Lim, the nuclear norm of a Symmetric Tensor can be achieved at a Symmetric decomposition. We discuss how to compute Symmetric Tensor nuclear norms, depending on the Tensor order and the ground field. Lasserre relaxations are proposed for the computation. The theoretical properties of the relaxations are studied. For Symmetric Tensors, we can compute their nuclear norms, as well as the nuclear decompositions. The proposed methods can be extended to nonSymmetric Tensors.
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Generating Polynomials and Symmetric Tensor Decompositions
Foundations of Computational Mathematics, 2015Co-Authors: Jiawang NieAbstract:This paper studies Symmetric Tensor decompositions. For Symmetric Tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the Tensor. A Symmetric Tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a Symmetric Tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing Symmetric Tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.
Yue Zhang - One of the best experts on this subject based on the ideXlab platform.
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robust and fast extraction of 3d Symmetric Tensor field topology
IEEE Transactions on Visualization and Computer Graphics, 2019Co-Authors: Lawrence Roy, Prashant Kumar, Yue Zhang, Eugene ZhangAbstract:3D Symmetric Tensor fields appear in many science and engineering fields, and topology-driven analysis is important in many of these application domains, such as solid mechanics and fluid dynamics. Degenerate curves and neutral surfaces are important topological features in 3D Symmetric Tensor fields. Existing methods to extract degenerate curves and neutral surfaces often miss parts of the curves and surfaces, respectively. Moreover, these methods are computationally expensive due to the lack of knowledge of structures of degenerate curves and neutral surfaces. In this paper, we provide theoretical analysis on the geometric and topological structures of degenerate curves and neutral surfaces of 3D linear Tensor fields. These structures lead to parameterizations for degenerate curves and neutral surfaces that can not only provide more robust extraction of these features but also incur less computational cost. We demonstrate the benefits of our approach by applying our degenerate curve and neutral surface detection techniques to solid mechanics simulation data sets.
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feature surfaces in Symmetric Tensor fields based on eigenvalue manifold
IEEE Transactions on Visualization and Computer Graphics, 2016Co-Authors: Jonathan Palacios, Yue Zhang, Harry Yeh, Wenping Wang, Robert S Laramee, Ritesh Sharma, Thomas Schultz, Eugene ZhangAbstract:Three-dimensional Symmetric Tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D Symmetric Tensor fields focus on degenerate Tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces , into Tensor field analysis, based on the notion of eigenvalue manifold . Neutral surfaces are the boundary between linear Tensors and planar Tensors, and the traceless surfaces are the boundary between Tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete Tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of Tensor modes, Tensor isotropy, and Tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches , to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.
