The Experts below are selected from a list of 231 Experts worldwide ranked by ideXlab platform
Jaehyun Yang - One of the best experts on this subject based on the ideXlab platform.
-
invariant metrics and laplacians on siegel jacobi space
Journal of Number Theory, 2007Co-Authors: Jaehyun YangAbstract:Abstract In this paper, we compute Riemannian metrics on the Siegel–Jacobi space which are invariant under the natural action of the Jacobi group explicitly and also provide the Laplacians of these invariant metrics. These are expressed in terms of the Trace Form.
-
Invariant metrics and Laplacians on Siegel–Jacobi space☆
Journal of Number Theory, 2007Co-Authors: Jaehyun YangAbstract:Abstract In this paper, we compute Riemannian metrics on the Siegel–Jacobi space which are invariant under the natural action of the Jacobi group explicitly and also provide the Laplacians of these invariant metrics. These are expressed in terms of the Trace Form.
Alexander Premet - One of the best experts on this subject based on the ideXlab platform.
-
Vanishing of Trace Forms in low characteristics
Algebra & Number Theory, 2009Co-Authors: Skip Garibaldi, Alexander PremetAbstract:A finite-dimensional representation of an algebraic group G gives a Trace symmetric bilinear Form on the Lie algebra of G. We give a criterion in terms of the root system data for this Form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called “quotient Trace Form”, answering a question posed in the 1960s. Let G be an algebraic group over a field F , acting on a finite-dimensional vector space V via a homomorphism ρ : G → GL(V ). The differential dρ of ρ maps the Lie algebra Lie(G) of G into gl(V ), and we put Trρ for the symmetric bilinear Form Trρ(x, y) := Trace(dρ(x) dρ(y)) for x, y ∈ Lie(G). We call Trρ a Trace Form of G. Such Forms appear, for example, in the hypotheses for the Jacobson-Morozov Theorem [Ca, 5.3.1]. We prove: Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent: (a) The characteristic of F is a torsion prime for G. (b) Every Trace Form of G is zero. The set of torsion primes for G is given by the following table, cf. e.g. [St 75, 1.13]: type of G torsion primes An, Cn none Bn (n ≥ 3), Dn (n ≥ 4), G2 2 F4, E6, E7 2, 3 E8 2, 3, 5 A prime p is called a torsion prime for G if the corresponding group G(C) over C (or, equivalently, its compact Form) is such that one of its homology groups, with coefficients in Z, contains an element of order p. We also prove a generalization of Theorem A that removes the hypotheses “simply connected” and “split”; it is somewhat more complicated, so we leave the statement until Th. D (and Remark 4.6). Replacing the simply connected group G with a nontrivial quotient G changes the situation in two ways: the group G has “fewer” representations and the Lie algebras of G and G may be different. These two changes are reflected in the integers N(G) and E(G) defined below. As a particular example of Th. A, for G of type E8 over a field of characteristic 2, 3, or 5, Trρ is zero for every representation ρ of G. One may ask whether the same is true for the representations of the Lie algebra Lie(G). That is, for a representation ψ of Lie(G), we write Trψ for the bilinear Form (x, y) 7→ Trace(ψ(x)ψ(y)), and ask 2000 Mathematics Subject Classification. 20G05 (17B50, 17B25).
-
Vanishing of Trace Forms in low characteristics
arXiv: Representation Theory, 2007Co-Authors: Skip Garibaldi, Alexander PremetAbstract:Every finite-dimensional representation of an algebraic group G gives a Trace symmetric bilinear Form on the Lie algebra of G. We give criteria in terms of root system data for the existence of a representation such that this Form is nonzero or nondegenerate. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called "quotient Trace Form", answering a question posed in the 1960s.
-
Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture
Inventiones mathematicae, 1995Co-Authors: Alexander PremetAbstract:Let g be the Lie algebra of a connected reductive group G over an algebraically closed field of characteristic p >0. Suppose that G ^(1) is simply connected and p is good for the root system of G . If p =2, suppose in addition that g admits a nondegenerate G -invariant Trace Form. Let V be an irreducible and faithful g -module with p -character χ∈ g ^*. It is proved in the paper that dim V is divisible by p ^1/2dimΩ(χ) where Ω(χ) stands for the orbit of χ under the coadjoint action of G .
Guillermo Mantilla-soler - One of the best experts on this subject based on the ideXlab platform.
-
The spinor genus of the integral Trace
Transactions of the American Mathematical Society, 2016Co-Authors: Guillermo Mantilla-solerAbstract:Let K be a number field of degree at least 3. In this article we show that the genus of the integral Trace Form of K contains only one spinor genus. Additionally we show that exactly 43% (resp. 29%, resp. 58%) of quadratic (resp. real quadratic, resp. imaginary quadratic) fields have the same property. Introduction Let K be a number field. The rational quadratic Form x 7→ trK/Q(x) has been extensively studied by several authors, see for example [Ba], [BaLe], [C-P], [Ep], [Ga], [Mau], [MiRe] and [S]. For arithmetic purposes, a finer invariant of a number field is its integral Trace Form i.e., the integral quadratic Form obtained by restricting trK/Q(x ) to the maximal order in K. Recent applications of the integral Trace Form can be found in the work of Bhargava and Shnidman (see [BhSh]) where they count cubic orders using the shape, an invariant closely related to the integral Trace Form. Some other applications of the integral Trace on cubic fields can be found in [Man] and [Man2]. Given a non-degenerate integral quadratic Form q, it is of great interest to study the number of spinor classes on its genus. This is explained by the famous result of Eichler [Eich] which says that for an indefinite Form q of dimension of at least 3 the spinor genus and the isometry class coincide. In this article we analyze the spinor genus of the integral Trace Form of a number field. Our main theorem is the following: Theorem (cf.Theorem 2.12). Let K be a non-quadratic number field. Then, the genus of the integral Trace Form of K contains only one proper spinor genus. In the case of quadratic fields it is not necessary true that the genus and the proper spinor genus of the integral Trace coincide, however they agree and differ
-
ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL Trace FormS
International Journal of Number Theory, 2012Co-Authors: Guillermo Mantilla-solerAbstract:Let K be a number field. The integral Trace Form is the integral quadratic Form given by trk/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral Trace Forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral Trace Forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent Trace zero Forms if and only if they are conjugated.
-
Integral Trace Forms associated to cubic extensions
arXiv: Number Theory, 2011Co-Authors: Guillermo Mantilla-solerAbstract:Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the Trace Form $q_K:\text{tr}_{K/\mathbb{Q}}(x^2)|_{O^{0}_{K}}$ as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element $T_K$ in Bhargava's class group $\Cl(\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}; -3d)$ such that $q_K$ is completely determined by $T_K$. By using one of Bhargava's composition laws, we show that $q_K$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant
-
On number fields with equivalent integral Trace Forms
arXiv: Number Theory, 2011Co-Authors: Guillermo Mantilla-solerAbstract:Let $K$ be a number field. The \textit{integral Trace Form} is the integral quadratic Form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral Trace Forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral Trace Forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent Trace zero Forms if and only if they are conjugated.
Thomas Huang - One of the best experts on this subject based on the ideXlab platform.
-
Trace Ratio vs. Ratio Trace for Dimensionality Reduction
2007 IEEE Conference on Computer Vision and Pattern Recognition, 2007Co-Authors: Huan Wang, Dong Xu, Xiaoou Tang, Thomas HuangAbstract:A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the Form of arg maxW Tr(WT SPW)/Tr(WT SIW)1, which is generally transFormed into the corresponding Ratio Trace Form arg maxW Tr[ (WTSIW)-1 (WTSPW) ] for obtaining a closed-Form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio problem. In each step, a Trace Difference problem arg maxW Tr [WT (SP - lambdaSI) W] is solved with lambda being the Trace ratio value computed from the previous step. Convergence of the projection matrix W, as well as the global optimum of the Trace ratio value lambda, are proven based on point-to-set map theories. In addition, this procedure is further extended for solving Trace ratio problems with more general constraint WTCW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed solution, as well as its superiority in classification capability over corresponding solutions to the ratio Trace problem.
Huan Wang - One of the best experts on this subject based on the ideXlab platform.
-
Trace Ratio vs. Ratio Trace for Dimensionality Reduction
2007 IEEE Conference on Computer Vision and Pattern Recognition, 2007Co-Authors: Huan Wang, Dong Xu, Xiaoou Tang, Thomas HuangAbstract:A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the Form of arg maxW Tr(WT SPW)/Tr(WT SIW)1, which is generally transFormed into the corresponding Ratio Trace Form arg maxW Tr[ (WTSIW)-1 (WTSPW) ] for obtaining a closed-Form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio problem. In each step, a Trace Difference problem arg maxW Tr [WT (SP - lambdaSI) W] is solved with lambda being the Trace ratio value computed from the previous step. Convergence of the projection matrix W, as well as the global optimum of the Trace ratio value lambda, are proven based on point-to-set map theories. In addition, this procedure is further extended for solving Trace ratio problems with more general constraint WTCW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed solution, as well as its superiority in classification capability over corresponding solutions to the ratio Trace problem.
-
CVPR - Trace Ratio vs. Ratio Trace for Dimensionality Reduction
2007 IEEE Conference on Computer Vision and Pattern Recognition, 2007Co-Authors: Huan Wang, Dong Xu, Xiaoou Tang, Thomas S. HuangAbstract:A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the Form of arg maxW Tr(WT SPW)/Tr(WT SIW)1, which is generally transFormed into the corresponding Ratio Trace Form arg maxW Tr[ (WTSIW)-1 (WTSPW) ] for obtaining a closed-Form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio problem. In each step, a Trace Difference problem arg maxW Tr [WT (SP - lambdaSI) W] is solved with lambda being the Trace ratio value computed from the previous step. Convergence of the projection matrix W, as well as the global optimum of the Trace ratio value lambda, are proven based on point-to-set map theories. In addition, this procedure is further extended for solving Trace ratio problems with more general constraint WTCW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed solution, as well as its superiority in classification capability over corresponding solutions to the ratio Trace problem.