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Symmetric Product
Symmetric product may refer to: * The product operation of a symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ... * The symmetric product of tensors * The symmetric product of an algebraic curve * The Symmetric product (topology), \operatorname^n(X) or infinite symmetric product \operatorname^\infty(X) of a space ''X'' in algebraic topology {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal property: for every linear map from to a commutative algebra , there is a unique algebra homomorphism such that , where is the inclusion map of in . If is a basis of , the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring , where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over . The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form . All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring. Construction From tensor algebra It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Tensor
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :T_ = T_. The space of symmetric tensors of order ''r'' on a finite-dimensional vector space ''V'' is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. Definition Let ''V'' be a vector space and :T\in V^ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :\tau_\sigma T = T\, for the braiding ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Product Of An Algebraic Curve
In mathematics, the ''n''-fold symmetric product of an algebraic curve ''C'' is the quotient space of the ''n''-fold cartesian product :''C'' × ''C'' × ... × ''C'' or ''C''''n'' by the group action of the symmetric group ''S''''n'' on ''n'' letters permuting the factors. It exists as a smooth algebraic variety denoted by Σ''n''''C''. If ''C'' is a compact Riemann surface, Σ''n''''C'' is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on ''C'' of degree ''n'', that is, formal sums of points with non-negative integer coefficients. For ''C'' the projective line (say the Riemann sphere \mathbb ∪ ≈ ''S''''2''), its nth symmetric product Σ''n''''C'' can be identified with complex projective space \mathbb^n of dimension ''n''. If ''G'' has genus ''g'' ≥ 1 then the Σ''n''''C'' are closely related to the Jacobian variety ''J'' of ''C''. More accurately for ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Product (topology)
In algebraic topology, the ''n''th symmetric product of a topological space consists of the unordered ''n''-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor. From an algebraic point of view, the infinite symmetric product is the free commutative monoid generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the James reduced product. One of its essential applications is the Dold-Thom theorem, stating that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as the reduced homology groups of that complex. That way, one can give a homotopical definition of homology. Definition Let ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |