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Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.
Among the properties of Hodge numbers are Hodge symmetry h p,q = h q,p (because H p,q (X) is the complex conjugate of H q,p (X)) and h p,q = h n−p,n−q (by Serre duality). The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the Hodge diamond (shown in the case of complex dimension 2):
In 1991 the paper [5] about mirror symmetry on the quintic threefold in from the string theoretical viewpoint gives numbers of degree d rational curves on for all >. Prior to this, algebraic geometers could calculate these numbers only for d ≤ 5 {\displaystyle d\leq 5} .
The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other. [6]
Along with the homological mirror symmetry conjecture, it is one of the most explored tools applied to understand mirror symmetry in mathematical terms. While the homological mirror symmetry is based on homological algebra, the SYZ conjecture is a geometrical realization of mirror symmetry.
Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols consists of ...
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.
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