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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. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose .
The space of symmetric tensors of rank r on a finite-dimensional vector space 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.
An example of such a Koszul algebra A is the symmetric algebra on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra , i.e., of the form A = T ( V ) / R , {\displaystyle A=T(V)/R,}
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group whose natural action on tensor products of a complex vector space has as image an irreducible representation of the group of invertible linear transformations ().
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V ∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V ∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).
Hilbert (1890) proved that if V is a finite-dimensional representation of the complex algebraic group G = SL n (C) then the ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used the Reynolds operator ρ from R to R G with the properties ρ(1) = 1; ρ(a + b) = ρ(a) + ρ(b)