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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.
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).
If k is infinite, then k[V] is the symmetric algebra of the dual space . In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies.
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.
The symmetric group S n has order n!. Its conjugacy classes are labeled by partitions of n . Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations , over the complex numbers , is equal to the number of partitions of n .
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)
In linear algebra, the n-th symmetric power of a vector space V is the vector subspace of the symmetric algebra of V consisting of degree-n elements (here the product is a tensor product). In algebraic topology , the n -th symmetric power of a topological space X is the quotient space X n / S n {\displaystyle X^{n}/{\mathfrak {S}}_{n}} , as in ...