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From the formulas above, the Hilbert–Poincaré series of a direct sum and of a tensor product of graded vector spaces (finite dimensional in each degree) are respectively the sum and the product of the corresponding Hilbert–Poincaré series.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
The direct sum is a submodule of the direct product of the modules M i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M i with α(i)∈M i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product ...
The Whitney sum (named for Hassler Whitney) or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum E x ⊕ F x of the vector spaces E x and F x. The tensor product bundle E ⊗ F is defined in a similar way, using fiberwise tensor product of vector spaces.
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification: V C := V ⊕ V , {\displaystyle V^{\mathbb {C} }:=V\oplus V,} where V C {\displaystyle V^{\mathbb {C} }} is given a linear complex structure by the operator J defined as J ( v , w ) := ( − w , v ) , {\displaystyle J(v,w ...