15.19.3 Tensor Symmetric Power Notation

Tensor Symmetric Power Notation is the specific notational device, written S superscript d applied to a vector space V, used to denote the space of order-d symmetric tensors as a distinguished algebraic construction in its own right, together with the ring and functorial structure this notation is built to carry.

The Basic Symbol and What It Denotes

Reading the Notation

The expression S^d V denotes the d-th symmetric power of the vector space V: the subspace of the full tensor power V superscript tensor d consisting of tensors invariant under every permutation of the d tensor factors, exactly the space characterized by the Component Constraint and shown, under Subspace Invariance, to be a well-defined, basis-independent construction. The superscript d in this notation plays the same role as an ordinary exponent, but applied to the operation of forming symmetric tensors rather than to ordinary multiplication.

Small Values of d

For d equal to zero, S^0 V is conventionally taken to be the base field itself, regarded as the space of constants; for d equal to one, S^1 V is simply V, since a single vector trivially satisfies any permutation-invariance condition on one index; for d equal to two, S^2 V is the space of symmetric matrices studied throughout the Matrix Case, giving the notation its first structurally rich instance.

The Symmetric Algebra Structure

Direct Sum Across All Orders

Collecting the symmetric powers for every order d from zero upward into a single direct sum produces the symmetric algebra of V, denoted by S(V) or equivalently by the direct sum over d of S^d V, and this direct sum carries a natural multiplication, induced by the tensor product followed by symmetrization, under which the symmetric algebra becomes a graded commutative ring, with S^d V as its degree-d graded piece.

Multiplication Within the Notation

The product of an element of S^p V and an element of S^q V under this ring structure lands in S^{p+q} V, obtained by taking the tensor product of the two elements and then applying the symmetrization operator to the combined p plus q indices; this operation is precisely what allows a pure power form v superscript tensor d to be written unambiguously as v to the d-th power within the symmetric algebra, treating the tensor power notation and ordinary polynomial exponent notation as interchangeable once the symmetric algebra structure is in place.

Universal Property and Functorial Behavior

Characterization by a Universal Property

The symmetric power S^d V is characterized, up to canonical isomorphism, by a universal property: it is the target of a universal symmetric multilinear map from d copies of V, meaning any symmetric multilinear map from V to d copies into another vector space factors uniquely through the canonical map from V to d copies into S^d V. This universal property is the abstract justification for using the same construction, denoted uniformly by the S^d notation, regardless of the particular vector space V to which it is applied.

Functoriality Under Linear Maps

The Symmetric Power Notation extends naturally to linear maps: given a linear map from V to another vector space W, there is an induced linear map from S^d V to S^d W, obtained by applying the original map to each of the d factors of a pure power form and extending linearly; this functorial behavior is the notation-level expression of the Transformation Preservation theorem, since a change of basis is itself a linear map from V to itself, and the induced map on S^d V is exactly the Component Transformation formula acting on symmetric tensor components.

Relation to Polynomial and Divided Power Notation

Identification with Homogeneous Polynomials

When V is finite-dimensional, S^d applied to the dual space of V is naturally identified with the space of homogeneous polynomials of degree d on V, recovering the correspondence used throughout the Tensor Quadratic Form Relation and its generalizations; under this identification, the Symmetric Power Notation and the notation for homogeneous polynomials become two labels for the same underlying object, chosen according to whether the tensorial or the polynomial perspective is more convenient in a given context.

Divided Powers in Positive Characteristic

In settings of positive characteristic, particularly where the characteristic divides d factorial, the symmetric power S^d V and the closely related divided power construction, often denoted using a Greek gamma superscript d, can fail to coincide, reflecting the same breakdown of clean symmetrization discussed under Tensor Symmetric Type Preservation; the divided power notation exists specifically to provide a well-behaved substitute for S^d V in exactly the characteristic regimes where the ordinary Symmetric Power Notation loses some of its expected properties.

Practical Use of the Notation

Compact Expression of Dimension and Structure Results

Because S^d V is treated as a single algebraic object rather than merely a set of components, results such as the dimension formula for symmetric tensors, expressed as the binomial coefficient of n plus d minus one choose d for an n-dimensional V, and the classification results of the Alexander-Hirschowitz theorem concerning secant varieties built from S^d V, can be stated compactly and manipulated algebraically using standard operations on graded rings and their associated projective varieties, a level of compactness unavailable if symmetric tensors were described only through raw indexed components.