1 Basic definitions and notation

1.1 Tensor symmetry and index permutations

Let \(V\) be a vector space over a field (commonly \(\mathbb{R}\) or \(\mathbb{C}\)). An order-\(n\) tensor is an element of \(V^{\otimes n}\), the \(n\)-fold tensor product. For \(T\in V^{\otimes n}\), symmetry is expressed by how \(T\) responds when two tensor factors are interchanged.

A tensor \(T\in V^{\otimes n}\) is symmetric if it is unchanged under every permutation of the tensor factors. Equivalently, for any permutation \(\sigma\in S_n\), \[ T = \sigma\cdot T, \] where \(\sigma\cdot T\) denotes the tensor with indices permuted according to \(\sigma\). In index notation relative to a basis \(\{e_i\}\), this means that components satisfy \[ T_{i_1 i_2 \dots i_n} = T_{i_{\sigma(1)} i_{\sigma(2)} \dots i_{\sigma(n)}} \quad \text{for all } \sigma\in S_n. \]

1.2 Symmetric tensors of low order

1.2.1 Symmetric vectors (order 1)

For \(n=1\), there is only one index, so every tensor in \(V^{\otimes 1}\cong V\) is automatically symmetric. Thus the notion of symmetry becomes nontrivial starting at order \(2\).

1.2.2 Symmetric bilinear forms (order 2)

An order-2 tensor \(T\in V\otimes V\) can be identified with a bilinear form once a choice of dual pairing is made. In a basis, symmetry means \[ T_{ij}=T_{ji}. \] For bilinear forms \(B:V\times V\to \Bbb{F}\), this is precisely the classical condition \(B(u,v)=B(v,u)\).

1.2.3 Symmetric multilinear maps (order \(\ge 3\))

For \(n\ge 3\), symmetry corresponds to invariance under any rearrangement of the \(n\) arguments. If \(M:V^n\to \Bbb{F}\) is multilinear, symmetry means \[ M(v_1,\dots,v_n)=M(v_{\sigma(1)},\dots,v_{\sigma(n)}) \quad \text{for all } \sigma\in S_n. \] Under the usual identification between multilinear maps and tensors in \(V^{*\otimes n}\), this is the same as requiring the associated tensor in the tensor product to be fixed by all index permutations.

1.3 Equivalent characterizations

1.3.1 Invariance under the action of the symmetric group

The symmetric group \(S_n\) acts naturally on \(V^{\otimes n}\) by permuting the tensor factors. A tensor \(T\in V^{\otimes n}\) is symmetric exactly when it is fixed by this action for every element of \(S_n\).

Concretely, if \(T=v_1\otimes \cdots \otimes v_n\), then for \(\sigma\in S_n\), \[ \sigma\cdot T = v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(n)}. \] By linearity, this extends to all tensors. Symmetric tensors are those lying in the common fixed space of the entire group action.

1.3.2 Symmetric subspace of \(V^{\otimes n}\)

Define the symmetric subspace \[ \mathrm{Sym}^n(V)\subseteq V^{\otimes n} \] as the set of all tensors fixed by every permutation of tensor factors. This subspace is well-defined and is stable under any linear maps induced on \(V^{\otimes n}\).

1.3.3 Symmetrization operator (averaging over permutations)

Given any tensor \(T\in V^{\otimes n}\), the symmetrization is obtained by averaging over the action of \(S_n\): \[ \mathrm{Sym}(T)=\frac{1}{n!}\sum_{\sigma\in S_n}\sigma\cdot T. \] This element is symmetric, and it depends linearly on \(T\). Moreover, \(\mathrm{Sym}(T)=T\) exactly when \(T\) is already symmetric.

2 Algebraic structure of symmetric tensors

2.1 Symmetric powers of vector spaces

2.1.1 Definition of \(\mathrm{Sym}^n(V)\)

The \(n\)-th symmetric power of \(V\), denoted \(\mathrm{Sym}^n(V)\), is the subspace of \(V^{\otimes n}\) consisting of symmetric tensors: \[ \mathrm{Sym}^n(V)=\{T\in V^{\otimes n}: \sigma\cdot T=T \text{ for all }\sigma\in S_n\}. \] This construction is standard in algebra and category theory because it behaves well under linear transformations.

2.1.2 Functorial properties

If \(f:V\to W\) is linear, then \(f^{\otimes n}:V^{\otimes n}\to W^{\otimes n}\) is defined by acting with \(f\) on each tensor factor. Because permutation of factors commutes with applying \(f\) in each slot, \(f^{\otimes n}\) maps symmetric tensors to symmetric tensors. Hence it restricts to a map \[ \mathrm{Sym}^n(f):\mathrm{Sym}^n(V)\to \mathrm{Sym}^n(W). \] Thus \(\mathrm{Sym}^n\) is a functor from vector spaces to vector spaces.

2.2 Natural maps and universal properties

2.2.1 Symmetric product and multilinearization

There is a natural symmetric product that combines symmetric tensors in different degrees. At the level of tensor products, one can form products in \(V^{\otimes m}\otimes V^{\otimes n}\cong V^{\otimes (m+n)}\) and then symmetrize. The result lands canonically in \(\mathrm{Sym}^{m+n}(V)\).

Relatedly, symmetric tensors correspond to symmetric multilinear forms and allow one to define multilinearization: from a homogeneous function or polynomial in \(n\) inputs that is symmetric, one obtains a symmetric \(n\)-linear map.

2.2.2 Relationship with polynomial rings

A classical bridge connects symmetric tensors with homogeneous polynomials. If \(V\) is finite-dimensional with dual \(V^*\), then elements of \(\mathrm{Sym}^n(V^*)\) correspond to homogeneous degree-\(n\) polynomials on \(V\). Under this identification, evaluating a symmetric tensor on \(n\) vectors matches the standard evaluation of a homogeneous polynomial’s associated \(n\)-linear form.

2.3 Basis and dimension

2.3.1 Monomial bases and counting arguments

Assume \(V\) has basis \(\{e_1,\dots,e_d\}\). Symmetric tensors can be represented by symmetrized monomials in basis elements. For nonnegative integers \((a_1,\dots,a_d)\) with \(\sum_{i=1}^d a_i=n\), one defines a symmetrized basis element corresponding to \[ e_1^{\otimes a_1}\otimes \cdots \otimes e_d^{\otimes a_d} \] after averaging over permutations that rearrange identical factors. These yield a spanning set that is also linearly independent, providing a basis indexed by exponent vectors of monomials of total degree \(n\).

2.3.2 Dimension formula in terms of binomial coefficients

The number of exponent vectors \((a_1,\dots,a_d)\) with \(\sum a_i=n\) is the “stars and bars” count, giving \[ \dim \mathrm{Sym}^n(V)=\binom{n+d-1}{n}=\binom{n+d-1}{d-1}. \] This formula is widely used to quantify the size of spaces of symmetric tensors in applications.

2.4 Direct sums and gradings

2.4.1 Graded algebra \(\bigoplus_{n\ge 0}\mathrm{Sym}^n(V)\)

The direct sum \[ \mathrm{Sym}(V)=\bigoplus_{n\ge 0}\mathrm{Sym}^n(V) \] forms a graded object, with degree \(n\) part equal to \(\mathrm{Sym}^n(V)\). By convention \(\mathrm{Sym}^0(V)\cong \Bbb{F}\).

2.4.2 Commutative multiplication of symmetric tensors

There is a natural multiplication \[ \mathrm{Sym}^m(V)\times \mathrm{Sym}^n(V)\to \mathrm{Sym}^{m+n}(V) \] coming from symmetrized tensor products. It is commutative and associative, so \(\mathrm{Sym}(V)\) is naturally a commutative graded algebra, often interpreted as the algebra of polynomial functions on \(V^*\) (or polynomial expressions in a coordinate-free form).

3 Symmetry operations and projections

3.1 The symmetrizer idempotent

3.1.1 Construction using permutation sums

Define the symmetrizer \(P_{\mathrm{sym}}\in \mathrm{End}(V^{\otimes n})\) by \[ P_{\mathrm{sym}}=\frac{1}{n!}\sum_{\sigma\in S_n}\sigma\cdot . \] That is, \(P_{\mathrm{sym}}(T)=\frac{1}{n!}\sum_{\sigma\in S_n}\sigma\cdot T\). This operator depends only on the permutation action, not on any additional structure such as an inner product.

3.1.2 Idempotence and projection onto \(\mathrm{Sym}^n(V)\)

A key property is idempotence: \[ P_{\mathrm{sym}}^2=P_{\mathrm{sym}}. \] This follows because averaging twice over the same group action returns the same average. Consequently, \(P_{\mathrm{sym}}\) acts as a projection: its image is precisely \(\mathrm{Sym}^n(V)\), and it fixes every symmetric tensor while sending non-symmetric tensors to their symmetric component.

3.2 Group actions and representation-theoretic viewpoint

3.2.1 Action of \(S_n\) on tensors

The permutation action of \(S_n\) on \(V^{\otimes n}\) makes \(V^{\otimes n}\) into a representation of \(S_n\). Symmetric tensors correspond to the subrepresentation consisting of vectors invariant under the group.

3.2.2 Symmetric representation and invariants

Within representation theory, the symmetric tensors form the invariant subspace \[ (V^{\otimes n})^{S_n}=\{T\in V^{\otimes n}:\sigma\cdot T=T\ \forall\sigma\in S_n\}. \] This description clarifies why symmetrization involves averaging: the fixed subspace of a group action is obtained by applying the averaging operator (when the field characteristic does not divide \(n!\)).

3.3 Decomposition with other tensor symmetries

3.3.1 Symmetric vs. alternating tensors

Two fundamental symmetry types on \(n\) indices are:

  • symmetric tensors fixed by all permutations,
  • alternating (antisymmetric) tensors that change sign under odd permutations.

These correspond to different subspaces of \(V^{\otimes n}\). While the symmetric subspace corresponds to invariants of the trivial representation of \(S_n\), the alternating subspace corresponds to invariants of the sign representation.

3.3.2 Orthogonal complements under natural inner products (when available)

If \(V\) carries an inner product and thus \(V^{\otimes n}\) has an induced inner product (for example, one making pure tensors orthonormal in a basis), then the symmetric and alternating subspaces have a natural relationship via orthogonality. More generally, projections onto symmetry types become orthogonal projections with respect to such an inner product, when the action is unitary.

4 Operations involving symmetric tensors

4.1 Symmetric tensor product

4.1.1 Product \(\mathrm{Sym}^m(V)\times \mathrm{Sym}^n(V)\to \mathrm{Sym}^{m+n}(V)\)

Given \(S\in \mathrm{Sym}^m(V)\) and \(T\in \mathrm{Sym}^n(V)\), one defines their symmetric tensor product by embedding \(S\otimes T\) into \(V^{\otimes(m+n)}\) and then symmetrizing: \[ S\odot T := P_{\mathrm{sym}}(S\otimes T)\in \mathrm{Sym}^{m+n}(V). \] This construction respects the symmetry in each factor and produces a fully symmetric tensor of combined order.

4.1.2 Associativity and commutativity

The operation \(\odot\) is associative and commutative. Commutativity reflects the fact that in the fully symmetrized tensor, exchanging the \(m\) and \(n\) blocks corresponds to a permutation of the \(m+n\) factors, which leaves the result invariant.

4.2 Contraction and trace (when dual spaces are involved)

4.2.1 Contraction maps from \(V^*\otimes V\)

Contraction uses the natural pairing between a vector space and its dual. If \(V^*\) is the dual of \(V\), contraction identifies one covariant slot with one contravariant slot, producing a lower-order tensor. In the symmetric setting, contraction reduces degree and typically interacts with symmetrization in a controlled way, yielding maps such as \[ \mathrm{Sym}^n(V)\to \mathrm{Sym}^{n-2}(V) \] when traced over appropriate index pairs (the precise form depends on how covariant/contravariant types are arranged).

4.2.2 Trace of symmetric tensors

For order-2 tensors interpreted as linear maps or bilinear forms, the trace is the contraction of the tensor against an identity-like pairing. For higher-order symmetric tensors, “trace” generally means contracting multiple index pairs to produce a symmetric tensor of lower order, sometimes used to define harmonic or trace-free components.

4.3 Raising/lowering indices (algebraic form)

4.3.1 Use of bilinear forms to relate covariant/contravariant indices

To relate covariant and contravariant indices, one needs additional structure such as a nondegenerate bilinear form \(g\) (or an inner product). This provides an isomorphism between \(V\) and \(V^*\), allowing indices to be moved. When symmetric tensors are raised or lowered, the symmetry property is preserved if the identification between slots respects the permutation action.

4.3.2 Effect on symmetry

The symmetry of components under index interchange is compatible with raising or lowering as long as the bilinear form is itself symmetric (or otherwise consistent with the index permutation). Under such assumptions, operations on each index do not disturb the invariant behavior under permutations.

5 Coordinate and polynomial interpretations

5.1 Components and transformation under change of basis

5.1.1 Tensor component symmetries in coordinates

Given a basis \(\{e_i\}\), a symmetric order-\(n\) tensor is characterized by components \(T_{i_1\cdots i_n}\) that are invariant under any reshuffling of indices. In practice, it suffices to specify components where indices are in nondecreasing order (e.g., \(i_1\le i_2\le \cdots \le i_n\)).

5.1.2 Covariance under linear transformations

Under a change of basis given by an invertible matrix \(A\), tensor components transform according to the tensor product rule. Because the symmetry condition is phrased in terms of invariance under index permutations, it is preserved by these transformations: if \(T\) is symmetric in one basis, it remains symmetric in every basis.

5.2 Polynomial correspondence

5.2.1 Homogeneous polynomials and symmetric tensors

Homogeneous degree-\(n\) polynomials on \(V\) correspond to elements of \(\mathrm{Sym}^n(V^*)\). This is because any such polynomial determines an \(n\)-linear symmetric form via polarization, and any symmetric \(n\)-linear form defines a homogeneous polynomial by evaluation on repeated arguments.

5.2.2 Coefficients as tensor components

In coordinates, a polynomial \[ p(x)=\sum_{i_1,\dots,i_n} T_{i_1\cdots i_n}\, x_{i_1}\cdots x_{i_n} \] has coefficients that form a symmetric tensor \(T\). The requirement that \(p\) be well-defined as a polynomial forces the coefficient array to be symmetric, since monomials \(x_{i_1}\cdots x_{i_n}\) are invariant under permutation of indices.

5.3 Symmetric tensors as coefficients of expansions

5.3.1 Multivariate Taylor expansions

Multivariate Taylor formulas expand a smooth function around a point into sums of homogeneous polynomials. The coefficients in these expansions can be interpreted as symmetric tensors, representing the derivatives in symmetric multilinear form.

5.3.2 Power series and symmetric multilinear forms

In analytic contexts, the same correspondence extends to power series: the coefficient of \((x-x_0)^n\) is determined by an \(n\)-linear symmetric map. This viewpoint aligns naturally with the algebraic structure of \(\mathrm{Sym}^n(V^*)\).

6 Canonical forms and examples

6.1 Diagonalization for symmetric bilinear forms (structural results)

6.1.1 Spectral decomposition (context and assumptions)

For order-2 symmetric tensors corresponding to self-adjoint operators (or symmetric bilinear forms), one often has a spectral theorem under appropriate hypotheses (e.g., positive-definite inner product over \(\mathbb{R}\), or complex Hermitian cases). Then the tensor can be represented in a basis where cross-terms vanish, yielding a diagonal form.

6.1.2 Quadratic forms derived from symmetric tensors

Given a symmetric bilinear form \(B\), the associated quadratic form \(q(v)=B(v,v)\) encodes the same data. Diagonalization of \(B\) corresponds to rewriting \(q\) as a sum of weighted squares (or their bilinear analogues), clarifying geometric or analytic properties tied to \(q\).

6.2 Explicit examples in small dimensions

6.2.1 Order-2 tensors in \(\mathbb{R}^2\) or \(\mathbb{R}^3\)

In \(\mathbb{R}^2\), a symmetric order-2 tensor has components \[ T=\begin{pmatrix} a & b\\ b & c\end{pmatrix}, \] so only three independent parameters determine it. In \(\mathbb{R}^3\), a symmetric matrix has six independent entries, reflecting the constraint \(T_{ij}=T_{ji}\).

6.2.2 Order-3 symmetric tensors and component constraints

For \(n=3\) in dimension \(d\), the number of independent components is \(\binom{3+d-1}{3}=\binom{d+2}{3}\). In a basis, components must satisfy invariance under any permutation of three indices, so values depend only on how many times each basis index appears.

6.3 Tensor symmetries from symmetrized rank-one tensors

6.3.1 Symmetrization of \(v_1\otimes\cdots\otimes v_n\)

Given vectors \(v_1,\dots,v_n\), the symmetrized tensor is \[ \mathrm{Sym}(v_1\otimes\cdots\otimes v_n)=\frac{1}{n!}\sum_{\sigma\in S_n} v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)}. \] This produces a symmetric tensor even when the original factors are not equal or already symmetric.

6.3.2 Rank-one and sum-of-rank-one constructions (overview)

A symmetric tensor can often be expressed as a finite sum of symmetrized rank-one tensors built from repeated vectors, which relates to decompositions used in algebraic geometry and numerical methods. While uniqueness is not generally guaranteed, such representations motivate computational approaches to storing and manipulating symmetric tensors.

7 Further algebraic topics (optional advanced threads)

7.1 Symmetric algebra and invariant theory (high-level)

7.1.1 Invariants under group actions

When a group \(G\) acts linearly on \(V\), it induces actions on each \(\mathrm{Sym}^n(V)\) and on the full symmetric algebra \(\mathrm{Sym}(V)\). Elements fixed by the action (invariants) form subalgebras that often capture geometric data, such as orbit structure and classification problems.

7.1.2 Hilbert basis viewpoint (overview)

Invariant theory includes results asserting that, over certain fields and under suitable conditions, the invariant algebra is finitely generated. This perspective motivates studying symmetric tensors not only as linear spaces but as building blocks of invariant polynomial expressions.

7.2 Relation to Young diagrams and Schur functors (overview)

7.2.1 Symmetric representation as a special case

In the representation theory of the symmetric group, partitions of \(n\) label irreducible components. The symmetric tensors correspond to the partition \((n)\), i.e., the fully symmetric representation, while other shapes correspond to tensors with other index symmetries.

Schur-Weyl duality relates representations of \(S_n\) and \(\mathrm{GL}(V)\) acting on \(V^{\otimes n}\). The symmetric subspace appears as one particular isotypic component, and this framework organizes the decomposition of general tensor spaces into symmetry types.

7.3 Computational aspects

7.3.1 Efficient storage of symmetric tensor components

Because symmetry collapses many degrees of freedom, a symmetric tensor can be stored by recording only independent components. This reduces memory requirements compared with storing the full array in \(V^{\otimes n}\).

A typical indexing scheme stores components by exponent tuples \((a_1,\dots,a_d)\) with \(\sum a_i=n\), matching monomials of total degree \(n\). Such schemes align naturally with the polynomial correspondence.

7.3.2 Algorithmic symmetrization and contraction (outline)

To symmetrize computationally, one can average over permutations, but for large \(n\) this may be costly. Practical algorithms often exploit the fact that many permutations yield identical results for repeated indices, replacing the full group sum with combinatorial weighting. Contraction in symmetric contexts similarly benefits from counting-based formulas that account for multiplicities of index coincidences, avoiding redundant operations.