Symmetric Tensors

Symmetric Tensors are mathematical objects that generalize symmetric properties in multilinear algebra, playing a key role in geometry and physics.

Symmetric Tensors is the collection of elements of a tensor power $V^{\otimes k}$ that remain unchanged under every permutation of the $k$ tensor factors, forming a distinguished subspace that captures exactly the information invariant under reordering of the inputs to a multilinear construction.

Definition

The Permutation Action on a Tensor Power

For a vector space $V$ over a field $K$ and a positive integer $k$ , the symmetric group $S_{k}$ acts on the tensor power $V^{\otimes k}$ by permuting the tensor factors of a simple tensor,

σ \cdot (v_{1} \otimes \dots \otimes v_{k}) = v_{σ^{- 1} (1)} \otimes \dots \otimes v_{σ^{- 1} (k)}

extended linearly to all of $V^{\otimes k}$ , for each permutation $σ \in S_{k}$ .

The Symmetric Tensors Are the Fixed Points

An element $t \in V^{\otimes k}$ is called a symmetric tensor if

σ \cdot t = t

for every $σ \in S_{k}$ , and the set of all such tensors forms a linear subspace of $V^{\otimes k}$ , called the $k$ -th symmetric power of $V$ and denoted ${Sym}^{k} (V)$ .

The Symmetrization Operator

Averaging Over All Permutations

When the characteristic of $K$ does not divide $k!$ , the symmetrization operator

Sym (t) = \frac{1}{k!} \sum_{σ \in S_{k}} σ \cdot t

is well defined on all of $V^{\otimes k}$ , produces a symmetric tensor for every input, and acts as the identity on tensors that are already symmetric, so it is a projection of $V^{\otimes k}$ onto ${Sym}^{k} (V)$ .

Symmetric Product Notation

Applying symmetrization to a simple tensor $v_{1} \otimes \dots \otimes v_{k}$ produces the symmetric product, written

v_{1} ⊙ \dots ⊙ v_{k} = \frac{1}{k!} \sum_{σ \in S_{k}} v_{σ (1)} \otimes \dots \otimes v_{σ (k)}

which is unaffected by reordering the vectors $v_{1}, \dots, v_{k}$ being multiplied.

Basis and Dimension

Symmetric Products of a Basis

If ${e_{1}, \dots, e_{n}}$ is a basis of $V$ , the symmetric power ${Sym}^{k} (V)$ has a basis consisting of the distinct symmetric products $e_{i_{1}} ⊙ \dots ⊙ e_{i_{k}}$ with $i_{1} \leq \dots \leq i_{k}$ , since permuting a nondecreasing sequence of indices produces the same symmetric product.

Dimension Formula

Counting these nondecreasing index sequences of length $k$ from $n$ symbols gives the dimension

dim ({Sym}^{k} (V)) = (\begin{matrix} n + k - 1 \\ k \end{matrix})

the standard multiset coefficient, strictly smaller than $n^{k}$ , the dimension of the full tensor power, for $k \geq 2$ and $n \geq 2$ .

Relation to Polynomials

The Symmetric Algebra as a Polynomial Ring

The direct sum $⨁_{k \geq 0} {Sym}^{k} (V)$ , equipped with the symmetric product as multiplication, forms the symmetric algebra of $V$ , which is isomorphic to the polynomial ring $K [x_{1}, \dots, x_{n}]$ in variables corresponding to a basis of $V$ , under which ${Sym}^{k} (V)$ corresponds precisely to the homogeneous polynomials of degree $k$ .

Symmetric Tensors as Homogeneous Polynomial Coefficients

Under this identification, a symmetric tensor is exactly the coefficient data of a homogeneous polynomial of degree $k$ , with the symmetric product $e_{i_{1}} ⊙ \dots ⊙ e_{i_{k}}$ corresponding to the monomial $x_{i_{1}} \dots x_{i_{k}}$ .

Contrast With Antisymmetric Tensors

Complementary Behavior Under Permutation

Symmetric tensors satisfy $σ \cdot t = t$ for every permutation, in direct contrast with antisymmetric, or alternating, tensors, which instead satisfy $σ \cdot t = sgn (σ) t$ , changing sign under an odd permutation; the two subspaces overlap only in the zero tensor whenever $k \geq 2$ , and together they represent two of the simplest ways a tensor power can decompose under the action of the symmetric group, with the full tensor power generally containing further, more complicated pieces beyond these two extremes once $k \geq 3$ .