15 Symmetric Tensors

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,

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

The Symmetric Tensors Are the Fixed Points

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

for every $\sigma \in S_k$, 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 ${\text{Sym}}^k\left(V\right)$.

The Symmetrization Operator

Averaging Over All Permutations

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

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 ${\text{Sym}}^k\left(V\right)$.

Symmetric Product Notation

Applying symmetrization to a simple tensor $v_1\otimes \dots \otimes v_k$ produces the symmetric product, written

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 ${\text{Sym}}^k\left(V\right)$ has a basis consisting of the distinct symmetric products $e_{i_1}\odot \dots \odot e_{i_k}$ with $i_1\le \dots \le i_k$, 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

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

Relation to Polynomials

The Symmetric Algebra as a Polynomial Ring

The direct sum $\underset{k\ge 0}{⨁}{\text{Sym}}^k\left(V\right)$, 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 ${\text{Sym}}^k\left(V\right)$ 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 $e_{i_1}\odot \dots \odot e_{i_k}$ corresponding to the monomial $x_{i_1}\cdots x_{i_k}$.

Contrast With Antisymmetric Tensors

Complementary Behavior Under Permutation

Symmetric tensors satisfy $\sigma \cdot t=t$ for every permutation, in direct contrast with antisymmetric, or alternating, tensors, which instead satisfy $\sigma \cdot t=\text{sgn}\left(\sigma \right)t$, changing sign under an odd permutation; the two subspaces overlap only in the zero tensor whenever $k\ge 2$, 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 $k\ge 3$.