4.21.3 Tensor Symmetric Component Pattern

The Tensor Symmetric Component Pattern organizes tensor components using symmetry, revealing structural properties essential in advanced algebraic applications.

Tensor Symmetric Component Pattern is the structure exhibited by the coordinate array of a symmetric multilinear map once a basis is chosen: the array is completely invariant under any permutation of its indices, and consequently is determined entirely by its values on non-decreasing tuples of indices, with every other entry simply equal, with no sign correction, to the value at the corresponding sorted tuple.

The Component Array of a Symmetric Map

Full Invariance Under Reindexing

For a symmetric multilinear map f: V × ... × V → W and a basis {e₁,...,e_d} of V, the component array

T_{i_{1} \dots i_{n}} = f (e_{i_{1}}, \dots, e_{i_{n}})

satisfies

T_{i_{σ (1)} \dots i_{σ (n)}} = T_{i_{1} \dots i_{n}}

for every permutation σ, with no sign factor ever appearing, unlike the corresponding alternating component pattern. This holds regardless of whether the indices i₁,...,iₙ are distinct, unlike the alternating case, where any repeated index forces the entry to zero.

No Vanishing on Repeated Indices

In sharp contrast to the alternating component pattern, entries with repeated indices are not forced to vanish; a symmetric map's array can, and generically does, have nonzero entries even when several indices coincide, for instance T_{iii...i} = f(eᵢ,...,eᵢ) need not be zero.

Reduction to Non-Decreasing Index Tuples

One Representative Per Multiset

Since every entry is equal to the entry obtained by sorting its indices into non-decreasing order, the entire component array is determined by its values on non-decreasing tuples i₁ ≤ i₂ ≤ ... ≤ iₙ. Any other ordering of the same multiset of indices is recovered simply by copying the value at the sorted representative, with no sign or other correction applied.

Counting Independent Entries

The number of non-decreasing tuples i₁ ≤ ... ≤ iₙ drawn from {1,...,d} equals the number of multisets of size n from d elements, C(d+n-1, n). This is the number of independent scalars needed to specify a symmetric n-linear map on a d-dimensional space, larger than the C(d,n) count for the alternating case, since repeated indices are now permitted rather than forbidden.

Basis of the Symmetric Power From This Pattern

Symmetric Products of Non-Decreasing Index Tuples

The reduction to non-decreasing tuples matches the standard basis of the symmetric power Symⁿ(V), given by the symmetric products e_{i₁}⋯e_{iₙ} for i₁ ≤ ... ≤ iₙ; these C(d+n-1,n) elements form a basis precisely because any other symmetric product of basis vectors, with indices out of order, reduces without any sign change to one of these, by the symmetric component pattern.

Matching Dimension Counts

The dimension of Symⁿ(V), equal to C(d+n-1,n), agrees with the count of independent entries in the component array of a symmetric n-linear map on a d-dimensional space, confirming from two directions that symmetric multilinear maps of arity n correspond to a space of this dimension.

Relation to Monomials in the Symmetric Algebra

Multinomial Coefficients Linking Components to Polynomial Coefficients

Under the identification Sym(V) ≅ F[x₁,...,x_d], the symmetric product e_{i₁}⋯e_{iₙ} corresponds to the monomial x_{i₁}⋯x_{iₙ}, which, when the indices include repeats, say index k appearing mₖ times, is more familiarly written x₁^{m₁}⋯x_d^{m_d}; converting between the "component array" convention used for symmetric tensors and the "polynomial coefficient" convention used for homogeneous polynomials introduces multinomial coefficients n!/(m₁!⋯m_d!), accounting for the number of distinct orderings collapsed onto the same non-decreasing tuple.

The Symmetric Power Polynomial

For v = ∑ᵢ vⁱeᵢ, the value f(v,...,v) of a symmetric multilinear map on n identical copies of v expands, using the reduction to non-decreasing tuples and accounting for the multinomial multiplicities, as

f (v, \dots, v) = \sum i_{1} \leq \dots \leq i_{n} \frac{n!}{m_{1}! \dots m_{d}!} T_{i_{1} \dots i_{n}} v^{i_{1}} \dots v^{i_{n}}

recovering the expansion of the corresponding homogeneous polynomial in the standard monomial basis, with the multinomial coefficients arising directly from the symmetric component pattern's collapse of repeated-index orderings.

Contrast With the Alternating Case in Practice

Storage Requirements

Because entries outside non-decreasing order are redundant but not zero, storing a symmetric multilinear map's component array requires C(d+n-1,n) values, more than the C(d,n) values needed for an alternating map of the same arity and dimension, since the symmetric case permits, rather than forbids, coincidences among indices.

Reconstructing Any Entry From the Stored Ones

Given the stored values on non-decreasing tuples, any entry T_{i₁...iₙ} with indices in an arbitrary order is reconstructed simply by sorting the indices and reading off the corresponding stored value directly, with no sign tracking required, in contrast to the alternating case where a parity computation determines a possible sign flip.

Verifying a Component Array Represents a Symmetric Map

The Adjacent-Swap Test

A proposed component array T_{i₁...iₙ} is confirmed to represent a symmetric multilinear map exactly when it is unchanged under swapping any two adjacent indices; this single-transposition test suffices to guarantee invariance under every permutation, since adjacent transpositions generate the full symmetric group acting on the index positions.