15.19.1 Tensor Symmetric Index Notation

Tensor Symmetric Index Notation is the compact multi-index labeling scheme that replaces the d separate, individually ordered index slots of a symmetric tensor's components with a single multi-index recording only how many times each basis direction occurs, exploiting the fact that, for a symmetric tensor, the order of the original indices carries no information beyond their multiplicities.

From Ordered Indices to Multi-Indices

The Redundancy in Ordered Index Notation

An order-d symmetric tensor's component T subscript i_1 through i_d, written with d explicitly ordered indices, takes the same value for every rearrangement of those indices, by the Component Constraint; consequently, the ordered list i_1 through i_d carries strictly more information than the tensor actually distinguishes, since only the multiset of index values, not their order, affects the component's value.

Definition of the Multi-Index

The Symmetric Index Notation collapses this redundancy by recording, for a space with basis directions labeled one through n, a multi-index alpha equal to the tuple alpha_1 through alpha_n, where alpha_j counts how many of the d original index slots take the value j, so that the entries of alpha are non-negative integers summing to d:

A symmetric tensor's component is then written T subscript alpha, using this single multi-index in place of the d separate ordered indices, and this notation is well-defined precisely because symmetry guarantees the value depends only on the multiplicities alpha, not on which particular ordering of indices produced them.

Correspondence with Monomial Notation

Multi-Index Exponents on Variables

Under the standard identification of symmetric tensors with homogeneous polynomials, the multi-index alpha corresponds directly to the exponents of a monomial: the monomial x_1 to the alpha_1, times x_2 to the alpha_2, and so on up to x_n to the alpha_n, has total degree d exactly when alpha satisfies the multi-index constraint above, and T subscript alpha is precisely the coefficient of this monomial in the homogeneous polynomial associated with the tensor, connecting the Symmetric Index Notation directly to the polynomial notation used throughout the Tensor Quadratic Form Component Expression.

Multinomial Coefficient Corrections

Because a single multi-index alpha corresponds to a whole orbit of d factorial divided by alpha_1 factorial times ... times alpha_n factorial ordered index tuples, converting between a sum written with ordered indices and a sum written with multi-indices requires inserting exactly this multinomial coefficient, a bookkeeping detail already encountered, for the special case of order two, under the Tensor Quadratic Form Component Expression, where off-diagonal components acquired a factor of two relative to diagonal ones.

Multi-Index Notation in Calculus and Symmetrization

Multi-Index Derivative Notation

The same multi-index notation used to label symmetric tensor components is used, in multivariable calculus, to label mixed partial derivatives: the partial derivative operator corresponding to a multi-index alpha applies alpha_1 derivatives with respect to the first variable, alpha_2 with respect to the second, and so on, and this shared notation is not a coincidence, since the Tensor Quadratic Form Polarization Relation and its higher-order generalizations identify symmetric tensor components directly with (normalized) mixed partial derivatives of the associated homogeneous polynomial.

Compact Statement of the Symmetrization Sum

Using multi-index notation, sums that would otherwise require explicit reference to all d factorial permutations, such as the definition of the symmetrization operator, can instead be written as a single sum over multi-indices alpha weighted by multinomial coefficients, considerably shortening expressions that appear throughout apolarity computations and catalecticant matrix constructions used in Reconstruction.

Practical Advantages of the Notation

Reduced Redundancy in Symbolic and Numerical Storage

Because the number of distinct multi-indices for given n and d equals exactly the dimension of the space of symmetric tensors, given by the binomial coefficient of n plus d minus one, choose d, storing a symmetric tensor's components indexed by multi-indices, rather than by all n to the power d ordered tuples, avoids storing the same value many times over, directly reflecting the same efficiency gain exploited procedurally by the Component Equality Check when it groups ordered tuples into orbits.

Clarity in Stating General-Order Formulas

Multi-index notation allows formulas that would otherwise require explicit, order-dependent index lists, such as the general polarization identity recovering a symmetric tensor of arbitrary order from evaluations of its associated polynomial, to be stated uniformly for every order d at once, without separately writing out the order-two and order-three cases as done, for concreteness, under the ordinary indexed notation discussed in Tensor Symmetric Tensor Notation.

Relationship to Other Notational Conventions

A Compression, Not a Replacement, of Ordered Index Notation

The Symmetric Index Notation does not introduce any new mathematical content beyond ordinary indexed notation; it is a compression that becomes available specifically because of the symmetry established under Transformation Preservation and Subspace Invariance, and every formula expressible using multi-indices can, in principle, be unpacked back into the fully explicit, ordered-index form used elsewhere throughout the description of symmetric tensor components, with the two notations chosen interchangeably according to whether compactness or explicitness better serves the context.