15.12.5 Tensor Symmetric Basis Expansion Role

Tensor Symmetric Basis Expansion Role is the function performed by the multi-index-labeled basis of Sym^n(V) when an arbitrary symmetric tensor, given by its raw components or by some other implicit description, is expressed explicitly as a linear combination of basis elements e^alpha, assigning a specific coefficient to each multi-index and thereby converting the tensor into coordinate form with respect to the chosen basis. This expansion role is the operational purpose the basis exists to serve: a basis is useful precisely because it allows every element of the space it spans to be written down concretely as a finite, coefficient-weighted sum of a fixed, known set of vectors.

Carrying out this expansion correctly requires care about the exact relationship between a symmetric tensor's raw indexed components and the coefficients that appear when the tensor is written in the e^alpha basis, since these two sets of numbers differ by the multinomial scaling factor already identified when discussing the symmetric power polynomial relation, making the expansion role not merely a relabeling exercise but one requiring an explicit combinatorial correction.

The Expansion Formula

Coefficients From Components

For a symmetric tensor T of rank n, its expansion in the basis e^alpha, with alpha ranging over degree-n multi-indices, takes the form:

where the coefficient c_alpha is related to the raw component T_{i1...in} at the canonical index tuple corresponding to alpha by the multinomial scaling factor n! divided by alpha!, reflecting how many orbit members that single stored component represents.

Justification for the Scaling Factor

The scaling factor arises because the basis element e^alpha, expanded back into raw tensor components via the symmetrization operator, itself carries a built-in normalization from the permutation averaging process; matching T's actual raw component value against this pre-normalized basis element requires compensating with the multinomial factor, ensuring the expansion, once fully carried out, reconstructs T's true component array rather than a rescaled version of it.

Expansion in Low Rank for Concreteness

Rank Two Illustration

For a rank-two symmetric tensor with components T_{ij}, the coefficient of the basis element e_i^{odot 2} in the expansion equals T_{ii} directly, since the multinomial factor for alpha equal to (2, 0, ..., 0) reduces to 2!/2! equal to one, while the coefficient of the basis element e_i odot e_j for i not equal to j equals 2 T_{ij}, since the multinomial factor for alpha with two entries equal to one reduces to 2!/(1!1!) equal to two.

Connection to the Quadratic Form Expansion

This rank-two expansion pattern, with diagonal coefficients matching components directly and off-diagonal coefficients carrying a factor of two, exactly reproduces the earlier observed structure of the associated quadratic form, where cross terms in the expansion of B(v, v) appeared with a coefficient of two relative to the underlying off-diagonal component, confirming that the basis expansion role and the quadratic form relation are two views of the identical underlying combinatorics.

Expansion as an Inverse to Coefficient Extraction

Recovering Coefficients From a Given Tensor

Given a symmetric tensor already known in raw component form, extracting its expansion coefficients is a direct, mechanical process: for each degree-n multi-index alpha, compute the multinomial factor n!/alpha! and multiply it by the raw component stored at the canonical index tuple corresponding to alpha, yielding the coefficient c_alpha needed for the expansion.

Reconstructing a Tensor From Given Coefficients

Conversely, given an arbitrary assignment of coefficients c_alpha to each multi-index, the expansion formula defines a unique symmetric tensor, obtained by summing c_alpha times e^alpha over all alpha; this direction of the expansion role is what allows a symmetric tensor to be specified directly by choosing basis coefficients, rather than by specifying raw components, whenever that is the more natural starting point for a given application.

Expansion Role Within the Broader Framework

Bridge Between Raw Components and Coordinate Representation

The expansion role functions as the explicit bridge connecting the raw, index-based description of a symmetric tensor, used throughout the initial development of the equality constraint and independent component selection, to the coordinate-based description used once a specific basis has been fixed, making explicit the translation that has been implicitly assumed whenever tensor components and basis coefficients were discussed together.

Foundation for Polynomial-Style Computation

Because the expansion coefficients c_alpha correspond directly, under the polynomial analogy, to the coefficients of the associated homogeneous polynomial's monomials, carrying out the basis expansion role is precisely the step that converts a symmetric tensor into its most computationally convenient polynomial form, ready for further manipulation using ordinary polynomial arithmetic.