15.11.2 Tensor Independent Symmetric Multi Index

Tensor Independent Symmetric Multi Index is the compact notational device that replaces an explicit, sorted index tuple representing an independent component of a symmetric tensor with a single vector of non-negative integers recording how many times each basis direction appears, providing a more efficient and more directly polynomial-compatible way of labeling the independent components than writing out the full canonical index tuple. Where a canonical index tuple such as (1, 1, 2, 3) lists every occurrence of every index explicitly, the corresponding multi-index alpha equal to (2, 1, 1, 0, ..., 0) records only the count of occurrences of each of the d possible index values, achieving the same labeling with a fixed-length vector of length d regardless of the tensor's rank n.

This notation is standard throughout the theory of partial differential equations and multivariable calculus for exactly the same reason it is useful here: it condenses repetitive index information into a compact form well suited to algebraic manipulation, and it aligns exactly with the exponent tuples already used in the symmetric power basis construction and in the polynomial analogy for the symmetric algebra.

Definition of the Multi-Index Notation

From Canonical Index Tuple to Multi-Index

Given a canonical, non-decreasing index tuple (i_1, ..., i_n) with entries drawn from 1 through d, the associated multi-index alpha equal to (alpha_1, ..., alpha_d) is defined by setting alpha_k equal to the number of times the value k appears among i_1 through i_n:

so that alpha records the multiplicity of each index value rather than the ordered sequence of values itself.

The Degree of a Multi-Index

The sum of the entries of a multi-index, denoted |alpha| and equal to alpha_1 plus alpha_2 plus ... plus alpha_d, always equals the rank n of the original tensor, since every occurrence among the n index slots is counted exactly once across the d components of alpha; this sum recovers directly the degree already associated with symmetric powers and homogeneous polynomials.

Correspondence With Other Symmetric Tensor Constructions

Multi-Indices and the Symmetric Power Basis

The exponent tuples (k_1, ..., k_d) used to index the symmetric power basis construction are precisely multi-indices in this sense, with each basis element e_1^{odot k_1} odot ... odot e_d^{odot k_d} labeled by exactly the multi-index recording how many times each basis vector appears; multi-index notation is therefore not a new concept introduced here but a formal name for the labeling scheme already used implicitly in that construction.

Multi-Indices and Monomials

Under the polynomial analogy, a multi-index alpha corresponds directly to the monomial x_1^{alpha_1} x_2^{alpha_2} ... x_d^{alpha_d}, commonly abbreviated x^alpha in multi-index notation, providing the exact dictionary entry linking an independent symmetric tensor component to its associated monomial coefficient.

Counting and Factorial Notation for Multi-Indices

The Multi-Index Factorial

A useful auxiliary quantity is the multi-index factorial, defined as alpha! equal to alpha_1! times alpha_2! times ... times alpha_d!, which appears as the denominator in the multinomial coefficient governing how many distinct orderings of the original index tuple collapse onto the single multi-index alpha:

recovering the multinomial coefficient that governs the orbit size and the associated coefficient scaling used when converting between tensor components and polynomial coefficients.

Counting All Multi-Indices of a Given Degree

The total number of distinct multi-indices alpha with d components satisfying |alpha| equal to n is exactly the independent symmetric count relation, C(d, n), since enumerating all such alpha is combinatorially identical to enumerating all canonical, non-decreasing index tuples of length n drawn from d values.

Practical Advantages of Multi-Index Notation

Rank-Independent Fixed Length

Unlike a canonical index tuple, whose length grows with the rank n, a multi-index always has fixed length d, the dimension of the underlying vector space, regardless of rank; this makes multi-index notation particularly convenient when working across tensors of several different ranks simultaneously, since every independent component, of any rank, is labeled by a vector of the same fixed length.

Direct Compatibility With Polynomial Manipulation

Because multi-indices correspond exactly to monomial exponents, arithmetic operations on multi-indices, such as componentwise addition corresponding to monomial multiplication, translate directly into corresponding operations on symmetric tensors under the polynomial analogy, making multi-index notation the natural bridge for translating symmetric tensor manipulations into polynomial manipulations and back.