15.12.2 Tensor Symmetric Basis Multi Index Label

Tensor Symmetric Basis Multi Index Label is the specific assignment of a unique multi-index to each basis element e^alpha of a graded piece Sym^n(V) of the symmetric algebra, functioning as the naming scheme that lets any basis vector be referred to, retrieved, and manipulated by its label alone rather than by an explicit written-out symmetric product expression. While multi-index notation itself was introduced as a general compact way of recording index multiplicities, the labeling role examined here is the more specific claim that this notation gives a complete, unambiguous, and bijective naming system for basis elements, a property that must be verified rather than assumed.

Treating the multi-index as a label, rather than merely as a bookkeeping convenience, emphasizes the functional requirement that distinct basis elements receive distinct labels and that every valid label corresponds to exactly one basis element, precisely the two properties required of any naming scheme intended to support reliable lookup and indexing.

The Labeling Bijection

From Basis Elements to Labels

Every basis element e^alpha of Sym^n(V), constructed as the symmetric product e_1^{odot alpha_1} odot ... odot e_d^{odot alpha_d}, is assigned the label alpha, a vector of d non-negative integers summing to n; this assignment is well-defined precisely because the symmetric product's commutativity and associativity guarantee that the multi-index alpha alone, without any record of construction order, fully determines the resulting basis element.

From Labels to Basis Elements

Conversely, every multi-index alpha of degree n, meaning every vector of d non-negative integers summing to n, determines a unique basis element by direct substitution into the symmetric product formula, so the labeling map is invertible: distinct multi-indices always produce distinct basis elements, and the earlier linear independence argument, transferred through the polynomial correspondence, confirms that no two different labels can name the same basis vector.

Properties Required of a Valid Labeling Scheme

Injectivity

The labeling scheme must be injective, meaning no two distinct multi-indices label the same basis element; this holds because distinct multi-indices correspond, under the polynomial analogy, to distinct monomials, and distinct monomials are linearly independent, ruling out the possibility that two different labels could refer to the same vector.

Surjectivity Onto the Basis

The labeling scheme must also be surjective onto the chosen basis, meaning every basis element of Sym^n(V) receives some multi-index label; this follows directly from the construction, since the basis itself is defined as the set of e^alpha ranging over every degree-n multi-index alpha, leaving no basis element unlabeled by construction.

Enumerating and Ordering Labels

Lexicographic Ordering of Multi-Indices

For practical indexing, multi-indices of a fixed degree n are commonly arranged in lexicographic order, comparing alpha_1 first, then alpha_2 if the first entries tie, and so on; this ordering provides a canonical linear sequence of labels, associating each multi-index with a definite position, useful for mapping the abstract set of labels onto a concrete, linearly addressed storage array.

Graded Lexicographic Ordering Across Ranks

When labels from multiple ranks must be compared or enumerated together, a graded lexicographic ordering is typically used, first comparing total degree |alpha| and only then, for labels of equal degree, falling back to ordinary lexicographic comparison of the individual entries; this ordering matches the natural progression through the graded structure of the symmetric algebra, listing lower-degree labels before higher-degree ones.

Multi-Index Labels in Relation to Canonical Index Tuples

Two Equivalent Naming Schemes

The multi-index label alpha and the canonical, non-decreasing index tuple used in independent component selection are two equivalent naming schemes for the same underlying orbit of index tuples, related by a direct, easily computed conversion: expanding alpha into a sorted list of index values reproduces the canonical tuple, and counting repetitions within the canonical tuple reproduces alpha.

Choosing Between the Two Schemes

The canonical index tuple scheme is more directly tied to the original tensor index notation, while the multi-index label scheme is more compact, of fixed length d regardless of rank, and aligns more directly with polynomial exponent notation; the choice between the two is a matter of which downstream use, direct tensor index manipulation or polynomial-style computation, is more prominent in a given context.

Practical Role of the Labeling Scheme

Supporting Direct Array Indexing

Once a fixed ordering of multi-index labels is established, such as lexicographic order, each label can be mapped to a single integer position, allowing the independent components of a symmetric tensor to be stored in an ordinary linear array indexed by this position, with the multi-index label serving as the intermediate, human-readable identifier bridging the abstract basis element and its concrete storage location.

Supporting Symbolic and Algebraic Manipulation

Beyond storage, the multi-index label supports direct symbolic manipulation of basis elements, since operations such as multiplying two basis elements e^alpha and e^beta correspond simply to adding their labels componentwise to produce e^{alpha+beta}, a direct reflection of the underlying polynomial multiplication rule for monomials expressed entirely in terms of the labels themselves.