15.12.4 Tensor Symmetric Basis Dimension Count

Tensor Symmetric Basis Dimension Count is the specific act of determining the dimension of a graded piece Sym^n(V) by directly counting the elements of its constructed multi-index-labeled basis, rather than deriving the dimension abstractly through orbit counting or through the general symmetric power dimension relation established earlier. Because the basis elements e^alpha are in exact bijection with degree-n multi-indices, counting the basis reduces to the purely combinatorial task of counting how many vectors of d non-negative integers sum to n, and carrying out this count explicitly, using standard combinatorial techniques, provides a basis-first confirmation of the dimension formula already known from other approaches.

Approaching the dimension count from the basis side specifically, rather than repeating the orbit-based derivation, reinforces that the multi-index labeling scheme is not merely a convenient notation but a construction whose cardinality can be independently verified, closing the loop between the abstract dimension formula and its concrete realization as an actual, countable list of basis vectors.

Counting via Stars and Bars

The Combinatorial Setup

Counting the number of multi-indices alpha equal to (alpha_1, ..., alpha_d) of non-negative integers summing to n is a classical combinatorial problem solvable by the stars and bars technique: representing n as a row of n stars and inserting d minus 1 dividing bars among them to split the stars into d groups, with the sizes of the groups giving the values alpha_1 through alpha_d.

Resulting Count

The number of ways to arrange n stars and d minus 1 bars in a row is the number of ways to choose the positions of the bars among the n plus d minus 1 total symbols, giving:

exactly reproducing the count relation established previously, now derived directly from the structure of the basis labels rather than from orbit counting.

Cross-Checking the Count Against Small Cases

Direct Enumeration for Small Rank and Dimension

For d equal to two and n equal to three, direct enumeration of multi-indices (alpha_1, alpha_2) with alpha_1 plus alpha_2 equal to three lists exactly four possibilities, (3,0), (2,1), (1,2), and (0,3), matching the stars and bars formula's prediction of four choose one equals four, providing a concrete, hand-checkable instance of the basis dimension count.

Consistency Across Multiple Small Cases

Repeating this direct enumeration for other small values of d and n, such as d equal to three and n equal to two, yielding the six multi-indices (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1), consistently matches the formula's prediction, reinforcing confidence in the general stars and bars derivation before it is applied to larger, less easily enumerated cases.

Relation Between the Basis Count and Other Dimension Derivations

Agreement With Orbit-Based Counting

The stars and bars count of multi-indices coincides exactly with the earlier orbit-based count of non-decreasing index tuples, since both are counting the identical combinatorial object from two different but equivalent descriptions, one phrased in terms of distributing indistinguishable repetitions and the other in terms of sorted sequences.

Agreement With Monomial Counting

The same count also matches the number of degree-n monomials in d variables under the polynomial analogy, since each monomial x_1^{alpha_1} ... x_d^{alpha_d} corresponds to exactly one multi-index alpha, reaffirming that the basis dimension count, the orbit count, and the monomial count are three names for the same underlying combinatorial quantity approached from three different constructions.

Significance of Deriving the Count From the Basis Directly

Confirming Completeness of the Basis Construction

Deriving the dimension independently from the basis itself, rather than relying solely on the general formula, confirms that the symmetric power basis construction produces neither too few nor too many linearly independent elements: the count matches the previously established dimension exactly, verifying that no basis element was missed and none was redundantly included.

A Template for Verifying Other Combinatorial Bases

The technique of counting a basis directly via its label set, and cross-checking that count against an independently derived dimension formula, serves as a template applicable whenever a new basis or labeling scheme is proposed for a graded piece of the symmetric algebra, providing a standard method for validating that any newly constructed basis has the correct size before it is put to further use.