LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Finite dimensionality of fixed-order local operator spaces

Statement

Let $J = J_{\overset{ˉ}{0}} \oplus J_{\overset{ˉ}{1}}$ be a finite-dimensional $Z / 2$ -graded complex vector space with a finite homogeneous basis $z_{1}, \dots, z_{n}$ , each generator assigned a rational engineering weight. Form $P = Sym (J_{\overset{ˉ}{0}}^{*}) \otimes Λ (J_{\overset{ˉ}{1}}^{*})$ . For finite $q \in Z_{\geq 0}$ and any $D \in Q$ , let $P_{q, D}$ be the span of supercommutative monomials of total generator degree at most $q$ and total engineering weight $D$ . Then $P_{q, D}$ is finite-dimensional. If a group acts linearly on $J$ preserving parity and the weight grading, its invariant subspace $P_{q, D}^{G}$ is also finite-dimensional.

Statement references

Burgess, §§2.2–3

Neubert, §2

Facts & Assumptions

Given: A finite homogeneous basis of $J$ , a finite degree bound $q$ , a rational target weight $D$ , and a linear parity- and weight-preserving group action on $J$ .

[F1]

Finite-jet local operators and engineering dimension: for a finite jet representation, the polynomial algebra is $Sym (J_{\overset{ˉ}{0}}^{*}) \otimes Λ (J_{\overset{ˉ}{1}}^{*})$ , and the selected operator space has an explicit finite maximum polynomial field degree.

Proof

1.1givenF1algebra

If $n = 0$ , the polynomial algebra consists only of constants, so its degree-at-most- $q$ subspace has dimension at most one. If $n > 0$ , every monomial is indexed by an exponent vector $(a_{1}, \dots, a_{n})$ with $0 \leq a_{j} \leq q$ for each $j$ , since $\sum_{j} a_{j} \leq q$ ; hence the full degree-at-most- $q$ algebra is spanned by at most $(q + 1)^{n}$ monomials. The exterior generators only reduce this finite set because their exponents are restricted to $0$ or $1$ .

2.1givenstep 1.1algebra∎

The subspace $P_{q, D}$ is spanned by the subset of those finitely many monomials whose assigned weights sum to $D$ , so it is finite-dimensional; if no monomial has weight $D$ , it is the zero space. The invariant vectors $P_{q, D}^{G}$ form a linear subspace of $P_{q, D}$ because the action preserves parity and weight, and every subspace of a finite-dimensional vector space is finite-dimensional.

Depends on

Finite-jet local operators and engineering dimension