Functionally Independent Invariants

• Structure of Functionally Independent Invariants is defined by predictable grades in the Hilbert series, revealing the degrees and multiplicities of basic invariant generators.
• The methodology uses tensor constructions and spinorial techniques to explicitly form invariants, with key steps outlined by contraction rules and representation theory.
• Analysis of syzygies and extended Lie algebras demonstrates how algebraic relations and dimension counting constrain the independence and completeness of invariant sets.

A functionally independent invariant is an invariant quantity or function constructed from a set of variables—typically arising in physical or mathematical structures with underlying symmetries—such that none of the invariants can be expressed as a function of the others. Understanding the structure of such invariants is fundamental in fields ranging from classical and quantum integrable systems, differential geometry, and invariant theory, to the classification of tensor fields and symmetry reduction in field theory. The structure of functionally independent invariants has both algebraic and geometric underpinnings, often illuminated through the technology of representation theory, Hilbert series, and tensor calculus.

1. Algebraic Framework and Hilbert Series

The construction of functionally independent invariants frequently begins by considering the polynomial algebra over a vector space $V$ carrying a (semi-simple) Lie algebra $\mathfrak{g}$-module structure. The subalgebra of invariants is defined as:

$$S^\mathfrak{g} = \{ P(X) \in S = \operatorname{Sym}^\bullet(V) \;|\; P(g \cdot X) = P(X) \; \forall g \in G \}$$

The structure of $S^\mathfrak{g}$ is analyzed through the Hilbert (partition) function:

$$P(t) = \sum_{k=0}^\infty \dim S^\mathfrak{g}_k \; t^k$$

The multiplicative decomposition of $P(t)$, when available,

$P(t) = \prod_{i=1}^s (1 - t^{d_i})^{-m_i}$

directly reveals the degrees $d_i$ and multiplicities $m_i$ of algebraically independent generators ("basic invariants")—the functionally independent invariants—at each order. In favorable cases, $S^\mathfrak{g}$ is freely generated, meaning there are no algebraic (syzygy) relations and the number and degrees of generators are completely predicted by $P(t)$. In many situations, $S^\mathfrak{g}$ is only a quotient $R/I$, with $I$ encoding the relations among generators.

When dealing with higher-order tensor fields, especially those with constraints—such as self-duality or tracelessness—the partition function often includes "ghost" factors or deficiencies at high orders, indicating subtle or nontrivial syzygies.

2. Construction of Explicit Invariants: Building Blocks and Tensor Methods

After obtaining the structure of the ring via the Hilbert series, explicit construction of invariants proceeds by identifying tensorial building blocks conforming to the symmetry and constraint properties of the system. For a given tensor field $F$ (of rank $k$), one constructs lower-rank symmetric (or antisymmetric) tensors by contraction:

• Quadratic invariants: $$M_{\mu\nu} = F_{\mu\lambda_1...\lambda_{k-1}} F_\nu^{\lambda_1...\lambda_{k-1}}$$
• Higher-degree invariants: Contracting multiple $M$ tensors, or forming higher-symmetry objects like $$N_{\mu_1...\mu_r}^{\nu_1...\nu_r} = F_{\mu_1...\mu_r\lambda_1...\lambda_{k-r}} F^{\nu_1...\nu_r\lambda_1...\lambda_{k-r}}$$
• In the context of self-dual forms (e.g., self-dual $5$-forms in $10$D), duality and tracelessness further refine the permitted contractions.

The combination and symmetrization of these building blocks, usually demanding explicit consideration of their irreducible decomposition under $\mathfrak{g}$ (e.g., via Young tableaux or character methods), yield all candidate invariants. The dimension of the space of invariants at each order computed from $P(t)$ constrains which contractions yield functionally independent invariants.

Explicit formulas are often provided for the lowest orders:

• Fourth order: $I_4 = M_{\mu\nu} M^{\nu\mu}$
• Sixth order: $I_6^{(1)} = \mathrm{tr}[M^3]$, $I_6^{(2)}$ a specific triple $N$-tensor contraction
• Higher orders: Increasingly numerous and complex combinations, as predicted by the Hilbert series

This process is illustrated, for instance, in the analysis of self-dual 5-forms in 10D, where a dimension count shows $126-45=81$ functionally independent invariants (Cederwall et al., 17 Sep 2025).

3. Syzygies and Freeness: Relations Among Invariants

The existence of functional independence among invariants requires that no nontrivial algebraic relations (syzygies) exist among the chosen generators. In practice, not all calculated invariants at a given order are algebraically independent; syzygies manifest as polynomial equations among the generators, reducing the maximal independent set.

When $S^\mathfrak{g}$ is freely generated as a polynomial ring, the Hilbert series product expression confirms the absence of such relations. When relations exist, $S^\mathfrak{g}$ is isomorphic to $R/I$ with $I$ nontrivial. For example, the Hilbert series may not be a product of elementary denominators, and zeros in the numerator (a "deficiency" in the dimension count) signal relations.

A further diagnostic is provided by explicit computation or constraint equations in tensor language; for the self-dual 5-form case, identities such as $$M_{[\mu_1}{}^\lambda F_{\mu_2\mu_3\mu_4\mu_5]\lambda} = 0$$ restrict the form of higher-order invariants, enforcing syzygies among them.

4. Tools: Extended Lie Algebras and Dimension Counting

Extended Lie algebras $\mathfrak{g}^+$ provide a theoretical tool for deducing properties of the invariant ring $S^\mathfrak{g}$. When $\mathfrak{g}^+$ is affine or infinite-dimensional, this may force $S^\mathfrak{g}$ to be trivial or very limited. In more favorable (finite-dimensional) circumstances, this approach is a sufficient criterion for invariants' freeness and can predict the generator structure in cases where direct calculation is infeasible.

Simple dimension counting also provides an upper bound on the number of functionally independent invariants: for a $V$ of dimension $n$ under $\mathfrak{g}$ of dimension $d$, the deficit $n-d$ gives the expected maximal number in generic situations. For the self-dual 5-form in 10D, this yields 81 functionally independent invariants.

5. Spinorial Versus Tensorial Methods

In cases where the tensor field admits a spinorial representation (for instance, by mapping an antisymmetric $k$-form to a symmetric spinor via Dirac or Majorana matrices), the same invariants can be constructed using spin-tensor contractions. This formalism is especially suitable for computer-algebra implementations, as it can streamline symmetrization and contraction steps. For self-dual 5-forms, the spin-tensor approach confirms and complements tensorial results for the construction of basic invariants and their dependencies.

6. Applications and Broader Implications

The classification and explicit determination of functionally independent invariants underlie many fundamental operations in theoretical and mathematical physics. Concrete applications include:

• Building invariant Lagrangians in high-dimensional field theories, particularly for tensor gauge fields of arbitrary rank or with self-duality constraints.
• Determining the space of local observables or conserved quantities in quantum field theories and integrable models, guided by invariants' algebraic structure.
• Systematic construction of interacting actions for higher-order tensor fields, where invariants provide the only nontrivial local terms allowed by symmetry.
• Extending the classification of invariants when the symmetry group is enlarged (e.g., to extended or affine Lie algebras), where the methods discussed provide sufficiency criteria for triviality or freeness of the invariant ring.

Rigorous methodology, combining representation-theoretic, combinatorial (Hilbert series), and explicit tensor calculations, is indispensable for advancing both the general theory and its physical applications.

7. Summary Table: Key Steps in Classifying Functionally Independent Invariants

Step	Description	Output
Define $S$ and $S^\mathfrak{g}$	Polynomial algebra and its invariant subalgebra under Lie group action	Ring $S^\mathfrak{g}}$
Compute Hilbert series $P(t)$	Encodes graded dimensions of $S^\mathfrak{g}$, predicts generator degrees	$P(t)=\prod_{i} (1-t^{d_i})^{-m_i}$
Construct building blocks	Define tensors $M_{\mu\nu}$, $N_{\mu_1...\mu_r}^{\nu_1...\nu_r}$, etc., respecting symmetry and constraints	Candidate invariants
Determine syzygies	Identify relations, possibly through vanishing Hilbert numerator or explicit contraction identities	Minimal set of independent invariants
(Optional) Use extended $\mathfrak{g}^+$	Use structure of extended algebra as criterion for triviality/freeness	Sufficiency for trivial/freely generated ring
Spinor/tensor formalism	Re-express invariants using spinorial notation for computational or structural advantages	Alternative explicit invariant bases

This approach provides a systematic means of classifying and explicitly constructing functionally independent invariants for tensor fields under Lie algebra actions, with broad utility in both pure mathematics and theoretical physics contexts (Cederwall et al., 17 Sep 2025).