On cyclic invariants of the free associative algebra

1. Problem Statement and Context

The paper addresses a central problem in noncommutative invariant theory: describing the structure and generators of the algebra of invariants $K\langle X_d \rangle^G$ under the action of a finite group $G$ on the free associative algebra $K\langle X_d \rangle$ (the algebra of non-commuting polynomials in $d$ variables).

• Background: While classical invariant theory (commutative case) is well-developed (e.g., Noether's theorem, Chevalley-Shephard-Todd theorem), the noncommutative case presents significant differences.
  • Finite Generation: Unlike the commutative case where invariants are often finitely generated, $K\langle X_d \rangle^G$ is finitely generated as an associative algebra only if $G$ acts by scalar multiplication (Dicks-Formanek, Kharchenko). For cyclic groups $C_d$ ($d \ge 2$) acting by permuting variables, the algebra is not finitely generated as an associative algebra.
  • S-Algebra Structure: Kharchenko introduced an action of the symmetric group $\text{Sym}(n)$ on the homogeneous component of degree $n$ by permuting variable positions. This equips the algebra with the structure of an S-algebra (an algebra with a compatible $\text{Sym}(n)$-action). Koryukin proved that for reductive groups, the algebra of invariants is finitely generated as an S-algebra.
• Specific Goal: The authors focus on the cyclic group $C_d$ acting on $K\langle X_d \rangle$ by cyclically shifting the variables ($x_1 \to x_2 \to \dots \to x_d \to x_1$). They aim to:
  1. Compute the Hilbert series of the invariant algebra.
  2. Construct a vector space basis and a set of free generators.
  3. Explicitly describe the generators of the invariant algebra as an S-algebra (specifically for $d=3$ and general $d$ over $\mathbb{C}$).

2. Methodology

The authors employ a combination of representation theory, combinatorial algebra, and basis transformations.

• Basis Transformation (Fourier Transform): To simplify the action of the cyclic group $C_d$, the authors perform a change of basis from the standard variables $X_d = \{x_1, \dots, x_d\}$ to a new basis $Y_d = \{y_0, y_1, \dots, y_{d-1}\}$ using roots of unity ($\varepsilon = e^{2\pi i/d}$).
  • The new basis elements are eigenvectors of the cyclic shift operator $\rho$: $\rho(y_k) = \varepsilon^k y_k$.
  • In this new basis, the condition for a monomial to be invariant becomes a simple congruence on the sum of indices modulo $d$.
• Hilbert Series Calculation: They utilize the Dicks-Formanek formula (the noncommutative analogue of Molien's formula) to compute the Hilbert series:
  $$H(K\langle X_d \rangle^G, t) = \frac{1}{|G|} \sum_{g \in G} \frac{1}{1 - \text{tr}(g)t}$$
• Combinatorial Construction of Generators:
  • They identify the set of monomials $y_{i_1} \dots y_{i_n}$ where $\sum i_k \equiv 0 \pmod d$ as a basis for the invariant algebra.
  • They define a "primitive" condition (no proper prefix sums to $0 \pmod d$) to isolate the free generators of the algebra.
• S-Algebra Analysis: They investigate the generators of the S-algebra $(K\langle X_d \rangle^{C_d}, \circ)$. They prove that the generators of the commutative invariant algebra $K[Y_d]^{C_d}$ (which are known to be finite in number) also generate the noncommutative S-algebra, provided one considers the action of the symmetric group to permute positions.

3. Key Contributions and Results

A. Hilbert Series and Dimension

The authors derive a closed-form expression for the Hilbert series of the cyclic invariants:
$$H(K\langle X_d \rangle^{C_d}, t) = \frac{1 - (d-1)t}{1 - dt}$$
This implies that the dimension of the homogeneous component of degree $n$ is:
$$\dim((K\langle X_d \rangle^{C_d})^{(n)}) = d^{n-1}$$
This result holds for any field $K$ of characteristic 0.

B. Free Generators of the Associative Algebra

They provide an explicit algorithm to construct a free generating set for $K\langle X_d \rangle^{C_d}$:

• Theorem 3.2: A set of free generators consists of all monomials $y_{i_1} \dots y_{i_n}$ such that $\sum_{k=1}^n i_k \equiv 0 \pmod d$, subject to the condition that for all $m < n$, the partial sum $\sum_{k=1}^m i_k \not\equiv 0 \pmod d$.
• Count: There are exactly $(d-1)^{n-1}$ free generators of degree $n$.
• Algorithm 3.3: An inductive procedure is provided to construct these generators step-by-step, separating them into "generators" ($Z_n$) and "non-generators" ($U_n$) based on prefix sums.

C. Generators of the S-Algebra

The paper addresses the finite generation of the S-algebra $(K\langle X_d \rangle^{C_d}, \circ)$.

• Theorem 4.1: The S-algebra is generated by the same monomials that generate the commutative invariant algebra $K[Y_d]^{C_d}$.
• Explicit Sets:
  • For $d=3$: Generated by $y_0, y_1y_2, y_1^3, y_2^3$.
  • For $d=4$: Generated by $y_0, y_1y_3, y_2^2, y_1^2y_2, y_2y_3^2, y_1^4, y_3^4$.
  • General $d$: The set includes $y_0$ and monomials $u$ of degree $\le d$ satisfying the index sum condition, which cannot be factored into smaller invariant monomials.

D. Transfer to Canonical Variables ($x_i$)

The authors translate these results back to the original variables $x_1, \dots, x_d$ for the case $d=3$.

• They show that the S-algebra $(K\langle X_3 \rangle^{C_3}, \circ)$ is generated by the cyclic sums:
  • $v_1 = x_1 + x_2 + x_3$
  • $v_2 = x_1x_2 + x_2x_3 + x_3x_1$
  • $v_{31} = x_1^2x_2 + x_2^2x_3 + x_3^2x_1$
  • $v_{32} = x_1x_2^2 + x_2x_3^2 + x_3x_1^2$
• They demonstrate via direct computation that these generators are minimal and that the system is independent of the basis choice (Conjecture 5.2).

4. Significance

• Bridging Commutative and Noncommutative Theory: The paper successfully demonstrates that while the associative algebra of cyclic invariants is infinitely generated, its structure as an S-algebra is finite and closely mirrors the commutative case. This validates the utility of the Kharchenko-Koryukin S-algebra framework.
• Explicit Construction: Unlike many theoretical results in invariant theory that prove existence without construction, this paper provides explicit algorithms and closed-form formulas for dimensions and generators.
• Foundation for Further Research: The results for $d=3$ and the general algorithm for $d$ provide a concrete foundation for studying more complex group actions and the structure of noncommutative symmetric functions.
• Resolution of Specific Cases: It settles the structure of invariants for the cyclic group $C_d$ in the free associative setting, a case previously less understood than the symmetric group case.

In summary, Boumova and Drensky provide a complete structural description of cyclic invariants in free associative algebras, proving that while the algebra itself is free and infinite, its S-algebra structure is finitely generated by specific cyclic sums, with explicit formulas for their count and construction.