Nilpotent representations over equioriented cyclic quivers

Technical Summary: Nilpotent Representations Over Equioriented Cyclic Quivers

Problem Statement
This paper investigates the combinatorial and geometric properties of nilpotent representations of equioriented cyclic quivers. The study addresses two distinct algebraic contexts:

1. Linear Representations: Representations over a field $k$, specifically focusing on finite fields $\mathbb{F}_q$. A representation is defined as nilpotent if sufficiently long compositions of consecutive maps vanish.
2. Semirepresentations: Representations over the Boolean semiring $\mathbb{B} = \{0, 1\}$ (where $1+1=1$). Here, nilpotence is defined analogously via vanishing compositions.

The primary objectives are to establish explicit bijections between nilpotent representations and specific geometric or combinatorial structures, derive exact enumeration formulas for finite fields, and analyze the asymptotic probability of nilpotence in both settings.

Methodology
The authors employ a combination of linear algebraic bijections, combinatorial graph theory, and inclusion-exclusion principles.

• Geometric Bijections (Linear Case): Building on Leinster's bijection for nilpotent endomorphisms and its extension to pairs of maps, the authors construct an explicit bijection for cyclic quivers. They relate the set of nilpotent representations $N(\beta)$ to the full representation space $\text{Rep}(Q, \beta)$ and a specific subset $Z$ of tuples containing at least one zero vector. This relies on decomposing the representation space and utilizing the kernel-image structure of linear maps.
• Combinatorial Correspondence (Boolean Case): For representations over the Boolean semiring, the authors identify the representation space with the set of spanning subgraphs of a specific directed graph $G_\beta$. Under this identification, nilpotent semirepresentations correspond exactly to spanning directed acyclic graphs (DAGs).
• Enumeration Techniques:
  • In the finite field case, cardinality is derived by counting the size of the set $Z$ (using the principle of inclusion-exclusion on non-zero vectors) and applying the established bijection.
  • In the Boolean case, the authors derive a recursive formula for the number of DAGs by applying inclusion-exclusion over source vertices in the graph $G_\beta$. This reduces the problem to counting DAGs on induced subgraphs corresponding to strictly smaller dimension vectors.
• Asymptotic Analysis: The probability of nilpotence is analyzed by comparing the cardinality of nilpotent sets to the total representation space. For the Boolean case, bounds are established using properties of cycles in $G_\beta$ and strict total orders on vertices.

Key Contributions and Results

1. Explicit Bijection for Linear Representations:
   The paper establishes a bijection (Theorem 3.5):
   $$N(\beta) \times \prod_{i=1}^k V_i \cong \text{Rep}(Q, \beta) \times Z$$
   where $Z$ is the set of tuples $(v_1, \dots, v_k)$ such that at least one $v_i = 0$. This generalizes previous results for single endomorphisms and pairs of maps.

2. Enumeration over Finite Fields:
   Using the bijection, the authors derive a closed-form formula for the number of nilpotent representations $N(\beta)$ over $\mathbb{F}_q$ (Theorem 3.6):
   $$N(\beta) = q^{\beta \cdot \sigma \beta - \mathbf{1} \cdot \beta} \left( q^{\mathbf{1} \cdot \beta} - \prod_{i=1}^k (q^{\beta_i} - 1) \right)$$
   Consequently, the probability that a random representation is nilpotent is:
   $$\text{Prob}(N(\beta)) = 1 - \prod_{i=1}^k (1 - q^{-\beta_i})$$
   The paper also provides a refinement of this count by the rank of the representation using $q$-binomial coefficients (Theorem 3.9).

3. Enumeration over the Boolean Semiring:
   The authors identify nilpotent semirepresentations with spanning DAGs of $G_\beta$. They derive a recursive formula for the number of such representations $N_B(\beta)$ (Theorem 4.2):
   $$N_B(\beta) = \sum_{\gamma < \beta} (-1)^{\mathbf{1} \cdot (\beta - \gamma) - 1} \binom{\beta}{\gamma} 2^{(\beta - \gamma) \cdot \sigma \gamma} N_B(\gamma)$$
   where the sum is over dimension vectors $\gamma$ strictly less than $\beta$.

4. Asymptotic Behavior:

   • Finite Fields: As the dimension vector components grow, the probability of nilpotence approaches 1 if any $\beta_i$ grows, but the paper notes the probability tends to 0 if the sum of $q^{-\beta_i}$ tends to 0 (Corollary 3.14).
   • Boolean Semiring: The probability of nilpotence decays exponentially as the dimension vector scales. Specifically, if $\beta(n) = n\beta$, the asymptotic decay rate is determined by $m(\beta) = \min_i (\beta_i \beta_{i+1})$ (Corollary 4.7):
     $$\lim_{n \to \infty} \frac{\log_2(\text{Prob}(N_B(n\beta)))}{n^2} = -m(\beta)$$

5. Comparison with Set-Valued Representations:
   The paper relates linear nilpotence to "eventually constant" set-valued representations. It proves that a linear representation is nilpotent if and only if its underlying set-valued representation (via the forgetful functor) is eventually constant. Furthermore, the probability of nilpotence in the linear case over $\mathbb{F}_q$ is shown to be equal to the probability of eventual constancy in the set-valued case over sets of cardinality $q^{\beta_i}$ (Proposition 5.2).

Significance and Claims
The paper claims to provide a unified geometric and combinatorial framework for understanding nilpotence in cyclic quiver representations.

• Geometric Insight: By constructing explicit bijections, the work connects the abstract concept of nilpotence in cyclic quivers to the geometry of zero-fibers and specific subsets of vector tuples, extending known results for single endomorphisms to cyclic systems.
• Combinatorial Precision: The derivation of exact formulas for finite fields and recursive structures for the Boolean semiring offers precise counting tools where previously only qualitative or limited results might have existed.
• Asymptotic Understanding: The determination of the asymptotic decay rate for Boolean semirepresentations provides a quantitative measure of how "rare" nilpotence becomes as the system size increases in the Boolean setting.
• Interdisciplinary Connection: The work bridges linear representation theory, set-valued representations, and graph theory (DAGs), demonstrating that nilpotence in the linear context over finite fields is structurally equivalent to eventual constancy in the set-theoretic context.

The authors do not propose new experimental applications or future research directions beyond the mathematical results presented, maintaining a focus on the structural and enumerative properties of these algebraic objects.