Row and column detection complexities of character tables

TL;DR

The paper formulates row-column dual complexity measures for finite-group character tables within a 2D TQFT framework, defining circle and circle-and-handle generator complexities for both the class algebra Z(C[G]) and the fusion algebra R(G). It leverages DW-TQFT and fusion-TQFT structures to relate generating subsets of conjugacy classes or irreps to eigenvalue distinctions provided by characters, and introduces averages such as 𝒞_gens(G) and ℛ_gens(G) to study global behavior. An empirical study over character tables up to size 30 (via CTblLib) reveals that class algebras typically admit smaller circle generator complexity than fusion algebras, with several conjectures proposed to capture asymptotic trends; the work also analyzes average class size and irreducible-dimension statistics, revealing systematic biases that motivate further theoretical and data-driven investigations. Overall, the results connect representation-theoretic data to quantum-complexity notions, offering a framework for understanding the computational aspects of distinguishing algebraic sectors in AdS/CFT-inspired contexts and suggesting directions for extending these insights to larger groups and holographic models.

Abstract

Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define row and column versions of detection complexities for character tables, and investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.

Figures (4)

• Figure 1: In blue, we have the average circle generator complexity of class algebras $\mathbb{E}(K_n, N_{\text{cls}})$ of dimension $n = |\mathrm{Cl}(G)| \in\{2,\dots,30\}$. In red, we have the average circle generator complexity of fusion algebras $\mathbb{E}(K_n, N_{\text{fus}})$ of dimension $n = |\mathrm{Irr}(G)| \in \{2, \dots, 30\}$.
• Figure 2: In blue, we have the average circle-and-handle generator complexity of class algebras $\mathbb{E}(K_n, N^{\text{ch}}_{\text{cls}})$ of dimension $n = |\mathrm{Cl}(G)| \in\{2,\dots,30\}$. In red, we have the average circle-and-handle generator complexity of fusion algebras $\mathbb{E}(K_n, N^{\text{ch}}_{\text{fus}})$ of dimension $n = |\mathrm{Irr}(G)| \in \{2, \dots, 30\}$.
• Figure 3: (a) The blue points show the average conjugacy class size of generating sets for the class algebra, $\mathbb{E}(E_{30,n}, \mathcal{C}_{\text{gens}})$, as a function of $n=\mathcal{N}(G)$. The red line is $y(n)=n$. The number of points above the red line is $198$ while the number of points below the red line is $193$. The number of points on the red line is $3$. (b) This plot displays $F_{\text{cls}}(30,l)$, the cumulative ratio of number of points above and below the line $y(n)=n$ in (a) from $n=1$ to $n=l$.
• Figure 4: (a) The blue points show the average squared dimension of generating sets for the fusion algebra, $\mathbb{E}(E_{30,n}, \mathcal{R}_{\text{gens}})$, as a function of $n=\mathcal{N}(G)$. The red line is $y(n)=n$. The number of points above the red line is $179$ while the number of points below the red line is $212$ and the number of points on the red line is $3$. (b) This plot displays $F_{\text{fus}}(30,l)$, the cumulative ratio of number of points above and below the line $y(n)=n$ in (a) from $n=1$ to $n=l$.

Theorems & Definitions (25)

• Definition 1: Commutative Frobenius algebra
• Proposition 1
• proof
• Theorem 1
• Definition 2: Combinatorial basis
• Definition 3: Circle generator complexity
• Definition 4: Circle-and-handle generator complexity
• Proposition 2
• Proposition 3
• proof