character theory

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This work addresses the classification of Galois groups of irreducible septic polynomials over the rationals. We propose a neuro-symbolic hybrid method: leveraging algebraic invariants $J_0$–$J_4$ derived from binary transvectants, combined with explicit factorization, symbolic computation, and supervised learning, to construct the first million-scale database of normalized projected septic polynomials, each annotated with its exact Galois group. Our framework achieves high-accuracy identification of all seven transitive subgroups of $S_7$, notably improving discrimination of rare solvable groups. The database enables empirical analysis of subgroup distributions under constrained conditions. The resulting neuro-symbolic classifier balances interpretability and generalizability, and the methodology is extensible to higher-degree polynomials—advancing computational and data-driven research in constructive Galois theory.

This work establishes a prime-counting theory within graph homology classes, formulating a graph-theoretic analogue of the Dirichlet prime number theorem. Addressing the distribution of graph primes—i.e., irreducible closed walks—across homology classes, it introduces twisted adjacency matrices and conducts spectral analysis over the character group of the first homology. It rigorously proves that the associated spectrum is anti-symmetric about the origin and identifies the canonical character as yielding extremal spectral values. Building upon group representation theory and homological analysis, the paper develops a novel trace formula framework, yielding several exact trace formulas. This constitutes the first homologically refined characterization of prime distribution on graphs, bridging spectral graph theory, arithmetic geometry, and combinatorial topology through a unified paradigm.

This paper addresses the modeling challenge of string diagram rewriting in traced monoidal categories—categories supporting multi-output branching and input-output feedback connections. Methodologically: (1) it introduces and proves the hypergraph completeness of traced comonoid categories; (2) it adapts double-pushout (DPO) rewriting to traced string diagram grammars, ensuring locality and well-formedness of feedback operations; and (3) it axiomatizes traced structure to uniformly handle branching, merging, and cyclic connections. The contributions are threefold: (i) it establishes a unified formal foundation—combining equational theory and operational semantics—for dataflow and sequential circuits; (ii) every syntactic expression corresponds uniquely (up to isomorphism) to a hypergraph; and (iii) all rewrites preserve consistency with the traced axioms. This yields the first complete, hypergraph-based rewriting framework for traced monoidal structure.

This work addresses the problem of modeling invariant functions over symmetric matrices (under conjugation by permutations) and point clouds (under rotations, reflections, and point permutations). Methodologically, inspired by Galois theory, we construct the first lightweight universal approximator framework that yields separating invariant features of dimensionality only $O(n^2)$ for symmetric matrices and further optimizes to $O(n)$ for point clouds—breaking the bottleneck of traditional high-dimensional invariant representations. Our theoretical foundation integrates invariant algebra and generating sets of rational function fields, coupled with DeepSets architecture and orbit-separation analysis under group actions. Experiments on molecular property regression and point cloud distance prediction empirically validate almost-everywhere orbit separation, enabling universal approximation of weighted graph functions. The proposed framework significantly enhances both expressive power and computational efficiency of invariant representations.

This work addresses the reliable computation of Galois and monodromy groups for parametrized polynomial systems. To this end, it introduces a novel framework that integrates certified homotopy path tracking with homotopy graphs, enabling—for the first time—the rigorous numerical verification of monodromy group actions. By combining certified numerical algorithms with techniques from numerical algebraic geometry, the proposed method guarantees the mathematical correctness of its computational results. The approach has been successfully validated on a range of examples drawn from both pure and applied mathematics, demonstrating its effectiveness, reliability, and practical utility in analyzing the group-theoretic structures of complex polynomial systems.

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.

This work investigates how to directly learn algebraic properties of finite groups—such as commutativity, nilpotency, and solvability—from their Cayley graphs. To this end, we propose the first unified graph neural network (GNN) framework capable of end-to-end extraction of algebraic structure from Cayley graphs without requiring property-specific model customization. Employing a general-purpose GNN architecture and training protocol, our approach accurately predicts diverse algebraic properties across multiple families of finite groups. The results demonstrate that Cayley graph representations inherently encode rich algebraic information and establish a novel paradigm at the intersection of group theory and deep learning.

This study addresses the long-standing lack of explicit formulas and structural understanding of Chern classes expressed as symmetric polynomials across various bases of symmetric functions. By integrating multiple artificial intelligence systems with human mathematical insight, we establish a collaborative workflow that closes the research loop from experimental exploration and conjecture generation to symbolic proof. We demonstrate for the first time the feasibility of AI-augmented pure mathematical discovery, providing explicit expressions for the Chern and K-theoretic classes of $\mathrm{Sym}^d(\mathbb{C}^n)$. Furthermore, we prove refined positivity and a novel form of log-concavity for their Schur coefficients when expanded in the binomial basis, uncovering deep combinatorial structures in the rank-two case.

This work investigates the application of graph neural networks (GNNs) to determine the solvability of finite groups. Addressing this fundamental problem in abstract algebra, we propose the first approach that leverages GNNs on graph representations of finite groups—such as Cayley graphs—to learn from their structural properties and predict solvability. Experimental results demonstrate that the proposed model effectively distinguishes between solvable and non-solvable groups even on out-of-distribution instances not seen during training. These findings confirm that GNNs can capture deep algebraic properties inherent in group structures, thereby establishing a novel paradigm for integrating geometric representations of algebraic objects with machine learning techniques.