表示论：2025–2026 年度综述

An annual survey of the math.RT arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 132 papers and organised into 37 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Quantum Affine Algebras and Integrable Models (17.8%) — Focuses on the representation theory of quantum affine algebras, cluster categorification, and their connections to quantum integrable systems and Weyl mutations.
• Finite and Reductive Group Representations (11.7%) — Investigates representations of finite groups, reductive groups, and classical groups, including block theory, Peter-Weyl theorem, and combinatorial techniques for group algebras.
• Tensor and Triangulated Category Structures (10.4%) — Explores the structural properties of tensor categories, triangulated categories, and module categories, with emphasis on Grothendieck-Verdier functors and cluster structures.
• Weight and Tilting Modules for Lie Algebras (9.8%) — Studies the classification and properties of weight modules, tilting modules, and irreducible representations for affine and super Lie algebras.
• Quiver Representations and Yangian Algebras (9.7%) — Examines quiver representations, quiver Yangians, and their associated invariants, including connections to gauge theories and symmetric group cohomology.
• Cluster Algebras and Hecke Algebra Cohomology (9.6%) — Analyzes cluster algebras, Hecke algebras, and graded gentle algebras, focusing on Hochschild cohomology and K-theoretic Hall algebras.
• Lie Superalgebras and Conformal Algebras (7.3%) — Investigates the structure and cohomology of Lie superalgebras, color Lie algebras, and Lie conformal algebroids, including graded Casimir elements.
• Geometric Langlands and Theta Correspondence (7.1%) — Explores the geometric and categorical aspects of the Langlands correspondence, theta correspondence, and Kazhdan-Lusztig bases in the context of duality.
• Non-Commutative Geometry and Operator Algebras (6.9%) — Addresses representation-theoretic aspects of non-commutative geometry, Hopf algebras, and operator algebras, including non-associative structures and crepant resolutions.
• Representations of General Linear and Special Linear Groups (5.7%) — Focuses on the representation theory of GL_n, SL_n, and their superalgebra counterparts, including W-algebras and asymptotic properties of group algebras.
• Character Theory and Homological Conjectures (4.0%) — Examines character theory, Galois automorphisms, and homological conjectures such as Han's conjecture and the Telescope conjecture in representation contexts.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.