Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras

Abstract

We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresponding normalized R-matrices has a pole.

35 pages, Latex

Cited by in corpus (95)

• T-systems and Y-systems in integrable systems
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• Cluster algebras and quantum affine algebras
• Baxter's Relations and Spectra of Quantum Integrable Models
• Quantum Grothendieck rings and derived Hall algebras
• Baxter's Q-operators for supersymmetric spin chains
• Asymptotic representations and Drinfeld rational fractions
• Solutions of the T-system and Baxter equations for supersymmetric spin chains
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• On minimal affinizations of representations of quantum groups
• Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
• Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• Path description of type B q-characters
• Simple tensor products
• Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras
• Paths, tableaux, and q-characters of quantum affine algebras: the C_n case
• Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
• Langlands duality for finite-dimensional representations of quantum affine algebras
• Extended T-systems
• Q-data and representation theory of untwisted quantum affine algebras
• Finite type modules and Bethe ansatz equations
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Difference L operators related to q-characters
• Polynomial Relations for q-Characters via the ODE/IM Correspondence
• Local Weyl modules and cyclicity of tensor products for Yangians
• On irreducibility of tensor products of evaluation modules for the quantum affine algebra
• An Algebraic Bootstrap for Dimensionally Reduced Quantum Gravity
• Difference L operators and a Casorati determinant solution to the T-system for twisted quantum affine algebras
• Quantum loop modules
• Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types
• Schur Positivity and Kirillov-Reshetikhin Modules
• Integrable and Weyl modules for quantum affine sl_2
• A note on odd reflections of super Yangian and Bethe ansatz
• Weyl, Fusion and Demazure modules for the current algebra of sl_{r+1}
• M-systems and Cluster algebras
• Cyclicity and R-matrices
• Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity
• Graded quiver varieties and singularities of normalized R-matrices for fundamental modules
• Yangian characters and classical W-algebras
• A geometric $q$-character formula for snake modules
• Yangian doubles of classical types and their vertex representations
• A Drinfeld-type presentation of the orthosymplectic Yangians
• Quantum Grothendieck rings as quantum cluster algebras
• Extended $T$-System of Type $G_2$
• Braid group action and quasi-split affine $ı$quantum groups I
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Poles of finite-dimensional representations of Yangians
• Yangians and Baxter's relations
• On Hopf algebraic structures of quantum toroidal algebras
• Deformed Cartan matrices and generalized preprojective algebras I: Finite type
• Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
• Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras
• On the extended T-system of type $C_3$
• Deformations of $\mathcal{W}$ algebras via quantum toroidal algebras
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Affine Kac-Moody algebras, integrable systems and their deformations
• A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras
• Elliptic quantum groups and Baxter relations
• On Tensor Products of a Minimal Affinization with an Extreme Kirillov-Reshetikhin Module for type A
• Tensor K-matrices for quantum symmetric pairs
• Braid group actions, Baxter polynomials, and affine quantum groups
• Theta series for quantum loop algebras and Yangians
• Cluster realization of Weyl groups and $q$-characters of quantum affine algebras
• Graded limits of minimal affinizations and Beyond: the multiplicty free case for type E6
• On q,t-characters and the l-weight Jordan filtration of standard quantum affine sl2 modules
• Path description for $q$-characters of fundamental modules in type $C$
• Cluster algebra structure on the finite dimensional representations of $U_q(\widehat{A_3})$ for $l$=2
• On Frenkel-Mukhin algorithm for q-character of quantum affine algebras
• Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;\alpha )$
• t-analogs of q-characters of quantum affine algebras of type $A_n$, $D_n$
• Boundary transfer matrices arising from quantum symmetric pairs
• Intertwiners of representations of untwisted quantum affine algebras and Yangians revisited
• Intertwiners of representations of twisted quantum affine algebras
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Tensor products and $q$-characters of HL-modules and monoidal categorifications
• Kirillov-Reshetikhin crystals $B^{1,s}$ for ${\widehat{\mathfrak{sl}}}_n$ using Nakajima monomials
• On the minimal affinizations of type $F_4$
• Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs
• Non-Minimality of Certain Irregular Coherent Preminimal Affinizations
• Quantum affine algebras and Grassmannians
• A Representation-Theoretic Approach to $qq$-Characters
• Quantum periodicity and Kirillov-Reshetikhin modules
• A new description of the bicrystal $B(\infty )$ and the extended crystal
• Quantum toroidal algebras and their representations
• Hernandez-Leclerc modules and snake graphs
• Inflations for representations of shifted quantum affine algebras
• Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs
• On the properties of the density matrix of the ${\mathfrak{sl}}_{n+1}$-invariant model
• Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system
• Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux
• Inflations among quantum Grothendieck rings of type A