The Kirillov-Reshetikhin conjecture and solutions of T-systems

By D. Hernandez. 2005

173 citations recorded by the source corpus.

Abstract We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima in [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.], [H. Nakajima, t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Th. 7 (2003), 259–274.] with geometric arguments (main result of [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/326ddf4a4e104b55b7e9bf45741b646ff5f829ec

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Citations in the corpus

Path description of type B q-characters (2011, methodology): “(bg)-module can be computed recursively by an algorithm [FM01] starting from its highest l-weight. The algorithm works for the class of Kirillov-Reshetikhin modules, see [Nak04, Nak03] for ADE types, [Her06] for other types. It also works for some minimal aﬃnizations [Her07]. However, the straightfo”
Existence of Kirillov–Reshetikhin crystals for nonexceptional types (2007, background): “Quantum affine algebra.”
On Minimal Affinizations of Representations of Quantum Groups (2006, methodology): “The following result was proved in [Nak4, Nak5] for simply-laced types, and in full generality in [ H4 ] (see [FM1] for previous results).”
Cluster algebras and representation theory (2010, background): “This was conjectured by Kuniba, Nakanishi and Suzuki [44], and proved by Nakajima [51] (see also [31] for the non simply-laced cases).”
t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8 (2006, background): “By the property (1) we call the monomial mP corresponding to the Drinfeld polynomial P l–highest weight monomial.”
Quiver varieties and cluster algebras (2009, methodology): “The computation of the transition matrix is hard to use in practice, like the KazhdanLusztig polynomials.”
Quantum toroidal algebras and their representations (2008, background): “For quantum ane algebras, an analogous result holds with the usual tensor product, and was proved in [Nak5, Nak4] for simply-laced cases and in [ He5 ] for non-simply-laced cases.”
Demazure Modules, Fusion Products and Q-Systems (2013, background): “An obvious consequence of results proved in [23], [29],[30] is that the quantum Kirillov–Reshetikhin modules satisfy the required conditions on the dimensions in (5.”
Cluster algebras and quantum affine algebras (2009, unknown): “in [KNS] that the characters of the Kirillov-Reshetikhin modules are solutions of a T-system, and this was later proved by Nakajima [N2] in the simply-laced case, and by Hernandez in the general case [H3]. Now it is easy to notice that in the simply-laced case the equations of a T-system are exactly”
T-systems and Y-systems in integrable systems (2010, background): “ThisW (a) m (u) is equal toW (a) m,qtuq i in [67, 68].”
Periodicities of T-systems and Y-systems (2008, background): “Later this conjecture was proved for Uq(ˆ) by [N3, Her1 ].”
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules (2013, unknown): “r T (1) l,r+2, (l ≥1,r ∈Z), T(2) 2l+1,r+1T (2) 2l+1,r−1 = T (2) 2l,r+1T (2) 2l+2,r−1 +T (1) l+1,rT (1) l,r+2, (l ≥0,r ∈Z). 11 It was conjectured in [KNS1], and proved in [N2] (for gof type A,D,E) and [H] (general case), that the q-characters of the Kirillov-Reshetikhin modules ofUq(bg)satisfy the co”
BAXTER'S RELATIONS AND SPECTRA OF QUANTUM INTEGRABLE MODELS (2013, methodology): “on on i by using the following relation for 1 ≤ i ≤ n−3: [L(Y1,a)⊗L(Yi,aqi+1)] = [L(Y1,aYi,aqi+1)]+[L(Yi+1,aqi)]. This relation can be easily established following the proof of T-systems relations in [H2]. By [FM], it suﬃces to prove that dominant monomials have the same multiplicities when the q-ch”
Representations of Algebras and Related Topics (2011, background): “ of equations indexed by i∈I, k∈N∗, and a∈C∗, called the T-system4: [W(i) k,a][W (i) k,aq2] = [W (i) k+1,a][W (i) k−1,aq2] + Y j6= i [W(j) k,aq] −aij . (11) This was conjectured in [28] and proved in [34, 21]. Using these equations, one can calculate inductively the expression of any [W(i) k,a] as a”