Kirillov-Reshetikhin Modules Research Articles

Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification

Abstract We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 {\mathfrak{sl}_{n+1}} whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m {A_{m}} with m ≤ n {m\leq n} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category 𝒞 - {\mathscr{C}^{-}} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez, Geometric conditions for □ \square -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type D 4 {D_{4}} which do not arise from an embedding of A r {A_{r}} with r ≤ 3 {r\leq 3} in D 4 {D_{4}} .

Weak Faces of Highest Weight Modules and Root Systems

Chari and Greenstein (Adv. Math. 220(4), 1193–1221, 2009) introduced certain combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak {g}$ , which were important in studying Kirillov–Reshetikhin modules over $U_{q}(\widehat {\mathfrak {g}})$ and their specializations. Later, Khare (J. Algebra. 455, 32–76, 2016) studied these subsets for large classes of highest weight $\mathfrak {g}$ -modules (in finite type), under the name of weak- $\mathbb {A}$ -faces (for a subgroup $\mathbb {A}$ of $(\mathbb {R},+)$ ), and more generally, ({2};{1,2})-closed subsets. These notions extend and unify the faces of Weyl polytopes as well as the aforementioned combinatorial subsets. In this paper, we consider these “discrete” notions for an arbitrary Kac–Moody Lie algebra $\mathfrak {g}$ , in three prominent settings: (a) the weights of an arbitrary highest weight $\mathfrak {g}$ -module V; (b) the convex hull of the weights of V; (c) the weights of the adjoint representation. For (a) (respectively, (b)) for all highest weight $\mathfrak {g}$ -modules V, we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the sets of weights on the exposed faces (respectively, equal the exposed faces) of the convex hull of weights conv(wtV ). This completes the partial progress of Khare in finite type, and is novel in infinite type. Our proofs are type-free and self-contained. For (c) involving the root system, we similarly achieve complete classifications. For the weights of $\text {ad} \mathfrak {g}$ for any indecomposable Kac–Moody $\mathfrak {g}$ , we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the Weyl group translates of the sets of weights in certain “standard parabolic faces” (which also holds for highest weight modules). This was proved by Chari and her coauthors for Δ ⊔{0} in finite type, but is novel for other types.

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra mathfrak {g} and any height function xi adapted to an orientation Q of the Dynkin diagram of mathfrak {g}, Hernandez–Leclerc introduced a certain category mathcal {C}^{le xi } of representations of the quantum affine algebra U_q(widehat{mathfrak {g}}), as well as a subcategory mathcal {C}_Q of mathcal {C}^{le xi } whose complexified Grothendieck ring is isomorphic to the coordinate ring mathbb {C}[textbf{N}] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism {widetilde{D}}_{xi } on a torus mathcal {Y}^{le xi } containing the image of K_0(mathcal {C}^{le xi }) under the truncated q-character morphism. We prove that the restriction of {widetilde{D}}_{xi } to K_0(mathcal {C}_Q) coincides with the morphism overline{D} recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in mathcal {C}_Q, as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of overline{D} allows us to prove a conjecture by the first author on the distinguished values of overline{D} on the flag minors of mathbb {C}[textbf{N}]. We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra overline{mathcal {A}}_Q as a subquotient of K_0(mathcal {C}^{le xi }) naturally containing mathbb {C}[textbf{N}], and suggest the existence of an analogue of the Mirković–Vilonen basis in overline{mathcal {A}}_Q on which the values of {widetilde{D}}_{xi } may be interpreted as certain equivariant multiplicities.

Deformed Cartan Matrices and Generalized Preprojective Algebras I: Finite Type

AbstractWe give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiß–Leclerc–Schröer [33]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez–Leclerc [40] and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov–Reshetikhin modules.

Kirillov–Reshetikhin Modules of Generalized Quantum Groups of Type $A$

The generalized quantum group of type A is an affine analogue of the quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang–Baxter equation. In this paper we develop the crystal base theory for finite-dimensional representations of generalized quantum groups of type A . As the main result, we construct Kirillov–Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov–Reshetikhin modules, the combinatorial R matrix and the energy function on their tensor products.

Q-data and Representation Theory of Untwisted Quantum Affine Algebras

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $\Lambda(V,W)$ and $\Lambda^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.

Existence of Kirillov–Reshetikhin Crystals for Multiplicity-Free Nodes

We show that the Kirillov–Reshetikhin crystal B^{r,s} exists when r is a node such that the Kirillov–Reshetikhin module W^{r,s} has a multiplicity-free classical decomposition.

R matrix for generalized quantum group of type A

The generalized quantum group U(ϵ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra glM|N. We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U(ϵ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of glM|N. We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U(ϵ)-modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type AM−1(1) or AN−1(1).

Quantum Q-Systems and Fermionic Sums—The Non-Simply Laced Case

Abstract In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. [ 14] in its full generality, by extending the results of Di Francesco and Kedem [ 9] to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the Feigin–Loktev fusion product of Kirillov–Reshetikhin modules obtained by Chari and Venkatesh [ 5].

Uniform description of the rigged configuration bijection

We give a uniform description of the bijection \(\Phi \) from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form \(\bigotimes _{i=1}^N B^{r_i,1}\) in dual untwisted types: simply-laced types and types \(A_{2n-1}^{(2)}\), \(D_{n+1}^{(2)}\), \(E_6^{(2)}\), and \(D_4^{(3)}\). We give a uniform proof that \(\Phi \) is a bijection and preserves statistics. We describe \(\Phi \) uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that \(\Phi \) is a bijection for \(\bigotimes _{i=1}^N B^{r_i,s_i}\) when \(r_i\), for all i, map to 0 under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov–Reshetikhin crystals \(B^{r,1}\) using tableaux of a fixed height \(k_r\) depending on r in all affine types. Additionally, we are able to describe crystals \(B^{r,s}\) using \(k_r \times s\) shaped tableaux that are conjecturally the crystal basis for Kirillov–Reshetikhin modules for various nodes r.

Product Formula for the Limits of Normalized Characters of Kirillov–Reshetikhin Modules

Abstract The normalized characters of Kirillov–Reshetikhin modules over a quantum affine algebra have a limit as a formal power series. Mukhin and Young found a conjectural product formula for this limit, which resembles the Weyl denominator formula. We prove this formula except for some cases in type $E_8$ by employing an algebraic relation among these limits, which is a variant of $Q\widetilde{Q}$-relations.

On truncated Weyl modules

We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced . We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules.

LINEAR RECURRENCE RELATIONS IN Q-SYSTEMS VIA LATTICE POINTS IN POLYHEDRA

We prove that the sequence of the characters of the Kirillov–Reshetikhin (KR) modules $$ {W}_m^{(a)} $$ , m ∈ ℤm≥0 associated to a node a of the Dynkin diagram of a complex simple Lie algebra $$ \mathfrak{g} $$ satisfies a linear recurrence relation except for some cases in types E7 and E8. To this end we use the Q-system and the existing lattice point summation formula for the decomposition of KR modules, known as domino removal rules when $$ \mathfrak{g} $$ is of classical type. As an application, we show how to reduce some unproven lattice point summation formulas in exceptional types to finite problems in linear algebra and also give a new proof of them in type G2, which is the only completely proven case when KR modules have an irreducible summand with multiplicity greater than 1. We also apply the recurrence to prove that the function dim $$ {W}_m^{(a)} $$ is a quasipolynomial in m and establish its properties. We conjecture that there exists a rational polytope such that its Ehrhart quasipolynomial in m is dim $$ {W}_m^{(a)} $$ and the lattice points of its m-th dilate carry the same crystal structure as the crystal associated with $$ {W}_m^{(a)} $$ .

Randomized box–ball systems, limit shape of rigged configurations and thermodynamic Bethe ansatz

We introduce a probability distribution on the set of states in a generalized box–ball system associated with Kirillov–Reshetikhin (KR) crystals of type An(1). Their conserved quantities induce n-tuple of random Young diagrams in the rigged configurations. We determine their limit shape as the system gets large by analyzing the Fermionic formula by thermodynamic Bethe ansatz. The result is expressed as a logarithmic derivative of a deformed character of the KR modules and agrees with the stationary local energy of the associated Markov process of carriers.

Existence of Kirillov–Reshetikhin crystals of type [formula omitted] and [formula omitted

In this paper we prove that every Kirillov–Reshetikhin module of type G2(1) and D4(3) has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al.

$(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

Asymptotic Representations of Quantum Affine Superalgebras

We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of Frenkel-Hernandez for representations of the full quantum group.

Positivity and periodicity of Q-systems in the WZW fusion ring

We study properties of solutions of Q-systems in the WZW fusion ring obtained by the Kirillov–Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level k restricted Q-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba–Nakanishi–Suzuki on the level k restricted Q-systems.

Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

We study the classical limit of a tensor product of Kirillov-Reshetikhin modules over a quantum loop algebra, and show that it is realized from the classical limits of the tensor factors using the notion of fusion products. In the process of the proof, we also give defining relations of the fusion product of the (graded) classical limits of Kirillov-Reshetikhin modules.

A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$

We establish the equality of the specialization |$P_\lambda(x;q,0)$| of the Macdonald polynomial at |$t=0$| with the graded character |$X_\lambda(x;q)$| of a tensor product of “single-column” Kirillov–Reshetikhin modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated with Macdonald polynomials) and the quantum Lakshmibai–Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the |$t = 0$| limit of the symmetric Macdonald polynomial in the untwisted and non-simply laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.