We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite Q-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland-Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov-Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
We prove the Kirillov-Reshetikhin conjecture concerning certain finite dimensional representations of a quantum affine algebra $\Ua$ when $\hat\g$ is an untwisted affine Lie algebra of type $ADE$. We use $t$--analog of $q$--characters introduced by the author in an essential way.
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.
We study the graded limits of simple $U_q(\tilde{\mathfrak{sl}}_{n+1})$-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fix fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.
In this paper, we study a tensor product of perfect Kirillov-Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes isomorphic as a full subgraph to a certain disjoint union of Demazure crystals contained in a tensor product of highest weight crystals. Moreover, we show that this isomorphism preserves their gradings, where the grading on the tensor product of KR crystals is given by the energy function, and that on the other side is given by the minus of the action of the degr…
Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.
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.
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.
We show that a crystal base exists for any Kirillov-Reshetikhin module of type $D_n^{(1)}$, generalizing the result of [(KMN)^2] for the end nodes of the Dynkin diagram of $D_n$.
From a quantum $K$-matrix of the fundamental representation, we construct one for the Kirillov-Reshetikhin module by fusion construction. Using the $\imath$crystal theory by the last author, we also obtain combinatorial $K$-matrices corresponding to the symmetric tensor representations of affine type $A$ for all quasi-split Satake diagrams.
We give a new combinatorial model of the Kirillov-Reshetikhin crystals of type $A_n^{(1)}$ in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure without using a promotion operator. We also have a similar description of the Kirillov-Reshetikhin crystals associated to exceptional nodes in the Dynkin diagrams of classical affine or non-exceptional type.
In this paper we prove that every Kirillov-Reshetikhin module of type $G_2^{(1)}$ and $D_4^{(3)}$ has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al.
Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals B^{r,s} is established for all nonexceptional affine types. We also prove that the crystals B^{r,s} of type B_n^{(1)}, D_n^{(1)}, and A_{2n-1}^{(2)} are isomorphic to recently constructed combinatorial crystals for r not a spin node.
The conjecturally perfect Kirillov-Reshetikhin (KR) crystals are known to be isomorphic as classical crystals to certain Demazure subcrystals of crystal graphs of irreducible highest weight modules over affine algebras. Under some assumptions we show that the classical isomorphism from the Demazure crystal to the KR crystal, sends zero arrows to zero arrows. This implies that the affine crystal structure on these KR crystals is unique.
In this survey, we shall be concerned with the category of finite-dimensional representations of the untwisted quantum affine algebras when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and q-characters. We then concentrate on particular families of irreducible representations whose structure has recently been understood: Kirillov-Reshetikhin modules, minimal affinizations and beyond.
We give a proof of the periodicity of quantum $T$-systems of type $A_n\times A_\ell$ with certain spiral boundary conditions. Our proof is based on categorification of the $T$-system in terms of the representation theory of quantum affine algebras, more precisely on relations between classes of Kirillov-Reshetikhin modules and of evaluation modules.
The generalized quantum group of type $A$ is an affine analogue of 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 group of type $A$. As a 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-Reshetik…
This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of $\g[t]$-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.
We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras : we prove that the character 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 the 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 …
We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov--Reshetikhin crystals $B^{r,k}\otimes B^{1,l}$ of type $D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}$. The notion of $\pm$-diagrams is effectively used for the identification of classical highest weight elements in $B^{1,l}\otimes B^{r,k}$.
It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy functi…
We define and study the Kirillov--Reshetikhin modules for algebras of type $G_2$. We compute the graded character of these modules and verify that they are in accordance with the conjectures in math.QA/981202 and math.QA/0102113. These results give the first complete description of families of Kirillov--Reshetikhin modules whose isotypical components have multiplicity bigger than one.
We describe a cluster algebra algorithm for calculating q-characters of Kirillov-Reshetikhin modules for any untwisted quantum affine algebra. This yields a geometric q-character formula for tensor products of Kirillov-Reshetikhin modules. In simply laced type this formula extends Nakajima's formula for q-characters of standard modules in terms of homology of graded quiver varieties.
In this paper we give a direct proof of the equality of certain generating function associated with tensor product multiplicities of Kirillov-Reshetikhin modules for each simple Lie algebra g. Together with the theorems of Nakajima and Hernandez, this gives the proof of the combinatorial version of the Kirillov-Reshetikhin conjecture, which gives tensor product multiplicities in terms of restricted fermionic summations.
In this article, we show in the ADE case that the fusion product of Kirillov-Reshetikhin modules for a current algebra, whose character is expressed in terms of fermionic forms, can be constructed from one-dimensional modules by using Joseph functors. As a consequence, we obtain some identity between fermionic forms and Demazure operators. Since the same identity is also known to hold for one-dimensional sums of nonexceptional type, we can show from these results the X = M conjecture for type $A_n^{(1)}$ and $D_n^{(1)}$.
The fermionic formula conjectured by Kirillov and Reshetikhin describes the decomposition (as a module for $U_q(\frak g)$) of a tensor product of multiples of of fundamental representations $W(mλ_i)$ of the corresponding quantum affine algebras. In this paper, we show that the conjecture is true for the modules W(mλ_i), if $i$ is such that the corresponding simple root occurs in the highest root of the simple Lie algebra with multiplicity at most 2. In particular, the conjecture is established for all but a few nodes for the exceptional algebras.
The Kerov-Kirillov-Reshetikhin (KKR) bijection is the crux in proving fermionic formulas. It is defined by a combinatorial algorithm on rigged configurations and highest paths. We reformulate the KKR bijection as a vertex operator by purely using combinatorial R in crystal base theory. The result is viewed as a nested Bethe ansatz at q=0 as well as the direct and the inverse scattering (Gel'fand-Levitan) map in the associated soliton cellular automaton.
We prove the Kirillov-Reshetikhin (KR) conjecture in the general case : for all twisted quantum affine algebras we prove that the characters of KR modules solve the twisted Q-system and we get explicit formulas for the character of their tensor products (the untwisted simply-laced case was treated by Najakima, and the untwisted case by the author). The proof is uniform and provides several new developments for the representation theory of twisted quantum affine algebras, including twisted Frenkel-Reshetikhin q-characters (expected by Frenkel-Reshetikhin and Frenkel-Mukhin). We also prove the t…
We define a family of graded restricted modules for the polynomial current algebra associated to a simple Lie algebra. We study the graded character of these modules and show that they are the same as the graded characters of certain Demazure modules. In particular, we see that the specialized characters are the same as those of the Kirillov Reshetikhin modules for quantum affine algebras.
We establish the equality of the specialization $E_{wλ}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{wλ}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(λ)$ of a certain Demazure-type submodule $U_{w}^{+}(λ)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $λ$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_λ(x;q,0)$ of the symmetric Macdonald polynomial $P_λ(x;q,t)$ at $t=0$ and the grad…
We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang et.al. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules.
On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A^{(1)}_n$ via polytopes.
In proving the Fermionic formulae, combinatorial bijection called the Kerov--Kirillov--Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I (math.QA/0601630) written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial $R$ matrices on rigged configurations.
We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp. B_n(1) or A_{2n-1}(2)) Dynkin diagram that interchanges the 0 and 1 node. The involution sigma is defined in terms of new plus-minus diagrams that govern the D_n to D_{n-1} (resp. B_n to B_{n-1}, or C_n to C_{n-1}) branching. It is also shown that the crystal B^{r,s} is perfect. These crystals have been implemented in MuPAD-Combinat; the implementation is discuss…
Nakajima 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 th…
We study the polytope model for the affine type $A$ Kirillov-Reshetikhin crystals and prove that the action of the affine Kashiwara operators can be described in a remarkable simple way. Moreover, we investigate the combinatorial $R$-matrix on a tensor product of polytopes and characterize the map explicitly on the highest weight elements. We further give a formula for the local energy function and provide an alternative proof for the perfectness. We determine for any dominant highest weight element $Λ$ of level $\ell$ the elements $b_Λ, b^Λ$ involved in the definition of perfect crystals and …
We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a seco…
We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction …
We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi. We use this result to show that, given two tensor products of such KR crystals with the same maximal weight, after removing certain $0$-arrows, the two connected components containing the minimal/maximal elements are isomorphic. Based on the latter fact, we reduce a tensor product of higher level perfect KR crystals to one of single-column KR cry…
We establish the equality of the specialization $P_λ(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_λ(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) 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 to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy…
Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra ${\mathfrak g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its u…
Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of B^{r,s} (r not equal to n-1,n) in terms of tableaux of rectangular shape r x s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.
In this paper, we extend work of the first author on a crystal structure on rigged configurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged configurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in type $D_n^{(1)}$.
We prove that the specialization to q=1 of a Kirillov-Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov-Reshetikhin modules. We conjecture that these results holds for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q-characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating …
Naoi showed that tensor products of perfect Kirillov-Reshetikhin crystals are isomorphic to certain generalized Demazure crystals. We extend Naoi's results to address distinguished subsets of these tensor products. In type A, these are naturally described in terms of katabolizable tableaux which was key to resolving conjectures of Shimozono-Weyman and Chen-Haiman in arXiv:2007.04952.
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.
We study linear recurrence relations in the character solutions of $Q$-systems obtained from the Kirillov-Reshetikhin modules. We explain how known results on difference $L$-operators lead to a uniform construction of linear recurrences in many examples, and formulate certain conjectural properties predicted in general by this construcion.
Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger's slidi…
We compute the decomposition of representations of Yangians into g-modules for simply-laced Lie algebras g. The decomposition has an interesting combinatorial tree structure. Results depend on a conjecture of Kirillov and Reshetikhin.
A perfect crystal of any level is constructed for the Kirillov-Reshetikhin module of $U_q(D_4^{(3)})$ corresponding to the middle vertex of the Dynkin diagram. The actions of Kashiwara operators are given explicitly. It is also shown that this family of perfect crystals is coherent. A uniqueness problem solved in this paper can be applied to other quantum affine algebras.
The Kirillov--Reshetikhin modules W^{r,s} are finite-dimensional representations of quantum affine algebras U'_q(g), labeled by a Dynkin node r of the affine Kac--Moody algebra g and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^{2,s} corresponding to W^{2,s} for the algebra of type D_n^{(1)}.
Kerov, Kirillov, and Reshetikhin defined a bijection between highest weight vectors in the crystal graph of a tensor power of the vector representation, and combinatorial objects called rigged configurations, for type $A^{(1)}_n$. We define an analogous bijection for all nonexceptional affine types, thereby proving (in this special case) the fermionic formulas conjectured by Hatayama, Kuniba, Takagi, Tsuboi, Yamada, and the first author.
We study a family of power series characterized by a system of recursion relations (Q-system) with a certain convergence property. We show that the coefficients of the series are expressed by the numbers which formally count the off-diagonal solutions of the U_q(X^{(1)}_n) Bethe equation at q=0. The series are conjectured to be the X_n-character of a certain family of irreducible finite-dimensional U_q(X^{(1)}_n) -modules which we call the KR (Kirillov-Reshetikhin) modules. Under the above conjecture, these coefficients give a formula of the weight multiplicities of the tensor products of the …
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.
We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum dimension solution of the $Q$-system of type $D_r$ obtained by a certain specialization of classical characters of the Kirillov-Reshetikhin modules. To this end, we use various symmetries of quantum dimensions. As a result, we obtain an explicit formula for the positive solution of the level $k$ restricted $Q$-system of type $D_r$ which plays an important role in dilogarithm identities for conformal field theories.
