We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parametrize a basis for the tensor product of an irreducible representation with the space spanned by all partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.
Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and …
In this note we explain, in terms of finite dimensional representations of Lie algebras $\mathfrak{sp}_{2\ell}\subset\mathfrak{sl}_{2\ell}$, a combinatorial coincidence of difference conditions in two constructions of combinatorial bases for standard representations of symplectic affine Lie algebras.
We define a new family of symmetric functions which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalise the (dual of the) $k$-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the coh…
We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations complet…
In this paper we give a combinatorial parametrization of leading terms of defining relations for level $k$ standard modules for affine Lie algebra of type $C_{n}\sp{(1)}$. Using this parametrization we conjecture colored Rogers-Ramanujan type combinatorial identities for $n\geq 2$ and $k\geq 2$; the identity in the case $n=k=1$ is equivalent to one of Capparelli's identities.
The main result of this paper is a combinatorial description of a basis of standard level 1 module for the twisted affine Lie algebra $A_2^{(2)}.$ This description also gives two new combinatorial identities of Göllnitz (or Rogers--Ramanujan) type. Methods used through the paper are mainly developed by J. Lepowsky, R. L. Wilson, A. Meurman and M. Primc, and the crucial role in constructions plays a vertex operator algebra approach to standard representations of affine Lie algebras.
We consider principal subspaces $W_{L(kΛ_0)}$ and $W_{N(kΛ_0)}$ of standard module $L(kΛ_0)$ and generalized Verma module $N(kΛ_0)$ at level $k\geq 1$ for affine Lie algebra of type $B_2^{(1)}$. By using the theory of vertex operator algebras, we find combinatorial bases of principal ubspaces in terms of quasi-particles. From quasi-particle bases, we obtain character formulas for $W_{L(kΛ_0)}$ and $W_{N(kΛ_0)}$.
We give a realization of the infinity crystal for affine sl(2) using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of the infinity crystal in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the analogues in this case of the Mirkovic-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the only other rank two affine Kac-Moody algebra.
We give a combinatorial construction, not involving a presentation, of almost all untwisted affine Kac--Moody algebras modulo their one-dimensional centres in terms of signed raising and lowering operators on a certain distributive lattice ${\Cal B}$. The lattice ${\Cal B}$ is constructed combinatorially as a set of ideals of a ``full heap'' over the Dynkin diagram, which leads to a kind of categorification of the positive roots for the Kac--Moody algebra. The lattice ${\Cal B}$ is also a crystal in the sense of Kashiwara, and its span affords representations of the associated quantum affine a…
We give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type GL(m). Using quantum affine Schur-Weyl duality, this yields a combinatorial description of the composition factors of the tensor product of two evaluation modules of the quantum affine algebra of type sl(n).
We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.
There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and Drinfeld's unitarized R-matrix. We then give a new definition for the crystal commutor, which makes sense for any symmetrizable Kac-Moody algebra. We show that this new definition agrees with A. Henriques and J. Kamnitzer's definition in the finite type case, but we cannot prove our commutor remains a coboundary structure in other cases. Next, we extend these ide…
In this paper we construct bases of standard (i.e. integrable highest weight) modules $L(Λ)$ for affine Lie algebra of type $B_2\sp{(1)}$ consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces $W(Λ)$ of $L(Λ)$ by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence $W(kΛ_0)$ for $B_2\sp{(1)}$ and the integrable highest weight module $L(kΛ_0)$ for $A_1\sp{(1)}$ have the same parametrization of combinatorial bases and the same presentation $\mathcal P/\mathcal I$\…
J.~Lepowsky and R.~L.~Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via vertex operator constructions of standard (i.e. integrable highest weight) representations of affine Kac-Moody Lie algebras. A.~Meurman and M.~Primc developed further this approach for $\mathfrak{sl}(2,\mathbb C)\widetilde{}\ $ by using vertex operator algebras and Verma modules. In this paper we use the same method to construct combinatorial bases of basic modules for affine Lie algebras of type $C_{n}\sp{(1)}$ and, as a consequence, we obtain a series of Rogers-Ramanujan type identities.…
We use combinatorial description of bases of Feigin-Stoyanovsky's type subspaces of standard modules of level 1 for affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$ to obtain character formulas. These descriptions naturally lead to systems of recurrence relations for which we also find solutions.
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…
J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In a joint work with Arne Meurman this approach is developed further in the framework of vertex operator algebras. The main ingredients of that construction are defining relations for standard modules and relations among them. The arguments involve both representation theory and combinatorics, the final results hold only for affine Lie algebras $A_1^{(1)}$ and $A_2^{(1)}$. In the present paper some of those argume…
We first provide an explicit combinatorial description of the Auslander-Reiten quiver $Γ^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n \ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roo…
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 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 …
The natural representation of the quantized affine algebra of type A can be defined via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for a type A quantum group at a root of unity to the action of the quantized affine algebra as the rank tends towards infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional modules for the corresponding quatum group at a root of unity into Fock spaces of different charges and define an action of an affine quantum symm…
For an affine Lie algebra $\hat{\mathfrak g}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\hat{\mathfrak g}$-modules are the defining relations for level $k$ standard modules. In this paper we study a combinatorial structure of the leading terms of these relations for level $k=2$ standard $\hat{\mathfrak g}$-modules for affine Lie algebras of type $C_{n}\sp{(1)}$ and the main result is a construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will p…
The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall-Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C_n^{(1)}, A_{2n}^{(2)} and D_{n+1}^{(2)}. Through specialisation this yields generalisations for B_n^{(1)}, C_n^{(1)}, A_{2n-1}^{(2)}, A_{2n}^{(2)} an…
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of t…
For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.
We construct explicitly strong generators of the affine $\mathcal{W}$-algebra $\mathcal{W}^{K-N}(\mathfrak{sl}_N, f_{sub})$ of subregular type $A$. Moreover, we are able to describe the OPEs between them at critical level. We also give a description the affine $\mathcal{W}$-algebra $\mathcal{W}^{-N}(\mathfrak{sl}_N, f_{sub})$ in terms of certain fermionic fields, which was conjectured by Adamović.
We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.
We give a simple characterization of the highest weight vertices in the crystal graph of the level l Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words classically used in the decomposition of tensor products of fundamental $\mathfrak{sl}_{n}$-modules. This yields a combinatorial decomposition of the Fock spaces in their irreducible components and the branching law for the restriction of the irreducible highest weight $\mathfrak{sl}_{\infty}$-modules to $\hat{\mathfrak{sl}_{e}}$.
We consider principal subspace W(Λ) of integrable highest weight module L(Λ) for quantum affine algebra $U_q(\hat{\mathfrak{sl}}_{n+1})$. We introduce quantum analogues of the quasi-particles associated with the principal subspaces for $\hat{\mathfrak{sl}}_{n+1}$ and discover certain relations among them. By using these relations we find, for certain highest weight Λ, combinatorial bases of principal subspace W(Λ) in terms of monomials of quantum quasi-particles.
Let $\tilde{\mathfrak g}$ be an affine Lie algebra of the type $A_\ell^{(1)}$. We find a combinatorial basis of Feigin-Stoyanovsky's type subspace $W(Λ)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(Λ)$ is obtained as an "inductive limit" of the basis of $W(Λ)$.
A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\inft…
Let $\tilde{\mathfrak g}$ be an affine Lie algebra of type $A_\ell^{(1)}$. Suppose we're given a $\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus {\mathfrak g}_1$; then we also have the induced $\mathbb Z$-gradation of the affine Lie algebra $$\tilde{\mathfrak g}=\tilde{\mathfrak g}_{-1} \oplus \tilde{\mathfrak g}_0 \oplus \tilde{\mathfrak g}_1.$$ Let $L(Λ)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(Λ)$ is the $\tilde{\mathfrak g}_1$-submodule of $L(Λ)$ generated by the hi…
This is the first of a series of papers studying combinatorial (with no ``subtractions'') bases and characters of standard modules for affine Lie algebras, as well as various subspaces and ``coset spaces'' of these modules. In part I we consider certain standard modules for the affine Lie algebra $\ga,\;\g := sl(n+1,\C),\;n\geq 1,$ at any positive integral level $k$ and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky [FS]). The bases are given in terms of partitions: a color $i,\;1\leq i \leq n,$ and a charge $s,\; 1\leq s \leq k,$ are a…
In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra $\widehat{\mathfrak sl}(n+1,\mathbb C)$---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-…
We continue investigation of the universal weight function for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ started in arXiv:math/0610517 and arXiv:0711.2819. We obtain two recurrence relations for the universal weight function applying the method of projections developed in arXiv:math/0610398. On the level of the evaluation representation of $U_q(\hat{\mathfrak{gl}}_N)$ we reproduce both recurrence relations for the off-shell Bethe vectors calculated in arXiv:math/0702277 using combinatorial methods.
