We construct the Fock space representations of classical quantum affine algebras using combinatorics of Young walls. We also show that the crystal graphs of the Fock space representations can be realized as the abstract crystal consisting of proper Young walls. Finally, we give a generalized version of Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations of classical quantum affine algebras.
We deduce the Schaper formula for Hecke-algebras at root of unity from the Jantzen conjecture in the LLT-setup. This explains an observation due to R. Rouquier.
We prove that the Hall algebra U^-_n of the cyclic quiver of type A^(1)_{n-1} decomposes as a direct product of the quantum negative nilpotent subalgebra U_q^-(\hat{sl}_{n)) and C[q,q^{-1},z_1,z_2...]. We use this to prove a conjecture of Varagnolo and Vasserot in math/9803023 relating the canonical basis of U^-_n and the canonical basis of the level 1 Fock space representation of U_q(\hat{sl}_n) introduced by Leclerc and Thibon. This yields a proof of the positivity conjecture of Lascoux, Leclerc and Thibon, and a q-analogue of the Lusztig character formula for simple modules of the quantum g…
We construct the Fock space representations for the quantum affine algebra of type $C_2^{(1)}$ in terms of Young walls. Using this construction, we give a generalized Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations.
We prove the decomposition conjecture of Leclerc and Thibon for the Schur algebra. We also give a new approach to the Lusztig conjecture for the dimension of the simple U(sl_k)-modules at roots of unity via canonical bases of the Hall algebra.
We define canonical bases of the higher-level q-deformed Fock space modules of the affine Lie algebra sl(n)^. This generalizes the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of affine Kazhdan-Lusztig polynomials.
We deal with the representation theory of quantum groups and Hecke algebras at roots of unity. We relate the philosophy of Andersen, Jantzen and Soergel on graded translated functors to the Lascoux, Leclerc and Thibon-algorithm. This goes via the Murphy standard basis theory and the idempotents coming from the Murphy-Jucys operators. Our results lead to a guess on a tilting algorithm outside the lowest $p^2$ alcove, which at least in the $SL_2$-case coincides with Erdmann's results.
The aim of this paper is to generalize several aspects of the recent work of Leclerc-Thibon and Varagnolo-Vasserot on the canonical bases of the level 1 q-deformed Fock spaces due to Hayashi. Namely, we define canonical bases for the higher-level q-deformed Fock spaces of Jimbo-Misra-Miwa-Okado and establish a relation between these bases and (parabolic) Kazhdan-Lusztig polynomials for the affine Weyl group of type $A^{(1)}_{r-1}.$ As an application we derive an inversion formula for a sub-family of these polynomials. This article is an extended version of math.QA/9901032
We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and Thibon (proposed in the setting of Hecke algebras of type A, at roots of unity.)
We relate the canonical basis of the Fock space representation of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$, as defined by Leclerc and Thibon, to the canonical basis of its restriction to $U_q(\mathfrak{sl}_{n})$, regarded as a based module in the sense of Lusztig. More generally we consider the restriction to any parabolic subalgebra. We deduce results on decomposition numbers and branching coefficients of Schur algebras over fields of positive characteristic, generalising those of Kleshchev and of Tan and Teo.
We study a factor Hopf algebra $\mathfrak{PP}$ of the Malvenuto-Reutenauer convolution algebra of functions on symmetric groups ${\mathfrak{S}}=\oplus_{n\geq 0} \mathbb C[{\mathfrak{S}}_n] $ that we coined pre-plactic algebra. The pre-plactic algebra admits the Poirier-Reutenauer algebra based on Standard Young Tableaux as a factor and it is closely related to the quantum pseudo-plactic algebra introduced by Krob and Thibon in the non-commutative character theory of quantum group comodules. The connection between the quantum pseudo-plactic algebra and the pre-plactic algebra is similar to the …
The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with the Lie algebra $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. In this note, we formulate analogous conjectures for certain classes of irreducible representations of affine Hecke algebras of type B. We corrected typos.
In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type $B$ are described by using the lower global basis of symmetric crystals of $V_θ(λ)$. In this paper, we prove the existence of crystal bases and global bases of $V_θ(0)$ for any symmetric quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin diagram invo…
We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base for the Fock representation of quantum affine sl_n. We also show that the coefficients of the lower global crystal base coincide with certain affine Kazhdan-Lusztig polynomials. It is known that the coefficients of the global crystal base are q-analogues of decomposition numbers for Specht modules of the Hecke algebra of type A_n, and that the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum sl_k. Thus our al…
Lascoux, Leclerc and Thibon have introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these `ribbon functions' in analogy with Schur functions. In particular we will describe ribbon Pieri and Murnagham-Nakayama formulae, a ribbon Cauchy identity and an algebra involution which `conjugates' the ribbon functions. We will study these functions in the context of the action of the Heisenberg algebra on the Fock space representation of the quantum affine algebra U_q(sl_n)^, discovered by Kashiwara, Miwa and Stern. We wil…
The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $g$ where $g$ is $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type B. In this note, we present similar conjectures f…
Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_…
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov, and the Kac-Moody algebra A_l^(1) with the twisted algebra A_2l^(2). In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford supe…
These are lecture notes prepared for London Mathematical Society Symposium "Quantum Groups" held in Durham 19-29 July 1999. We give a survey on cyclotomic Hecke algebras. These algebras have been studied by Dipper, James, Malle, Mathas and the author. In lecture one, we start with definitions. We then go on to Specht module theory of cyclotomic Hecke algebras constructed by Dipper, James and Mathas. In lecture two, we explain Ginzburg and Lusztig's theory on affine Hecke algebras and its application to cyclotomic Hecke algebras due to the author. As Kazhdan-Lusztig bases of Hecke algebras desc…
We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from the quantum Fock space representation of $U_q(\widehat{\mathfrak{sl}}(n))$ introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering solvable lattice models known as `metaplectic ice' whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot. This allows us to expand the framework o…
