Bernard Leclerc, Hyohe Miyachi
We give some closed formulas for certain vectors of the canonical bases of the Fock space representation of U_v(sl^_n). As a result, a combinatorial description of certain parabolic Kazhdan-Lusztig polynomials for affine type A is obtained.
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In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula.
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 compare the canonical bases of level-$1$ quantised Fock spaces in affine types $A^{(1)}$ and $A^{(2)}$, showing how to derive the canonical basis in type $A^{(2)}_{2n}$ from the the canonical basis in type $A^{(1)}_n$ in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.
A symmetric bilinear form on a certain subspace $\widehat{\mathbb T}^{\bf b}$ of a completion of the Fock space $\mathbb T^{\bf b}$ is defined. The canonical and dual canonical bases of $\widehat{\mathbb T}^{\bf b}$ are dual with respect to the bilinear form. As a consequence, the inversion formula connecting the coefficients of the canonical basis and that of the dual canonical basis of $\widehat{\mathbb T}^{\bf b}$ expanded in terms of the standard monomial basis of $\mathbb T^{\bf b}$ is obtained. Combining with the Brundan's algorithm for computing the elements in the canonical basis of $\…
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.
A Fock space is introduced that admits an action of a quantum group of type A supplemented with some extra operators. The canonical and dual canonical basis of the Fock space are computed and then used to derive the finite-dimenisonal tilting and irreducible characters for the Lie superalgebra osp(2|2n). We also determine all the composition factors of the symmetric tensors of the natural osp(2|2n)-module.
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 define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra $\glchap_n$. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$ specialized to a $n$th root of unity.
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…
In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors' research on the subjects, stated in the title, is given. By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately. For Hilbert space realizability a detailed determinant computations (extending Zagier's one-parameter formulas) are carried out. The inversion problem of the associated Gram matrices on Fock weight spaces is completely solved (Extended Zagier's conjecture) and a counterexample (for $n=8$) to the origin…
We investigate an infinite dimensional analog of the theory of Lagrangian manifolds with complex germs. To such a manifold we assign a canonical operator that depends on creation and annihilation operators. This operator is by definition the geometrical quantization for these isotropic manifolds with complex germs. We prove that for secondary quantized equations this quantization is the asymptotics for the Cauchy problem. Results of Berezin are used thouroughly in the construction of the canonical operator and in proofs of the theorems.
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_…
A complete Fock space representation of the covariant differential calculus on quantum space is constructed. The consistency criteria for the ensuing algebraic structure, mapping to the canonical fermions and bosons and the consequences of the new algebra for the statistics of quanta are analyzed and discussed. The concept of statistical transmutation between bosons and fermions is introduced.
This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula for defect $2$ blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "$q$-decomposition numbers", i.e.\ the canonical basis coefficients in the level-$1$ $q$-deformed Fock space of type $A^{(2)}_{2n}$; a special case of a conjecture of Leclerc and Thibon asser…
We construct homomorphic images of $su(n,n)^{\mathbb C}$ in Weyl Algebras ${\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\mathcal H}_{2nr}$, $su(n,n)^{\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their l…
We provide closed formulas for a large subset of the canonical basis vectors of the Fock space representation of $U_q(\widehat{\mathfrak{sl}}_e)$. These formulas arise from parallelotopes which assemble to form polytopal complexes. The subgraphs of the $\mathrm{Ext}^1$-quivers of $v$-Schur algebras at complex $e$-th roots of unity generated by simple modules corresponding to these canonical basis vectors are given by the $1$-skeletons of the polytopal complexes.
We obtain some simple relations between decomposition numbers of quantized Schur algebras at an n-th root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space previously introduced in a joint work with Thibon. It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible sub-representation generated by the …
On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-modules
本文提出了一种新的方法，用于计算不可约最高权 $U_q (\mathfrak{gl}_{\infty})$-模的 canonical basis，并将其应用于 Calogero-Moser 细胞特征标和 Ariki-Koike 代数的分解矩阵。
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.
We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.
In this paper we study a collections of operators $a(k)$ satisfying the "$q_{kl} $-canonical commutation relations" $a(k)a^{+}(l)-q_{kl}a^{+}(l)a(k) =δ_{kl} $ (corresponding for $q_{kl}=q$ to Greenberg (infinite) statistics, for $q=\pm 1$ to classical Bose and Fermi statistics).We show that $n!\times n!$ matrices $A_{n}(\{q_{kl}\})$ of scalar products of n-particle states is positive definite for all n if $|q_{kl}|<1$, all k,l, so that the above commutation relations have a Hilbert space realization. This is achieved by explicit factorizations of $A_{n}(\{q_{kl}\})$ as a product of matrices of…
Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology classes of a configuration space. The main observation of the present paper is that the screening operators contain more information. Specifically, at the chain level, the screening operators provide a certain canonical cocycle of the Virasoro (resp. affine Kac-Moody) algebra with coefficients in the de Rham complex of an operator-valued local system on the …
Prefactorization algebras for the conformal Laplacian: Central charge and Hilbert Fock space
本文研究共形拉普拉斯算子的预因子化代数，探讨其在不同维数下的自然性及中心电荷作用，揭示其与希尔伯特福克空间的关系。
We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum univer…
We describe the Fock space representations and crystal bases for the affine Kac-Moody Lie algebra of type $C_n^{(1)}$ in terms of coloured Young diagrams.
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