Extremal Weight Module Research Articles

New structure on the quantum alcove model with applications to representation theory and Schubert calculus

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K -group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.

Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C

We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product eμgchVx-(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{\\mu } \ ext {gch} ~V_{x}^{-}(\\lambda )$$\\end{document} of the (one-dimensional) character eμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{\\mu }$$\\end{document}, where μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} is a (not necessarily dominant) minuscule weight, with the graded character gchVx-(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{x}^{-}(\\lambda )$$\\end{document} of the level-zero Demazure submodule Vx-(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{x}^{-}(\\lambda )$$\\end{document} over the quantum affine algebra Uq(gaf)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_{\ extsf{q}}(\\mathfrak {g}_{\ extrm{af}})$$\\end{document} as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant K-group of the semi-infinite flag manifold QG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{Q}_{G}$$\\end{document} associated to a connected, simply-connected and simple algebraic group G of type C. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} is a standard basis element εk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varepsilon }_{k}$$\\end{document} in the weight lattice P of G.

Polyhedral realization of the crystal bases for extremal weight modules over quantized hyperbolic Kac-Moody algebras of rank 2

Let g be a hyperbolic Kac-Moody algebra of rank 2. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight λ, where λ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight.

Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by the crystal of all Lakshmibai-Seshadri paths of shape λ. Let be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector of weight λ, and let be the crystal basis of with the element corresponding to We prove that the connected component of containing is isomorphic, as a crystal, to the connected component of containing the straight line Furthermore, we prove that if λ satisfies a special condition, then the crystal basis is isomorphic, as a crystal, to the crystal As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in where are the fundamental weights, in the case that is symmetric.

Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17], the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17], our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

Existence of Kirillov–Reshetikhin crystals for near adjoint nodes in exceptional types

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.

Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and set where are the fundamental weights. Denote by the extremal weight module of extremal weight λ with the extremal weight vector, and by the crystal basis of with the element corresponding to We prove that (i) is connected, (ii) the subset of elements of weight μ in is a finite set for every integral weight μ, and (iii) every extremal element in is contained in the Weyl group orbit of (iv) is irreducible. Finally, we prove that the crystal basis is isomorphic, as a crystal, to the crystal of Lakshmibai-Seshadri paths of shape λ.

Extremal loop weight modules and tensor products for quantum toroidal algebras

ABSTRACTWe define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin–Jimbo–Miwa–Mukhin [7] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.

Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(\lambda)$ over a quantum affine algebra $U_{q}$, where $\lambda$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(\lambda)$ of the level-zero extremal weight module $V(\lambda)$ and the crystal $\mathbb{B}^{\frac{\infty}{2}}(\lambda)$ of semi-infinite Lakshmibai-Seshadri paths of shape $\lambda$, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $V_{w_0}^{+}(\lambda)$ in terms of the specialization at $t=0$ of the symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$.

Semi-infinite Lakshmibai–Seshadri path model for level-zero extremal weight modules over quantum affine algebras

We introduce semi-infinite Lakshmibai–Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of the previous result due to Naito and Sagaki (which uses Littelmann's Lakshmibai–Seshadri paths), in which the level-zero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight.

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

We construct by fusion product new families of irreducible representations of the quantum affinization $\mathcal {U}_{q}(\hat {sl}_{\infty })$ . The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $\mathcal {U}_{q}(sl_{n+1}^{tor})$ : we prove the conjectural link between $\mathcal {U}_{q}(\hat {sl}_{\infty })$ and $\mathcal {U}_{q}(sl_{n+1}^{tor})$ stated in Hernandez (J. Algebra 329, 147–162, 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv: 1305.3481 ).

Quantum extremal loop weight modules and monomial crystals

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.

Modular invariant partition functions for noncompactG/Ad(H)models

We propose a spectrum for a class of gauged noncompact $G/Ad(H)$ Wess-Zumino-Novikov-Witten models, including spectrally flowed images of highest, lowest, and mixed extremal weight modules. These are combined into blocks whose characters, due to the Lorentzian signature of the target space, are divergent and treated as formal expressions in need of regularization. Assuming that this is possible, we show that these extended characters transform linearly under modular transformations, and can be used to write down modular invariant partition functions.

Differential operators and crystals of extremal weight modules

We give a combinatorial realization of extremal weight crystals over the quantum group of type A + ∞ and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight A + ∞ -crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight A + ∞ -crystals and generalized Verma A ∞ -crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.

Level 0 Monomial Crystals

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of. Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

Generalized Green Functions and Unipotent Classes for Finite Reductive Groups, I

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

Crystal Base Elements of an Extremal Weight Module Fixed by a Diagram Automorphism

In our previous work, we introduced a bijection between the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra \(U_{q}(\mathfrak{g})\) of a Kac–Moody algebra \(\mathfrak{g}\) that are fixed by a diagram automorphism and the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra \(U_{q}(\breve {\mathfrak{g}})\) of the orbit Lie algebra \(\breve {\mathfrak{g}}\) of \(\mathfrak{g}\) . In this paper, we prove that this bijection commutes with the *-operation. As an application of this result we show that there exists a canonical bijection between the elements ℬ0(λ) of the crystal base ℬ(λ) of an extremal weight module of extremal weight λ over \(U_{q}(\mathfrak{g})\) that are fixed by a diagram automorphism and the elements of the crystal base \(\breve {\mathcal{B}}(\widehat {\lambda})\) of an extremal weight module of extremal weight \(\widehat {\lambda}\) over \(U_{q}(\breve {\mathfrak{g}})\) , if the crystal graph of \(\breve {\mathcal{B}}(\widehat {\lambda})\) is connected.

Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules

Let W(\varpi_k) be the finite-dimensional irreducible module over a quantized affne algebra U'_q(\mathfrak g) with the fundamental weight \varpi_k as an extremal weight. We show that its crystal B(W(\varpi_k)) is isomorphic to the Demazure crystal B^−(−Λ_0 + \varpi_k) . This is derived from the following general result: for a dominant integral weight λ and an integral weight µ , there exists a unique homomorphism U^-q(\mathfrak g)(u_λ \otimes u_µ) → V(λ + µ) that sends u_λ \otimes u_µ to u_{λ+µ} . Here V(λ) is the extremal weight module with λ as an extremal weight, and u_λ \in V(λ) is the extremal weight vector of weight λ .