Weyl group
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Twofold quadruple Weyl nodes in chiral cubic crystals Open
Conventional Weyl nodes are twofold band crossings that carry a unit monopole\ncharge, which can exist in condensed matter systems with the protection of\ntranslation symmetry. Unconventional Weyl nodes are twofold/multifold band\ncrossing…
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Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds Open
We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold $G/B$ . In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a …
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Light leaves and Lusztig's conjecture Open
We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the …
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The dual group of a spherical variety Open
Let $X$ be a spherical variety for a connected reductive group $G$. Work of\nGaitsgory-Nadler strongly suggests that the Langlands dual group $G^\\vee$ of\n$G$ has a subgroup whose Weyl group is the little Weyl group of $X$.\nSakellaridis-…
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The shape of a random affine Weyl group element and random core partitions Open
Let $W$ be a finite Weyl group and ${\\hat{W}}$ be the corresponding affine Weyl group. We show that a large element in ${\\hat{W}}$, randomly generated by (reduced) multiplication by simple generators, almost surely has one of $|W|$-speci…
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Twisted orbital integrals and irreducible components of affine Deligne–Lusztig varieties Open
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functio…
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Remarks on quantum unipotent subgroups and the dual canonical basis Open
We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.
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Conditioned random walks from Kac-Moody root systems Open
Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit t…
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Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element Open
Chapuy and Stump have given a nice generating series for the number of factorizations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to …
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Lax pairs of discrete Painlevé equations: (<i>A</i><sub>2</sub>+<i>A</i><sub>1</sub>)<sup>(1)</sup>case Open
In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the -surface q -Painlevé system, which has the affine Weyl group…
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Young Diagrams and Intersection Numbers for Toric Manifolds associated with Weyl Chambers Open
We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial f…
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On <i>E</i>–discretization of tori of compact simple Lie groups. II Open
Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine, and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgro…
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Dini Lipschitz functions for the Dunkl transform in the Space $$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$ L 2 ( R d , w k ( x ) d x ) Open
Using a generalized spherical mean operator, we obtain an analog of Theorem 5.2 in Younis (J Math Sci 9(2),301–312 1986) for the Dunkl transform for functions satisfying the $$d$$ -Dunkl Dini Lipschitz condition in the space $$\mathrm {L}^…
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On involutions in Weyl groups Open
Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the …
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Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection Open
Let g be a complex, semisimple Lie algebra. We prove the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra of g, which binds the quasi-Coxeter structure underlying the Casimir connection of g …
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A <i>q</i>-Analogue of the Higher Order Painlevé Type Equations with the Affine Weyl Group Symmetry of Type <i>D</i> Open
We present a q-difference analogue of the higher order differential equations of Painlevé type whose symmetry can be described by the affine Weyl group of type D, so-called the Sasano system. It is derived from the birational realization o…
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Macdonald's formula for Kac–Moody groups over local fields Open
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Invariant theory for quantum Weyl algebras under finite group action Open
We study the invariant theory of a class of quantum Weyl algebras under group actions and prove that the fixed subrings are always Gorenstein. We also verify the Tits alternative for the automorphism groups of these quantum Weyl algebras.
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Hodge-Deligne polynomials of abelian character varieties Open
Let F be a finite group and X be a complex quasi-projective F-variety. For r in $\mathbb{N}$, we consider the mixed Hodge-Deligne polynomials of quotients $X^r/F$ where F acts diagonally, and compute them for certain classes of varieties X…
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Metaplectic Ice for Cartan Type C Open
We use techniques from statistical mechanics to provide new formulas for Whittaker coefficients of metaplectic Eisenstein series on odd orthogonal groups, matching Friedberg and Zhang. We study a particular variation/generalization of the …
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Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A Open
Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorph…
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Combinatorial Auslander-Reiten quivers and reduced expressions Open
In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\widetilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order $\prec_{[\widetilde{w}…
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THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS Open
This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$ , the field with one element . Based on Part I …
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Generalized Dual-Root Lattice Transforms of Affine Weyl Groups Open
Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute…
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Cordial elements and dimensions of affine Deligne–Lusztig varieties Open
The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${\mathbf G}$ depends on two parameters: the $\sigma $ -conjugacy class $[b]$ and the element w in the Iwahori–Weyl group $\tilde {W}$ of ${\mathbf…
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Coxeter combinatorics and spherical Schubert geometry Open
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
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Maximal Newton Points and the Quantum Bruhat Graph Open
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
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Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras Open
For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott\ndefined a subcategory $\\mathcal{F}(w)$ of a module category over a\npreprojective algebra of Dynkin type. This paper aims at studying categorical\nproperties of $…
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On the Canonical Representatives of a Finite Weyl Group Open
Let K be a field and G a split connected reductive affine algebraic K-group. Let T be a split maximal torus of G, W its finite Weyl group, and R its root system. After fixing a realization of R in G and choosing a simple system for R, one …
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Quantum Representation of Affine Weyl Groups and Associated Quantum Curves Open
We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau variab…