Affine Weyl Group Research Articles

Sharpness versus bluntness in affine Weyl groups

In this paper we give an explanation of the bijection between arithmetic and geometric diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra.

On the Finkelberg–Ginzburg Mirabolic Monodromy Conjecture

Abstract We compute the monodromy of the mirabolic $\mathcal{D}$-module for all values of the parameters $(\vartheta ,c)$ in rank 1 and outside an explicit codimension 2 set of values in ranks 2 and higher. This shows in particular that the Finkelberg–Ginzburg conjecture, which is known to hold for generic values of $(\vartheta ,c)$, fails at special values even in rank 1. Our main tools are Opdam’s shift operators and intertwiners for the extended affine Weyl group, which allow for the resolution of resonances outside the codimension two set.

On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras

AbstractWe introduce the combinatorial model of J-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.

Strange expectations in affine Weyl groups

Strange expectations in affine Weyl groups

Nonintegrability of coupled Painlevé systems with affine Weyl group symmetry of type $$A_4^{(2)}$$

Nonintegrability of coupled Painlevé systems with affine Weyl group symmetry of type $$A_4^{(2)}$$

Shi arrangements and low elements in Coxeter groups

AbstractGiven an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

A higher-order generalization of an A4(1) -surface type q-Painlevé equation with W˜((A2N ⋊A1)(1)×A1(1)) symmetry

Recently, a birational representation of an extended affine Weyl group of A2N⋊A11 -type, which gives a higher-order generalization of an A41 -surface type q-Painlevé equation, was obtained. In this paper, we extend it to a birational representation of an extended affine Weyl group of A2N⋊A11×A11 -type. Moreover, we provide conjectures on periodic reductions from systems of PΔEs with the CAC property to Painlevé type q-OΔEs.

Affine symmetries for ABJM partition function and its generalization

Partially motivated by the fact that the grand partition function of the ABJM theory or its generalization is expressed by a spectral operator enjoying symmetries of the Weyl group, it was found that the grand partition function satisfies the q-Painlevé equation, which is constructed from the affine Weyl group. In this paper we clarify the affine symmetries of the grand partition function. With the affine symmetries, we find that the grand partition function extends naturally outside the fundamental domain of duality cascades and once the Painlevé equation holds in the fundamental domain, so does it outside.

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type A1(1). As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

Two-fold degeneracy of a class of rational Painlev\'e V solutions

We present a construction of a class of rational solutions of the Painlev\'e V equation that exhibit a two-fold degeneracy, meaning that there exist two distinct solutions that share identical parameters. The fundamental object of our study is the orbit of translation operators of $A^{(1)}_{3}$ affine Weyl group acting on the underlying seed solution that only allows action of some symmetry operations. By linking points on this orbit to rational solutions, we establish conditions for such degeneracy to occur after involving in the construction additional B\"acklund transformations that are inexpressible as translation operators. This approach enables us to derive explicit expressions for these degenerate solutions. An advantage of this formalism is that it easily allows generalization to higher Painlev\'e systems associated with dressing chains of even period $N>4$.

Poset Structure Concerning Cylindric Diagrams

Cylindric diagrams admit the structure of infinite $d$-complete posets with natural ordering. The purpose of this paper is to provide a realization of a cylindric diagram as a subset of an affine root system of type A via colored hook lengths and to present several characterizations of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the affine Weyl group.

Infinite affine, hyperbolic and Lorentzian Weyl groups with their associated Calogero models

We propose generalizations of Calogero models that exhibit invariance with respect to the infinite Weyl groups of affine, hyperbolic, and Lorentzian types. Our approach involves deriving closed analytic formulas for the action of the associated Coxeter elements of infinite order acting on arbitrary roots within their respective root spaces. These formulas are then utilized in formulating the new type of Calogero models.

Affine Bruhat order and Demazure products

Abstract We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

On additional symmetry and bilinearization of the q-Painlevé systems associated with the affine Weyl group of type A

On additional symmetry and bilinearization of the q-Painlevé systems associated with the affine Weyl group of type A

A $3 \times 3$ Lax Form for the $q$-Painlevé Equation of Type $E_6$

For the $q$-Painlevé equation with affine Weyl group symmetry of type $E_6^{(1)}$, a $2\times 2$ matrix Lax form and a second order scalar lax form were known. We give a new $3\times 3$ matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed.

Geometrical description of equations related to the E8 affine Weyl group

We present a method for the construction of the trajectory of a discrete Painlev\'e equation associated with the affine Weyl group E$_8^{(1)}$ on the weight lattice of said group. The method is based on the geometrical description of the lattice and the construction of the fundamental Miura relation. To this end we introduce the relation between the nonlinear variables and the corresponding $\tau$ functions. Our approach is heuristic and makes use of some simple rules of thumb in order to derive the result. Once the latter is obtained, verifying that it does indeed correspond to the equation at hand is elementary. We apply our approach to the explicit construction of the trajectory of well-known, E$_8^{(1)}$ associated, discrete Painlev\'e equations derived in previous works of ours. For each of them we investigate the possibility of defining an evolution by periodically skipping up to four intermediate points in the trajectory and identifying the resulting equation to one previously obtained, whenever the latter exists.

On-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation

We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlevé VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions $\bigl(\mathcal{F}^{(1)},\mathcal{F}^{(2)}\bigr)$, which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.

Higher-order generalizations of the - and -surface type q-Painlevé equations

In this paper, we construct higher-order generalizations of the - and -surface type q-Painlevé equations from the system of partial difference equations with the consistency around a cube property by periodic reduction. Moreover, we also show their extended affine Weyl group symmetries and Lax pairs.

Fractal hypersurfaces, affine Weyl groups, and wavelet sets

In this expository paper, we present some fundamental connections between iterated function systems, in particular those whose attractors are the graphs of multivariate real-valued fractal functions, and foldable figures, affine Weyl groups, and wavelet sets.

Derivation of Painlev\'e type system with $D_4^{(1)}$ affine Weyl group symmetry in a self-similarity limit

We show how the zero-curvature equations based on a loop algebra of $D_4$ with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlev\'e III models with four canonical variables and $D_4^{(1)}$ affine Weyl group symmetry.