Affine Weyl Group Research Articles

Universal central extensions of linear groups over rings of non-commutative Laurent polynomials, associated $K_1$-groups and $K_2$-groups

We prove that linear groups over rings of non-commutative Laurent polynomials $D_{\tau}$ have Tits systems with the corresponding affine Weyl groups and have universal central extensions if $|Z(D)|\geq 5$ and $|Z(D)|\neq 9$. We also determine structures of $K_1$-groups and identify generators of $K_2$-groups.

A transitivity result for ad-nilpotent ideals in type [formula omitted

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal I meets a unique largest nilpotent orbit OI in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals I such that OI=O for a fixed nilpotent orbit O. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan–Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

Coupled Painleve systems with affine Weyl group symmetry of type Dl(1)

D_l(1) 型のaffine Weyl 群対称性をもつ微分方程式系について報告する. これらは, 適当な正準変換の下で正則性を保つ多項式Hamilton 系として特徴づけられ, 結合型Painleve V または結合型Painleve VI として記述できる.Garnier 系およびE_6(1) 型のaffine Weyl 群対称性をもつ微分方程式系についての新しい結果も紹介する.

Ricci curvature, graphs and eigenvalues

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjacency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups.

The Bruhat order, the lookup conjecture and spiral Schubert Varieties of type $Ã_2$

Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac–Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type Ã2 on a Euclidean space V≅ℝ2 to study the Bruhat order on W. We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of Boe and Graham (which is a conjectural simplification of the Carrell–Peterson criterion for rational smoothness) to type Ã2. Computational evidence suggests that the only Schubert varieties in type Ã2 where the “nontrivial” case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety X(w) of type Ã2. The proof uses descriptions we obtain of the elements x≤w and of the rationally smooth locus of X(w) in terms of the W-action on V. As a consequence we describe the maximal nonrationally smooth points of X(w). The results of this paper are used in a sequel to describe the smooth locus of X(w), which is different from the rationally smooth locus.

Extended ⋁-systems and trigonometric solutions to the WDVV equations

Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration—a configuration in a space one-dimension higher—satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281–4299 (1996)]. Symmetries of such resulting solutions to the WDVV equations are studied, in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions, and for certain values of the free data, one obtains a transformation from extended ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.

Affine Hecke algebras and their representations

This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations.Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with parameters in R≥1, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.

Maximal Newton Points and the Quantum Bruhat Graph

We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson’s isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne–Lusztig varieties in the affine flag variety.

Generalizedq-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

AbstractIn this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

Elliptic Double Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

Q-DISCRETE PAINLEVÉ EQUATIONS: THEIR HIERARCHIES AND PROPERTIES

The main objective of this thesis is to derive hierarchies of q-discrete PainelevA© equations. Some of the important properties of these hierarchies will also be given, namely Lax pairs, BA¤cklund transformations, solutions of their asso- ciated linear problems for special values of parameters and their symmetry groups. To construct these hierarchies, we apply a geometric reduction and a stair- case method on a multi-parameteric generalized lattice modified Korteweg-de Vries equation. In addition, the property of consistency around the cube is used in order to find BA¤cklund transformations. Starting with the base case of q-discrete second, third and fourth PainlevA© equations on A5 initial-values surface, new hierarchies of q-discrete third and fourth PainlevA© equations are discovered, and we also rediscover the hierarchy of q-discrete second PainlevA© equation. In this thesis, we provide the Lax pairs for each member in these hierarchies. Using the consistency around the cube, we also provide the BA¤cklund transformation for the entire hierarchy of q-discrete second and third PainlevA© hierarchies. We generate a hierarchy of special solutions starting with seed solutions for q-discrete second and third PainlevA© hierarchies. An assumption made is that particular parameter values would enable the ability to diagonalize the Lax pair. As a consequence, we found that the as- sociated linear problem for the three hierarchies can be solved in terms of q-Gamma function. Furthermore, the hierarchy of q-discrete fourth PainlevA© hierarchy can be reduced to one equation that can be linearlized to become Riccati equation which has hypergeometric special solutions. Finally, we investigated the affine Weyl group structure of the symmetry group for each hierarchy. In this thesis, we construct the explicit representation of the symmetry group for the first and second member of these hierarchies. The collection of new hierarchies, their Lax pairs, BA¤cklund transforma- tions, the resultant symmetry groups and special solutions comprise the new results of this thesis. This thesis contains material published in [10] in collabo- ration with N. Joshi and D. Tran and myself. The material of this paper is pre- sented in Chapter 3, and is related to qPII and qPIII hierarchies, their Lax pair and examples. In Chapter 4, BA¤cklund transformation of qPII and qPIII hierarchies, includes material from the above-mentioned paper. Similarly, Chapter 5 reports on results about solutions of the linear problem from the above paper. However, we emphasize that all the results about qPIV through out the thesis are completely new and unpublished. Chapter 6 includes unpublished material even for qPII and qPIII hierarchies.

Local finiteness of the twisted Bruhat orders on affine Weyl groups

Abstract In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra 163 1994, 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set B in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed B-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

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 the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

Oriented matroid structures from realized root systems

This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter system in real vector spaces. We settle the question for finite Coxeter systems, irreducible affine Weyl groups and all rank three Coxeter systems. In these cases, the oriented matroid structure is unique unless W is of type $${\widetilde{A}}_n, n\ge 3$$ , in which case there are three possibilities.

E8 spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B⊂G be a Borel subgroup. Then B acts with finitely many orbits on the variety N2⊂g of the nilpotent elements whose height is at most 2. We provide a parametrization of the B-orbits in N2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.

Strange Expectations and Simultaneous Cores

Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is (a2 −1)(b2 −1) 24, and showed that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is (a−1)(b−1)(a+b+1) 24. We apply P. Johnson's method to compute the variance and third moment. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of these formulae to simply-laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.