Minimal elements for the limit weak order on affine Weyl groups

We prove a property of left cells in certain crystallographic groups W , by which we formulate an algorithm to find a representative set of left cells of W in any given two-sided cell. As an illustration, we make some applications of this algorithm to the case where W is the affine Weyl group of type e F4. The cells of affine Weyl groups W , as defined by Kazhdan and Lusztig in [6], have been described explicitly in certain special cases: for W of rank 2, see [11]; for W of type An, see [16], [10]; for W of rank 3, see [1], [4]; for the cells with a-values 1, 2 and |Φ|/2 in a general W , see [2], [8], [9], [18], [19], where Φ is the root system determined by W . It is known that there exists a bijection between the set of two-sided cells in an affine Weyl group W and the set of unipotent classes in a certain complex reductive group G associated with W . It is also known that the value of the a-function on a two-sided cell of W is equal to the dimension of the variety of Borel subgroups of G containing an element of the corresponding unipotent class (see [14]). Thus for an affine Weyl group W , the two-sided cells of W are relatively well understood to certain extent. But the classification of left cells in a given two-sided cell of W is not known in general, even the number of these left cells. In the present paper, we shall introduce an algorithm to find a representative set of left cells of W ′ in a given two-sided cell, where W ′ is a group belonging to certain family of crystallographic

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p>0, and let g be its Lie algebra. The crucial Lie algebra representations to understand are those associated with the reduced enveloping algebra U_chi(g) for a nilpotent chi in g*. We conjecture that there is a natural assignment of simple modules in a regular block to left cells in the affine Weyl group (for the dual root system) lying in the two-sided cell which corresponds to the orbit of chi in Lusztig's bijection. This should respect the action of the component group of C_G(chi) and fit naturally into Lusztig's enriched bijection involving the characters of C_G(chi). Some evidence will be described in special cases.

The Lie algebra comes from P. Deligne’s work on the exceptional series of Lie groups. Using the triality algebra, J. Landsberg and L. Manivel construct this Lie algebra in 2006. In this paper, we study the structure of following B. Gross and N. Wallach’s work on the highest root and Heisenberg parabolic subalgebra. The process of removing the lowest root from the extended Dynkin diagram of will contribute to the components of . We also study the branching rule of to and . We use the computer algebra system SageMath to carry out the branching rule calculations. Then we calculate the Dynkin indices for . We find that the number 24 behaves like the ‘dual Coxeter number’ of .

Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [2], Berg et al. described a bijection between $$n$$n-cores with first part equal to $$k$$k and $$(n-1)$$(n-1)-cores with first part less than or equal to $$k$$k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper, we discuss how to generalize the bijection of Berg---Jones---Vazirani to parabolic quotients of affine Weyl groups in type $$C$$C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group.

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

The first, second and fourth Painlevé equations on weighted projective spaces

Random Walk in an Alcove of an Affine Weyl Group, and Non-colliding Random Walks on an Interval

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.

On the support varieties of tilting modules

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k >= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky-Kottwitz-MacPherson and Bezrukavnikov-Varshavsky. We first prove this result for the global Springer action, and then deduce the result for the local Springer action from that of the global one. The argument strongly relies on the fact that we can deform points on a curve, hence giving an example of using global Springer theory to solve more classical problems.

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.]

Using Buchberger–Shirshov Algorithm, Composition–Diamond Lemma and partitions of integers we obtain the reduced Gröbner–Shirshov basis of Ãn and classify all reduced words of the affine Weyl group Ãn.

Ricci curvature, graphs and eigenvalues