Two-sided Cells Research Articles

Kazhdan–Lusztig cells of a-value 2 in a(2)-finite Coxeter systems

A Coxeter group is said to be a(2)-finite if it has finitely many elements of a-value (in the sense of Lusztig) equal to 2. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan–Lusztig cells of a-value 2 in an irreducible a(2)-finite Coxeter group. In particular, we introduce elements we call stubs to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells of a-value 2 in irreducible a(2)-finite Coxeter groups.

Unipotent nearby cycles and the cohomology of shtukas

We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson–Drinfeld Grassmannian with parahoric reduction is a central functor together with a ‘Zorro's lemma’ argument similar to that of Xue [Smoothness of cohomology sheaves of stacks of shtukas, Preprint (2020), arXiv:2012.12833]. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langlands correspondence in terms of two-sided cells.

Sign insertion and Kazhdan–Lusztig cells of affine symmetric groups

Combinatorics of Kazhdan–Lusztig cells in affine type A was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson–Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan–Lusztig theory in affine type A was developed by Blasiak in his work on catabolism. He introduced a sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of the sign insertion process determines uniquely and is determined uniquely by the left cell. In this paper we unite these two approaches by proving Blasiak’s conjecture. In the process, we show that certain new operations we introduce called partial rotations connect the elements in the intersection of a left cell and a right cell. Lastly, we investigate the connection between Blasiak’s sign insertion and the standardization map acting on the set of semi-standard Young tableaux defined by Lascoux and Schützenberger.

The second lowest two-sided cell in the affine Weyl group [formula omitted

Let (Wa,Sa) be an irreducible affine Weyl group with W0 the associated Weyl group. In my previous paper [24], I proved that the number nqr of left cells of Wa in the second lowest two-sided cell Ωqr satisfies the inequality nqr⩽12|W0| and conjectured that the equality should actually hold. In the present paper, we verify the conjecture when Wa=B˜n; we prove that any left cell of B˜n in Ωqr is left-connected, verifying a conjecture of Lusztig in our case; we show that Ωqr consists of all the extensions of wJ in the set B˜n−W(ν), where W(ν) is the lowest two-sided cell of B˜n and J⊂Sa is such that wJ is the longest element in the subgroup WJ of B˜n of type Dn.

Cells in modified ıquantum groups of type AIII and related Schur algebras

We provide a combinatorial characterization of two-sided cells in modified ıquantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in ȷ-Schur algebras associated to Iwahori–Hecke algebras of type B. We further give a combinatorial description of two-sided cells in other two kinds of Schur-type algebras, namely ı-Schur algebras and ı˜-Schur algebras.

Some translations in the second lowest two-sided cell of an affine Weyl group

We are interested in cell properties of translations in affine Weyl groups. We find out some translations in the second lowest two-sided cell of an affine Weyl group and use the translations to formulate a refinement of Jianyi Shi's conjecture on the number of left cells in the second lowest two-sided cell. We verify the refinement for an affine Weyl group of type A ˜ n − 1 or type G ˜ 2 .

Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization

We study the classification problem of simple transitive 2-representations of the 2-category of finite-dimensional bimodules over a radical square zero Nakayama algebra. This results in a complete classification of simple transitive 2-representations whose apex is a finitary two-sided cell. We define a notion of localization of 2-representations. We construct previously unknown simple transitive 2-representations as localizations of cell 2-representations. Using the universal property of our construction we prove that any simple transitive 2-representation with finitary apex is equivalent to a localization of a cell 2-representation.

A parametrization of unipotent representations

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize the unipotent representations.

The based rings of two-sided cells in an affine Weyl group of type $${{\tilde B}_3}$$, I

For type $${{\tilde B}_3}$$ , we show that Lusztig’s conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp6(ℂ) with 3 equal Jordan blocks needs modification.

Some Kazhdan–Lusztig Coefficients of Affine Weyl Group of Type B~ 2

Let [Formula: see text] be the affine Weyl group of type [Formula: see text], on which we consider the length function [Formula: see text] from [Formula: see text] to [Formula: see text] and the Bruhat order [Formula: see text]. For [Formula: see text] in [Formula: see text], let [Formula: see text] be the coefficient of [Formula: see text] in Kazhdan–Lusztig polynomial [Formula: see text]. We determine some [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is the lowest two-sided cell of [Formula: see text] and [Formula: see text] is the higher one. Furthermore, we get some consequences using left or right strings and some properties of leading coefficients.

Bigrassmannian permutations and Verma modules

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by win S_n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

Relative hard Lefschetz for Soergel bimodules

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan–Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any two-sided cell in a Coxeter group is rigid and pivotal.

Conjectures P1-P15 for Coxeter groups with complete graph

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using decreasing induction on a-values and a kind of factorization formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, right, and two-sided cells. In the appendix, we prove P1-P15 for right-angled Coxeter groups by the same methods.

Fourier transform as a triangular matrix

Let V V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [ V ] [V] be the vector space of complex valued functions on V V , and let [ V ] Z [V]_{\mathbf {Z}} be the subgroup of [ V ] [V] consisting of integer valued functions. We show that there exists a Z \mathbf {Z} -basis of [ V ] Z [V]_{\mathbf {Z}} consisting of characteristic functions of certain isotropic subspaces of V V and such that the matrix of the Fourier transform from [ V ] [V] to [ V ] [V] with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group.

An asymptotic cell category for cyclic groups

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué, Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.

Cell structure of bimodules over radical square zero Nakayama algebras

In this paper, we describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description of left, right and two-sided cells.

Conjugacy Classes in Reductive Groups and Two-sided Cells

Let G' be a connected reductive group over the complex numbers. We show that the set of conjugacy classes of G' is in natural bijection with the set of two-sided cells associated to a certain algebra.

Photonic crystal elliptical-hole tapered low-index-mode nanobeam cavities for sensing.

A one-dimensional photonic crystal elliptical-hole tapered low-index-mode nanobeam cavity sensor fully encapsulated in a water environment is proposed. In the proposed structure, to confine the light in the low-index region and enhance the light-matter interaction, a tapered major axis of the elliptical hole away from the nanobeam cavities center is optimized. Through a three-dimensional finite-difference time-domain simulation, the results show that the low-index-mode of the middle geometry cell is confined in the photonic bandgap of two-sided cells. The highest quality factor of 6.04×105 is achieved when 13 tapered segments and 5 mirror segments are placed at both sides of the host waveguide. The proposed nanobeam structure theoretically possesses a sensitivity of 244.7nm/RIU (refractive index unit) in a water environment. Moreover, an ultra-compact footprint of 6.4 μm×0.85 μm is achieved, which is only half of the size compared to the best value reported for the nanobeam structure. The results indicate that it is a promising sensor for excellent on-chip sensing with respect to the very small footprint.

Canonical Left Cells and the Lowest Two-sided Cell in an Affine Weyl Group

We give some discussions to the relations between canonical left cells and the lowest two-sided cell of an affine Weyl group. In particular, we use the relations to construct irreducible modules attached to the lowest two-sided cell and some one dimensional representations of an affine Hecke algebra.

Simple transitive 2-representations of small quotients of Soergel bimodules

In all finite Coxeter types but I 2 ( 12 ) I_2(12) , I 2 ( 18 ) I_2(18) , and I 2 ( 30 ) I_2(30) , we classify simple transitive 2 2 -representations for the quotient of the 2 2 -category of Soergel bimodules over the coinvariant algebra which is associated with the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive 2 2 -representations are exhausted by cell 2 2 -representations. However, in Coxeter types I 2 ( 2 k ) I_2(2k) , where k ≥ 3 k\geq 3 , there exist simple transitive 2 2 -representations which are not equivalent to cell 2 2 -representations.