scholarly journals A GRAPHICAL DESCRIPTION OF (Dn,An−1) KAZHDAN–LUSZTIG POLYNOMIALS

2012 ◽  
Vol 55 (2) ◽  
pp. 313-340 ◽  
Author(s):  
TOBIAS LEJCZYK ◽  
CATHARINA STROPPEL
Keyword(s):  
Weyl Group ◽  
Parabolic Subgroup ◽  
Perverse Sheaves ◽  
Springer Theory ◽  
Brauer Algebras ◽  
Counting Formula ◽  
Lusztig Polynomials

AbstractWe give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.

scholarly journals The Leading Coefficients for an Affine Weyl Group of Type G 2 ˜ : The Lowest Two-Sided Cell Case

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Pengfei Guo ◽  
Zhu-Jun Zheng
Keyword(s):  
Weyl Group ◽  
Affine Weyl Group ◽  
Lusztig Polynomials

This study focusses on the leading coefficients μ u , w of the Kazhdan–Lusztig polynomials P u , w for the lowest cell c 0 of an affine Weyl group of type G 2 ˜ and gives an estimation μ u , w ≤ 3 for u , w ∈ c 0 .

scholarly journals Permutation characters of finite groups of Lie type

1979 ◽  
Vol 27 (3) ◽  
pp. 378-384 ◽  
Author(s):  
David B. Surowski
Keyword(s):  
Fixed Points ◽  
Algebraic Group ◽  
Weyl Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Semisimple Element ◽  
Frobenius Endomorphism ◽  
Groups Of Lie Type

AbstractLet g be a connected reductive linear algebraic group, and let G = gσ be the finite subgroup of fixed points, where σ is the generalized Frobenius endomorphism of g. Let x be a regular semisimple element of G and let w be a corresponding element of the Weyl group W. In this paper we give a formula for the number of right cosets of a parabolic subgroup of G left fixed by x, in terms of the corresponding action of w in W. In case G is untwisted, it turns out thta x fixes exactly as many cosets as does W in the corresponding permutation representation.

scholarly journals 2-row Springer Fibres and Khovanov Diagram Algebras for Type D

2016 ◽  
Vol 68 (6) ◽  
pp. 1285-1333 ◽  
Author(s):  
Michael Ehrig ◽  
Catharina Stroppel
Keyword(s):  
Weyl Group ◽  
Cohomology Ring ◽  
Main Tool ◽  
Type A ◽  
Point Of View ◽  
Perverse Sheaves ◽  
Type D ◽  
Diagram Algebras ◽  
Irreducible Components

AbstractWe study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections are iterated ℙ1-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type D diagram calculus labelling the irreducible components in a convenient way that relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types.

scholarly journals Elliptic Springer theory

2015 ◽  
Vol 151 (8) ◽  
pp. 1568-1584 ◽  
Author(s):  
David Ben-Zvi ◽  
David Nadler
Keyword(s):  
Elliptic Curve ◽  
Weyl Group ◽  
Principal Series ◽  
Degree Zero ◽  
Loop Groups ◽  
Characteristic Variety ◽  
Affine Weyl Group ◽  
Springer Theory ◽  
Character Sheaves

We introduce an elliptic version of the Grothendieck–Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the resolution of degree-zero, semistable $G$-bundles by degree-zero $B$-bundles over an elliptic curve $E$. From a representation theory perspective, they produce a full embedding of representations of the elliptic or double affine Weyl group into perverse sheaves with nilpotent characteristic variety on the moduli of $G$-bundles over $E$. The resulting objects are principal series examples of elliptic character sheaves, objects expected to play the role of character sheaves for loop groups.

Some Results on L-Indistinguishability for SL(r)

1983 ◽  
Vol 35 (6) ◽  
pp. 1075-1109 ◽  
Author(s):  
Freydoon Shahidi
Keyword(s):  
Positive Integer ◽  
Weyl Group ◽  
Recent Work ◽  
Eisenstein Series ◽  
Number Field ◽  
Parabolic Subgroup ◽  
Cusp Form ◽  
Constant Multiple ◽  
Levi Decomposition ◽  
Image Position

Fix a positive integer r. Let AF be the ring of adeles of a number field F. For a parabolic subgroup P of SLr, we fix a Levi decomposition P = MN, and we letLet be the Weyl group of . It follows from a recent work of James Arthur [1,2] (also cf. [3]) that, among the terms appearing in the trace formula for SLr(AF), coming from the Eisenstein series, are those which are a constant multiple (depending only on M and w) of1where σ is a cusp form on M(AF) satisfying wσ ≅ σ,and in the notation of [2, 3]).

scholarly journals Symmetry of Narayana Numbers and Rowvacuation of Root Posets

2021 ◽  
Vol 9 ◽  
Author(s):  
Colin Defant ◽  
Sam Hopkins
Keyword(s):  
Weyl Group ◽  
Catalan Number ◽  
Recent Past ◽  
Positive Roots ◽  
Narayana Numbers ◽  
The Subject

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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