Koszul duality and Soergel bimodules for dihedral groups

Marc Sauerwein

doi:10.1090/tran/7014

On monoidal Koszul duality for the Hecke category

Revista Colombiana de Matemáticas ◽

10.15446/recolma.v53nsupl.84084 ◽

2019 ◽

Vol 53 (supl) ◽

pp. 195-222 ◽

Cited By ~ 1

Author(s):

Shotaro Makisumi

Keyword(s):

Algebraic Model ◽

Koszul Duality ◽

Joint Work ◽

Soergel Bimodules

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work [1, 2] of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) SL2 using Soergel bimodules, a concrete algebraic model of the Hecke category.

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Power graphs and exchange property for resolving sets

Open Mathematics ◽

10.1515/math-2019-0093 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1303-1309 ◽

Cited By ~ 1

Author(s):

Ghulam Abbas ◽

Usman Ali ◽

Mobeen Munir ◽

Syed Ahtsham Ul Haq Bokhary ◽

Shin Min Kang

Keyword(s):

Present Article ◽

Linear Algebra ◽

Finite Groups ◽

Robot Navigation ◽

Simple Graph ◽

Exchange Property ◽

Metric Dimension ◽

Sufficient Condition ◽

Resolving Set ◽

Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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Affine category O, Koszul duality and Zuckerman functors

Advances in Mathematics ◽

10.1016/j.aim.2021.107921 ◽

2021 ◽

Vol 390 ◽

pp. 107921

Author(s):

Ruslan Maksimau

Keyword(s):

Koszul Duality ◽

Category O

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Regular Cayley maps for dihedral groups

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2020.12.002 ◽

2021 ◽

Vol 148 ◽

pp. 84-124

Author(s):

István Kovács ◽

Young Soo Kwon

Keyword(s):

Dihedral Groups ◽

Cayley Maps

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On *-clean non-commutative group rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501504 ◽

2016 ◽

Vol 15 (08) ◽

pp. 1650150 ◽

Cited By ~ 4

Author(s):

Hongdi Huang ◽

Yuanlin Li ◽

Gaohua Tang

Keyword(s):

Finite Field ◽

Local Ring ◽

Group Algebra ◽

Rational Numbers ◽

Semisimple Group ◽

Group Algebras ◽

Group Rings ◽

Dihedral Groups ◽

Commutative Local Ring ◽

Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

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A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups

College Mathematics Journal ◽

10.2307/2686678 ◽

1992 ◽

Vol 23 (2) ◽

pp. 150

Author(s):

David W. Jensen ◽

Eric R. Bussian

Keyword(s):

Theoretic Approach ◽

Dihedral Groups

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KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788710001497 ◽

2010 ◽

Vol 89 (1) ◽

pp. 23-49 ◽

Cited By ~ 9

Author(s):

VOLODYMYR MAZORCHUK

Keyword(s):

Complete Picture ◽

Projective Modules ◽

Koszul Duality ◽

Tilting Theory

AbstractWe give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.

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