Representations of the virtual braid groups to the rook algebras - PowerPoint PPT Presentation

Representations of the virtual braid groups to the rook algebras Konstantin Gotin Novosibirsk State University July 25, 2014 1 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Braid group on n strands, denoted by B n , is group generated by σ 1 , . . . , σ n − 1 satisfying the following relations: σ i σ j = σ j σ i if | i − j | > 1 σ i +1 σ i σ i +1 = σ i σ i +1 σ i if i = 1 , . . . , n − 1 2 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Virtual braid group on n strands, denoted by V B n , is group with generators: σ 1 , . . . , σ n − 1 , ρ 1 , . . . , ρ n − 1 and relations: σ i σ j = σ j σ i if | i − j | > 1 σ i +1 σ i σ i +1 = σ i σ i +1 σ i if i = 1 , . . . , n − 1 ρ 2 i = e if i = 1 , . . . , n − 1 ρ i +1 ρ i ρ i +1 = ρ i ρ i +1 ρ i if i = 1 , . . . , n − 1 ρ i ρ j = ρ j ρ i if | i − j | > 1 ρ i ρ i +1 σ i = σ i +1 ρ i ρ i +1 if i = 1 , . . . , n − 1 σ i ρ j = ρ j σ i if | i − j | > 1 3 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Let R n denote set of n × n matrices with entries from { 0, 1 } having at most one 1 in each row and in each column. Example for n = 2 �� 0 0 1 0 0 1 0 0 0 0 1 0 0 1 , , , , , , 0 0 0 0 0 0 1 0 0 1 0 1 1 0 4 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Rook diagram – bipartite graph on two rows of n vertices, one on top of the other forming the boundary of a rectangle, such that each vertex has degree either zero or one.  0 1 0 0 0 0  0 0 0 1 0 0     • • • • • • 0 0 0 0 0 0 . . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . ↔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . 0 0 1 0 0 0 . . . . . . . . . . . . . • . . . • • . . . . . . . . • • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   0 0 0 0 0 1   0 0 0 0 0 0 5 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition The product, d 1 d 2 , of two rook diagrams d 1 and d 2 is obtained by stacking d 1 on top of d 2 and deleting any edge from one that connects to a vertex zero degree end from the other. • • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d 1 = • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . • . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . = d 1 d 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • . . . . . . . . . . . . . . . . . . • • . . . . . . • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d 2 = • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Given two diagrams, a , b ∈ R n , we define the tensor product, denoted a ⊗ b , to be the result of appending of b to the right of a . • • • • • • • • • • • • • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⊗ • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . • . . . . • . . . . . . . . • • • . . . • . . . • • . . . . . • . . . . . • . . . . . . . . . . • • . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition Element of R n is planar if its diagram can be drawn (keeping inside of the rectangle formed by its vertices) without any edge crossings. • • • • • • • • • • • • • • • • • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . • . . . • . . . • • . . . • . . . • • . • . . . • • • • • • • . . . • . . . • . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Definition P n – planar rook monoid – set of planar diagrams of R n . Definition C R n ( C P n ) – (planar) rook algebra – C -algebra generated by R n ( P n ) with multiplication defined using the distributive law and multiplication in R n ( P n ). 9 / 15 Konstantin Gotin Virtual braid groups and rook algebras

Representations of the virtual braid groups to the rook algebras - PowerPoint PPT Presentation

Representations of the virtual braid groups to the rook algebras Konstantin Gotin Novosibirsk State University July 25, 2014 1 / 15 Konstantin Gotin Virtual braid groups and rook algebras Definition Braid group on n strands, denoted by B n ,

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Representations of the virtual braid groups to the rook algebras - PowerPoint PPT Presentation

Representations of the virtual braid groups to the rook algebras Konstantin Gotin Novosibirsk State University July 25, 2014 1 / 15 Konstantin Gotin Virtual braid groups and rook algebras Definition Braid group on n strands, denoted by B n ,

LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZATIONS Yu. A. Mikhalchishina Knots,

Non commutative representations of Torelli groups Christian Blanchet, Univ. Paris Diderot, IMJ

Finite Quotients of Braid Groups Lily Li Joint Work with Alice Chudnovsky, Caleb Partin, and

Garside structure and Dehornoy ordering of braid groups for topologist (mini-course II) Tetsuya

Garside structure and Dehornoy ordering of braid groups for topologist (mini-course I) Tetsuya

GROUPS Virtual Group Topics Overview of Virtual Groups Participating as a Virtual Group in

LDM70 TOPICS ON GROUPS AND THEIR REPRESENTATIONS Prova The Kummerian Property for pro- p -groups

Alex Suciu Northeastern University (joint work with He Wang) Sminaire dalgbre et de

Ceph storage with Rook Running Ceph on Kubernetes Alexander Trost, Rook Maintainer and DevOps

Representations of 2-Groups on Higher Hilbert Spaces Derek Wise Based on joint work with John

INVARIANTS OF VIRTUAL LINKS Julia Mikhalchishina NOVOSIBIRSK 2015 Braid group B n = 1

Decay of matrix coefficients of unitary representations of semisimple groups Michael G Cowling

On the Shape of a Set In Honour of Lino Di Martino Topics in Groups and their Representations

The 0-Rook Monoid Jo el Gay joint work with Florent Hivert e Paris-Sud, LRI &amp;

Square Dancing in the Pure Braid Group 8 1 2 7 3 6 5 4 Jon McCammond U.C. Santa Barbara

Rook polynomials Ira M. Gessel Department of Mathematics Brandeis University University of

Fourier transform for nilpotent Lie groups Index sets and representations Granada Index sets

Unitarizable representations and fixed points of groups of biholomorphic transformations of

Advanced Air Mobility Ecosystem Working Groups Virtual Kickoff AAM Ecosystem Working Groups

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Pattern avoidance Definitions in rook monoids Rook Monoids Avoidance 1d Avoidance All 0/No 0

On the arithmetic of integral representations of finite groups Dmitry Malinin Department of

Representations of quantum groups at p r th root of 1 over p -adic fields Zongzhu Lin Kansas

Irreducible representations of the classical algebraic groups with p-large highest weights:

The 0-Rook Monoid Jo el Gay joint work with Florent Hivert e Paris-Sud, LRI &