Monomial braidings

Seminar
Speaker
Jianrong Li (Weizmann Institute)
Date
17/06/2018 - 13:15 - 12:00Add to Calendar 2018-06-17 12:00:00 2018-06-17 13:15:00 Monomial braidings A braided vector space is a pair $(V, \Psi)$, where $V$ is a vector space and $\Psi: V \otimes V \to V \otimes V$ is an invertible linear operator such that $\Psi_1 \Psi_2 \Psi_1 = \Psi_2  \Psi_1  \Psi_2$. Given a braided vector space $(V, \Psi)$, we constructed a family of braided vector spaces $(V, \Psi^{(\epsilon)})$, where $\epsilon$ is a bitransitive function. Here a  bitransitive function is a function $\epsilon: [n] \times [n] \to \{1, -1\}$ such  that both of $\{(i,j) : \epsilon(i,j) = 1\}$ and $\{(i,j) : \epsilon(i,j) = -1\}$ are  transitive relations on $[n]$. The braidings $\Psi^{(\epsilon)}$ are monomials.  Therefore we call them monomial braidings.    We generalized this construction to the case of multi-colors. Given a braided  vector space $(V, \Psi)$, we used C-transitive functions to parametrize the  braidings on $V^{\otimes n}$ which come from $\Psi_1, \ldots, \Psi_{n-1}$.    Since $[n] \times [n]$ can be viewed as the set of edges of the bi-directed  complete graph with n vertices, a C-transitive function $\epsilon: [n] \times [n] \to C$  can be view as a C-transitive function on a bi-directed complete graph.  We generalized the concept of C-transitive functions to C-transitive functions on  any directed graphs. We showed that the number |\Epsilon_G(C)| of all C-transitive  functions on a directed graph G is a polynomial in |C|. This is a new invariant in graph  theory. It is analogue to the chromatic polynomial for an undirected graph in graph theory.    This talk is based on joint work with Arkady Berenstein and Jacob Greenstein.  Room 201 , Math and CS Building אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Room 201 , Math and CS Building
Abstract

A braided vector space is a pair $(V, \Psi)$, where $V$ is a vector space

and $\Psi: V \otimes V \to V \otimes V$ is an invertible linear operator

such that $\Psi_1 \Psi_2 \Psi_1 = \Psi_2  \Psi_1  \Psi_2$. Given a braided

vector space $(V, \Psi)$, we constructed a family of braided vector spaces

$(V, \Psi^{(\epsilon)})$, where $\epsilon$ is a bitransitive function. Here a 

bitransitive function is a function $\epsilon: [n] \times [n] \to \{1, -1\}$ such 

that both of $\{(i,j) : \epsilon(i,j) = 1\}$ and $\{(i,j) : \epsilon(i,j) = -1\}$ are 

transitive relations on $[n]$. The braidings $\Psi^{(\epsilon)}$ are monomials. 

Therefore we call them monomial braidings. 

 

We generalized this construction to the case of multi-colors. Given a braided 

vector space $(V, \Psi)$, we used C-transitive functions to parametrize the 

braidings on $V^{\otimes n}$ which come from $\Psi_1, \ldots, \Psi_{n-1}$. 

 

Since $[n] \times [n]$ can be viewed as the set of edges of the bi-directed 

complete graph with n vertices, a C-transitive function $\epsilon: [n] \times [n] \to C$ 

can be view as a C-transitive function on a bi-directed complete graph. 

We generalized the concept of C-transitive functions to C-transitive functions on 

any directed graphs. We showed that the number |\Epsilon_G(C)| of all C-transitive 

functions on a directed graph G is a polynomial in |C|. This is a new invariant in graph 

theory. It is analogue to the chromatic polynomial for an undirected graph in graph theory. 

 

This talk is based on joint work with Arkady Berenstein and Jacob Greenstein. 

Last Updated Date : 26/11/2019