Tuesday, July 17, 14:00, 7.527

Jianrong Li (Weizmann Institute of Science), Monomial braidings.

Abstract:

A braided vector space is a pair (V, Ψ), where V is a
vector space and Ψ: V ⊗ V → V ⊗ V is an
invertible linear operator such that Ψ_{1} Ψ_{2}
Ψ_{1} = Ψ_{2}
Ψ_{1} Ψ_{2}.
Given a braided vector space (V, Ψ), we constructed a family of
braided vector spaces

(V, Ψ^{(ε)}), where
ε is a bitransitive function. Here a bitransitive function is
a function ε: [n] × [n] → {1, -1}
such that both of

{(i,j): ε(i,j) = 1} and
{(i,j) : ε(i,j) = -1\} are transitive relations on
[n]. The braidings Ψ^{(ε)} are
monomials in Ψ_{i}. Therefore we call them monomial
braidings.

We generalized this construction to the case of multi-colors. Given a braided
vector space (V, Ψ), we used C-transitive functions to
parametrize the C-braidings on V^{⊗n} which come
from Ψ_{1}, ..., Ψ_{n-1}.

Since [n] × [n] can be viewed as the set of edges of the
bi-directed complete graph with n vertices, a C-transitive function

ε: [n] \times [n] →
C can be viewed 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
|Ε_{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.