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.