# Monomial braidings

Sun, 17/06/2018 - 12:00
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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.