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 Ψ₁ Ψ₂ Ψ₁ = Ψ₂ Ψ₁ Ψ₂. 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 Ψ₁, ..., 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.