Seminars "Combinatorics of invariants". Speaker: Mikhail Zaitsev

gl(N)-weight systems are naturally extended from chord diagrams to permutations. This extension allows one to use the powerful tools of quantum matrix theory, for which many convenient computational techniques are known.
I will introduce several sets of central elements in the universal enveloping algebra U(gl_n): Casimir elements, cut-and-join operators, quantum immanants. To establish their connections, we will need the notion of normal order on differential operators. Weight systems are closely related to symmetric functions and their normally ordered counterparts turn out to be shifted symmetric functions.
If time permits, I will explain how the quantization of the universal enveloping algebra clarifies these connections and allows us to ask new questions about the interaction of the described objects.