New approaches to $\mathfrak{gl}_N$ weight system

The present paper has been motivated by an aspiration for understanding the weight system corresponding to the Lie algebra $\mathfrak{gl}_N$. The straightforward approach to computing the values of a Lie algebra weight system on a general chord diagram amounts to elaborating calculations in the non-commutative universal enveloping algebra, in spite of the fact that the result belongs to the centre of the latter. The first approach is based on M. Kazarian's proposal to define an invariant of permutations taking values in the centre of the universal enveloping algebra of $\mathfrak{gl}_N$. The restriction of this invariant to involutions without fixed points (such an involution determines a chord diagram) coincides with the value of the $\mathfrak{gl}_N$ weight system on this chord diagram. We describe the recursion allowing one to compute the $\mathfrak{gl}_N$ invariant of permutations and demonstrate how it works in a number of examples. The second approach is based on the Harish-Chandra isomorphism for the Lie algebras $\mathfrak{gl}_N$. This isomorphism identifies the centre of the universal enveloping algebra $\mathfrak{gl}_N$ with the ring $\Lambda^*(N)$ of shifted symmetric polynomials in N variables. The Harish-Chandra projection can be applied separately for each monomial in the defining polynomial of the weight system; as a result, the main body of computations can be done in a commutative algebra, rather than non-commutative one.