[2202.12225] A new approach to the $\mathfrak{gl}_N$ weighting system

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overview: This paper was motivated by a desire to understand the weighting system corresponding to the Lie algebra $\mathfrak{gl}_N$. A naive approach to computing the values ​​of the Lie algebraic weight system on the general chord diagram is equivalent to complex computations in the non-commutative universal envelope algebra, despite the fact that the result belongs to the center of the latter . The first approach is based on a proposal by M. Kazarian in his work to define permutation invariants that take values ​​at the center of the universal envelope algebra of \mathfrak{gl}_N$. The restriction on invariants without fixed points of this invariant (such involution determines the chord diagram) is equal to the value of the $\mathfrak{gl}_N$ -weight system on this chord diagram. I will. $\mathfrak{gl}_N$ Describes recursion, which allows computation of permutation invariants, and gives many examples of how it works. The second approach is based on the Harish-Chandra isomorphism of the Lie algebra $\mathfrak{gl}_N$. This isomorphism locates the center of the universal envelope algebra $\mathfrak{gl}_N$ with the ring $\Lambda^*(N)$ of shift-symmetric polynomials in $N$ variables. The Harish-Chandra projection can be applied independently to each monomial of the defining polynomial of the weighting system. As a result, the main part of the computation can be done in commutative algebra instead of non-commutative algebra.