A Diagrammatic Construction of the (sl(N,C),ρ)-Weight System

Abstract

In this paper, we first give a diagrammatic analogue of the Young symmetrizer. By using this, the ðslðN; CÞ; Þ-weight system for an arbitrary finite-dimensional irreducible representation is formulated in a diagrammatic way. The formula is useful for the calculations of the ðslðN; CÞ; Þ-weight system in the sense that we do not need actual constructions of the representations of slðN; CÞ essentially. Hence by using this and the modified Kontsevich integral we can get the quantum ðslðN; CÞ; Þ-invariant for any finite-dimensional irreducible representation without actual constructions of the representations of slðN; CÞ. The diagrammatic construction is a generalization of the formula given in ''Remarks on the ðslðN; CÞ; adÞ-weight system''.

A note on weight systems which are quantum states

“…The weight system can be also defined as the following composition: where —cf. [6, 7]. Roughly speaking, we are writing each as the composition of the representation —encoded by , cf.…”

“…Example 2. 7 The canonical Young tableau of shape λ = (5, 3, 1, 1) is the following. Let T λ be the set of all tableaux of shape λ = (n 1 , .…”

“…[1]). There exists a similar description for more general gl n -weight system-compare with [6]. Given a vector of gl n -representations ρ = (λ 1 , .…”

“…(|λ 1 |,...,|λ N |) → A h |ρ| σ → C[S |ρ| ] cρ → C[S |ρ| ] wst → Cwhere |ρ| ∶= ∑ j≤N |λ j |-cf [6,7]…”

On a Background of the Existence of Multi-variable Link Invariants

“…We remark that the first author generalized the formula (3) to a universal sl(N, C)weight system via the Young symmetrizer in [10].…”