Tensor diagrams and Chebyshev polynomials

TL;DR

This paper explores a class of invariant polynomials in three-dimensional vector spaces, linking Chebyshev polynomials with Lusztig's dual canonical basis in quantum group representations.

Contribution

It introduces a new class of invariant polynomials related through Chebyshev polynomials and connects them to Lusztig's dual canonical basis in quantum algebra.

Findings

Identification of invariant polynomial elements in SL(V) functions

Establishment of an induction formula using Chebyshev polynomials

Relation between Chebyshev polynomials and Lusztig's dual canonical basis

Abstract

In this paper, we describe a class of elements in the ring of $\mathrm{SL}(V)$-invariant polynomial functions on the space of configurations of vectors and linear forms of a 3-dimensional vector space $V.$ These elements are related to one another by an induction formula using Chebyshev polynomials. We also investigate the relation between these polynomials and G. Lusztig's dual canonical basis in tensor products of representations of $U_q(\mathfrak{sl}_3(\mathbb C)).$