Narrow Results

We find explicitly the multiplicities in the (mixed) trace cocharacter sequence of two 3 × 3 matrices over a field of characteristic 0 and show that asymptotically they behave as polynomials of seventh degree. As a consequence we obtain also the multiplicities of certain irreducible characters in the cocharacter sequence of the polynomial identities of 3 × 3 matrices.

The action of the Bernstein operators on Schur functions was given in terms of codes by Carrell and Goulden (2011) and extended to the analog in Schur Q-functions in our previous work. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

In this paper, we relate Schur functions and a linear skein of annulus derived from the Homfly polynomail. Using this relations, we define topological invariants of 3-manifolds from the Homfly polynomial.