Monday, February 20, 2012

1201.3339 (A. Mironov et al.)

Interplay between MacDonald and Hall-Littlewood expansions of extended
torus superpolynomials

A. Mironov, A. Morozov, Sh. Shakirov, A. Sleptsov
In arXiv:1106.4305 extended superpolynomials were introduced for the torus
links T[m,mk+r], which are functions on the entire space of time variables and,
at expense of reducing the topological invariance, possess additional algebraic
properties, resembling those of the matrix model partition functions and the
KP/Toda tau-functions. Not surprisingly, being a suitable extension it actually
allows one to calculate the superpolynomials. These functions are defined as
expansions into MacDonald polynomials, and their dependence on k is entirely
captured by the action of the cut-and-join operator, like in the HOMFLY case.
We suggest a simple description of the coefficients in these character
expansions, by expanding the initial (at k=0) conditions for the k-evolution
into the new auxiliary basis, this time provided by the Hall-Littlewood
polynomials, which, hence, play a role in the description of the dual
m-evolution. For illustration we list manifest expressions for a few first
series, mk\pm 1, mk\pm 2, mk\pm 3, mk\pm 4. Actually all formulas were
explicitly tested up to m=17 strands in the braid.
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