12 research outputs found
Hunt for 3-Schur polynomials
This paper describes our attempt to understand the recent success of Na Wang
in constructing the 3-Schur polynomials, associated with the plane partitions.
We provide a rather detailed review and try to figure out the new insights,
which allowed to overcome the problems of the previous efforts. In result we
provide a very simple definition of time-variables and
the cut-and-join operator , which generates the set of -Schur
functions. Some coefficients in remain undefined and require more
effort to be fixed
On an alternative stratification of knots
We introduce an alternative stratification of knots: by the size of lattice
on which a knot can be first met. Using this classification, we find ratio of
unknots and knots with more than 10 minimal crossings inside different lattices
and answer the question which knots can be realized inside and
lattices. In accordance with previous research, the ratio of
unknots decreases exponentially with the growth of the lattice size. Our
computational results are approved with theoretical estimates for amounts of
knots with fixed crossing number lying inside lattices of given size.Comment: 12 page
Irreducible representations of simple Lie algebras by differential operators
We describe a systematic method to construct arbitrary highest-weight
modules, including arbitrary finite-dimensional representations, for any finite
dimensional simple Lie algebra . The Lie algebra generators are
represented as first order differential operators in variables. All rising
generators are universal in the sense that they do not depend on
representation, the weights enter (in a very simple way) only in the
expressions for the lowering operators . We present explicit formulas
of this kind for the simple root generators of all classical Lie algebras
3-Schurs from explicit representation of Yangian Y gl ̂ 1 . Levels 1–5
Abstract We suggest an ansatz for representation of affine Yangian Y gl ̂ 1 by differential operators in the triangular set of time-variables P a,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting “cut-and-join” generators ψ n of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods
Evolution properties of the knot's defect
The defect of differential (cyclotomic) expansion for colored HOMFLY-PT
polynomials is conjectured to be invariant under any antiparallel evolution and
change linearly with the evolution in any parallel direction. In other words,
each -matrix can be substituted by an entire 2-strand braid in two
different ways: the defect remains intact when the braid is antiparallel and
changes by half of the added length when the braid is parallel
Hunt for 3-Schur polynomials
This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables Pi⩾j and the cut-and-join operator Wˆ2, which generates the set of 3-Schur functions. Some coefficients in Wˆ2 remain undefined and require more effort to be fixed
Are Maxwell knots integrable?
We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. We also consider particular examples of the field lines for the particular family of finite energy source-free “knot” solutions, attempting to understand when the field lines are closed – and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada’s solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist