W-representation realizes partition functions by an action of a cut-and-join operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for β- and ...q,t-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the Wˆ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for Wˆ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between β- and q,t-deformations – which is not always seen at the accidentally simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots.
Differential expansion (DE) for a Wilson loop average in representation R is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes ...of the loop, e.g. of some knots in the case of 3d Chern–Simons theory. Especially simple is the relation between the DE for the trefoil 31 and for the figure eight knot 41. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso–Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfill this program for all rectangular representation R=rs, i.e. make a plausible conjecture for the rectangularly colored HOMFLY of the figure eight knot, which generalizes the old result for totally symmetric and antisymmetric representations.
An attempt is described to extend the notion of Schur functions from Young diagrams to plane partitions. The suggestion is to use the recursion in the partition size, which is easily generalized and ...deformed. This opens a possibility to obtain Macdonald polynomials by a change of recursion coefficients and taking appropriate limit from three to two dimensions – though details still remain to be worked out.
We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrixB, not just its eigenvalues Λ, and provide a universal formula ...for B, applicable to arbitrary rectangular representation R=rs. This expression is in terms of skew characters and it remains literally the same for the 4-graded rectangularly-colored hyperpolynomials, if characters are substituted by Macdonald polynomials. Due to additional factorization property of the differential-expansion coefficients for the double-braid knots, explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix S¯ in arbitrary rectangular representation R. We also relate matrix evolution to existence of a peculiar rotation U of Racah matrix, which diagonalizes the Z-factors in the differential expansion – what can be a key to further generalization to non-rectangular representations R.
We claim that the recently discovered universal-matrix precursor for the F functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended ...from rectangular to non-rectangular representations. This case is far more interesting, because it involves multiplicities and associated mysterious gauge invariance of arborescent calculus. In this paper we make the very first step – reformulate in this form the previously known formulas for the simplest non-rectangular representations r,1 and demonstrate their drastic simplification after this reformulation.
Equating Schur functions Morozov, A.
The European physical journal. C, Particles and fields,
03/2023, Volume:
83, Issue:
3
Journal Article
Peer reviewed
Open access
We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the ...corresponding measure is just Gaussian in times, which are all independent. The generating function for the number of Young diagrams does not straightforwardly appear as a product, but is reproduced in a non-trivial way.
Racah matrices and higher j-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually ...inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of 1 we obtain the exclusive Racah matrix S¯ for the currently-front-line case of representation R=3,1 with non-trivial multiplicities, where it is actually operator-valued, i.e. depends on the choice of bases in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of S¯ needs to be reconstructed from orthogonality conditions. Therefore we discuss the abundance of symmetric orthogonal matrices, to which S¯ belongs, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus the knowledge approximately matches the freedom and this explains why the method can work – with some limited addition of educated guesses. A similar calculation for R=r,1 for r>3 should also be doable.
Cauchy formula and the character ring Morozov, A.
The European physical journal. C, Particles and fields,
2019/1, Volume:
79, Issue:
1
Journal Article
Peer reviewed
Open access
Cauchy summation formula plays a central role in application of character calculus to many problems, from AGT-implied Nekrasov decomposition of conformal blocks to topological-vertex decompositions ...of link invariants. We briefly review the equivalence between Cauchy formula and expressibility of skew characters through the Littlewood–Richardson coefficients. As not-quite-a-trivial illustration we consider how this equivalence works in the case of plane partitions – at the simplest truly interesting level of just four boxes.
We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), ...when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in those of the original knot, where contributing are adjoint and other representations from the “E8-sector”, what makes the story closely related to Vogel's universality. We also point out a problem with lifting the decomposition rule to the level of superpolynomials — it looks like such rule, if any, should be different for positive and negative twistings.
A
bstract
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind ...of a double-evolution — that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations
R
=
r
s
we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of
R
= 33. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices
S
¯
and
S
— what allows to calculate 33-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.
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