Extension of KNTZ trick to non-rectangular representations

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.

and the second observation of [1,2], that for rectangular R: where B is a universal triangular "embedding" matrix with Y X ⊂ R ⊗R. In this paper we consider the possibility for (2) to hold also for non-rectangular R. We do not discuss what are the m-independent differential combinations Z X R , which is also a highly non-trivial story in this case, see [7] and a number of preceding papers, cited therein. This Z-story actually belongs to the theory of a single figure-eight knot, 4 1 and is well separated from the problem of m-dependence, which we address now -though both are equally relevant for the next step towards Racah matrices.
Representations X and Y in (1) are composite, see Fig.2. For rectangular representations R = [r s ] = s times [r, . . . , r] only very special diagonal composites (λ, λ) contribute to R ⊗R -and they are in one-to-one correspondence with the Young sub-diagrams of λ ⊂ R, and "embedding" for diagonal composites is understood as embedding of the corresponding λ: The entries of the matrix B in (2) are expressed through the skew Schur functions: where ∨ stands for transposition of the Young diagram, and Λ µ are the eigenvalues of R-matrix in the channel best expressed through the hook parameters of λ = (a 1 , b 1 |a 2 , b 2 |, . . .): Index • means that Schur functions are evaluated at the "unit" locus in the space of time-variables, At q = 1 this is equivalent to putting p k = δ k,1 , and there is even a a special notation for the result: The value of skew Schur at the unit locus at q = 1 can be also expressed through shifted Schur functions [21] where λ i denotes the lengths of l λ lines of the Young diagram λ. According to this definition, the shiftedχ µ {p λ } vanishes at the λ-locus (9) whenever µ is not a sub-diagram of λ. Since shifted Macdonald functions can be defined in just the same way as Schurs [22], eq.(8) can be immediately used to define a "refined" matrix B and thus, through (2), the hyper-polynomials (by definition of [12] they are result of a clever substitution of Schur by Macdonald functions in HOMPLY-PT polynomials, see also [18,23] and [24]). It was demonstrated in [1] that they are indeed positive Laurent polynomials, presumably in all rectangular representations and for all double twist knots.
For non-rectangular R expressions for Z X and F X become somewhat complicated, and one can expect that expression (2) of F X through an auxiliary(?) matrix B once again leads to drastic simplification. As we will see, this is indeed the case. Note that of the three properties for the figure-eight knot, unknot and trefoil respectively, the first one is automatic in (2), the second one requires that sum of the entries is zero along each line of B, and only the third one still remains a non-trivial constraint.
In this paper we consider the simplest case of R = [r, 1], for which the answers are already known from [7]. In this case in addition to the 2r + 1 diagrams X = (λ, λ) with λ ⊂ R = [r, 1], i.e. λ = ∅, each contributing once to the differential expansion. TheseX i contribute r − 1 additional lines to the matrix B, which thus becomes of the size 2r + 1 + r − 1 = 3r. Remarkably, B remains triangular, though a notion of embedding for generic composites X gets somewhat more subtle than (5). The first 2r + 1 lines remain as they were in (4). The new entries in the new r − 1 linesX i with i = 2, . . . , r are: In particular, As we see, these entries essentially depend on A, and are therefore sensitive to characters (or something else) beyond the unit locus (7).
In the simplest case of R = [2, 1] the matrix is The new one -revealed by consideration of the non-rectangular R -is the last line.
For R = [3, 1] the lineX 2 remains the same -this is the universality property of B,-and there is one more new, as compared to (4), line forX 3 : One can compare with the original formulas for F X in [7] to appreciate the simplification.