Extension of KNTZ trick to non-rectangular representations

TL;DR

This paper extends the KNTZ trick to non-rectangular representations in knot theory, simplifying formulas for complex representations and addressing multiplicities and gauge invariance issues.

Contribution

It introduces a reformulation of the universal-matrix precursor for non-rectangular representations, enabling simpler calculations in arborescent calculus.

Findings

Reformulated formulas for [r,1] representations

Demonstrated drastic simplification after reformulation

Addressed multiplicities and gauge invariance issues

Abstract

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.

Equations

$$\displaystyle\boxed{\begin{array}[]{ccc}{\cal H}_{R}^{(m,n)}=\sum_{X\subset R\otimes\bar{R}}Z^{X}_{R}\cdot{\cal F}_{X}^{(m,n)}=\sum_{X\subset R\otimes\bar{R}}Z^{X}_{R}\cdot{F_{X}^{(m)}F_{X}^{(n)}}\\ \\ F_{X}^{(m)}=\sum_{Y\triangleleft\,X}f_{XY}\cdot\Lambda_{Y}^{m}\end{array}}$$

$$\displaystyle\boxed{F_{X}^{(m)}=\sum_{Y\triangleleft\,X}\Big{(}{\cal B}^{m+1}\Big{)}_{XY}}$$

$$(\mu ,\mu )◃(\lambda ,\lambda )⟺\mu \subset \lambda$$

$$\boxed{{\mathcal{B}}_{\lambda \mu }=(-{)}^{|\lambda |-|\mu |}\cdot {\Lambda }_{\lambda }\cdot \frac{{\chi }_{\mu }^{\circ }\cdot {\chi }_{{\lambda }^{\vee }/{\mu }^{\vee }}^{\circ }}{{\chi }_{\lambda }^{\circ }}}$$

$$R\otimes \bar{R}={\oplus }_{\mu \in R}(\mu ,\mu )$$

$${\Lambda }_{\lambda }=\prod _i^{{\#}_{\mathrm{hooks}}(\lambda )}(q^{a_i-b_i}A{)}^{2(a_i+b_i+1)}$$

$$\displaystyle\chi_{\lambda}^{\circ}=\chi_{\lambda}\Big{\{}p_{k}=\frac{(q-q)^{k}}{q^{k}-q^{-k}}\Big{\}}$$

$$\frac{{\chi }_{\lambda /\mu }^{\circ }}{{\chi }_{\lambda }^{\circ }}={\bar{\chi }}_{\mu }\{{\bar{p}}^{\lambda }\}⟹{\mathcal{B}}_{\lambda \mu }\sim {\bar{\chi }}_{\mu }\{{\bar{p}}^{{\lambda }^{\vee }}\}$$

$$\displaystyle p_{k}=p_{k}^{\lambda}=\sum_{i=1}^{l_{\lambda}}\Big{(}(\lambda_{i}-i)^{k}-(-i)^{k}\Big{)}$$

$$F_X^{m=-1}=1,F_X^{m=0}=0,F_X^{m=1}=\prod _i^{{\#}_{\mathrm{hooks}}(X)}(-q^{a_i-b_i}{A}^2{)}^{a_i+b_i+1},$$

$${\tilde{\Lambda }}_i=(q^{i-2}A{)}^{2i},i=2,\dots ,r$$

$$\displaystyle\boxed{\begin{array}[]{ccl}{\cal B}_{\tilde{X}_{i},\emptyset}=&\frac{(-)^{i+1}\tilde{\Lambda}_{i}}{q^{(i-1)(i-2)}}\cdot A^{2}\\ \\ \phantom{.}\!{\cal B}_{\tilde{X}_{i},[j]}=&\frac{(-)^{i+j-1}\tilde{\Lambda}_{i}}{q^{(i-1)(i-j)}}\cdot\frac{[i-2]!}{[i-j]![j-1]!}\cdot\frac{[i-1]\cdot q^{3i+j-2}A^{2}-[i-j]\cdot q^{i-3}A^{2}-[j-1]}{q^{2j}-1}&{}_{j=1,\ldots,i}\\ \\ {\cal B}_{\tilde{X}_{i},[j,1]}=&\frac{(-)^{i+j-1}\tilde{\Lambda}_{i}}{q^{i^{2}-ij-2i-j+7}}\cdot\frac{[i-2]!}{[i-j-1]![j-1]!}\cdot\frac{(A^{2}-q^{2})(A^{2}-q^{6})}{(q^{2j+2}-1)(A^{2}q^{2j-4}-1)}&{}_{j=1,\ldots,i-1}\\ \\ {\cal B}_{\tilde{X}_{i},\tilde{X}_{j}}=&\frac{(-)^{i+j}\tilde{\Lambda}_{i}}{q^{(i-1)(i-j)}}\cdot\frac{[i-2]!}{[i-j]![j-2]!}\cdot\frac{A^{2}q^{2i-4}-1}{A^{2}q^{2j-4}-1}&{}_{j=2,\ldots,i}\\ \end{array}}$$

$$\displaystyle{\cal B}_{\tilde{X}_{i},[1]}=\ \$$

$$\displaystyle{\cal B}_{\tilde{X}_{i},[1,1]}=\ \$$

$$\displaystyle{\cal B}_{\tilde{X}_{i},[i]}=\ \$$

$$\displaystyle{\cal B}_{\tilde{X}_{i},[i,1]}=\ \$$

$$\displaystyle{\cal B}^{[2,1]}=\left(\begin{array}[]{c|cccccc}&\emptyset&[1]&[1,1]&[2]&[2,1]&\tilde{X}_{2}\\ &&&&&&\\ \hline\cr&&&&&&\\ \phantom{.}\emptyset&1&0&0&0&0&0\\ &&&&&&\\ \phantom{.}[1]&-A^{2}&A^{2}&0&0&0&0\\ &&&&&&\\ \phantom{.}[1,1]&\frac{A^{4}}{q^{2}}&-\frac{[2]A^{4}}{q^{3}}&\frac{A^{4}}{q^{4}}&0&0&0\\ &&&&&&\\ \phantom{.}[2]&q^{2}A^{4}&-[2]q^{3}A^{4}&0&q^{4}A^{4}&0&0\\ &&&&&&\\ \phantom{.}[2,1]&-A^{6}&[3]A^{6}&-\frac{[3]A^{6}}{[2]q}&-\frac{[3]qA^{6}}{[2]}&A^{6}&0\\ &&&&&&\\ \tilde{X}_{2}&-A^{6}&[3]A^{6}&\frac{(A^{2}-q^{6})A^{4}}{q^{2}(q^{4}-1)}&-\frac{(A^{2}q^{6}-1)A^{4}}{q^{4}-1}&0&A^{4}\end{array}\right)$$

$$Z_{[2,1]}^{\varnothing }=1,Z_{[2,1]}^{[1]}=\frac{[3]D_0^2+[3{]}^2{D}_2{D}_{-2}}{[2{]}^2},Z_{[2,1]}^{[2]}=\frac{[3]}{[2]}\cdot D_3{D}_2{D}_0{D}_{-2},Z_{[2,1]}^{[1,1]}=\frac{[3]}{[2]}\cdot D_2{D}_0{D}_{-2}{D}_{-3},$$

$$\displaystyle Z_{[2,1]}^{[2,1]}=D_{3}D_{2}D_{1}D_{-1}D_{-2}D_{-3},\ \ \ Z^{\widetilde{X_{2}}}_{[2,1]}=-[3]^{2}(q-q^{-1})^{4}\cdot D_{2}D_{-2}\ \ \ \ \$$

$$Z_{[r,1]}^{[k]}=\frac{[r+1]!}{[r][k]![r-k]!}\cdot \frac{D_{r+k-2}!D_{k-2}!}{D_{r-1}!}\left(D_{r-2}{D}_{k-1}-[r+1][k](q-q^{-1}{)}^2\right)=$$

$$=\frac{[r+1]}{[r{]}^2}\left(\frac{[r]!}{[k+1]![r-k-1]!}\cdot \frac{D_{r+k-2}!D_{k-1}!}{D_{r-1}!D_{-1}!}\cdot D_{r-2}+[k]\cdot \frac{[r+1]!}{[k+1]![r-k]!}\cdot \frac{D_{r+k-1}!D_{k-2}!}{D_{r-1}!D_{-1}!}\cdot D_{-2}\right)$$

$$Z_{[r,1]}^{[k,1]}=\frac{[k][r+1]!}{[r][k+1]![r-k]!}\cdot \frac{D_{r+k-1}!D_{k-1}!}{D_{r-1}!D_{k-2}}{D}_{-1}{D}_{-2}{D}_{-3}$$

$$Z_{[r,1]}^{\widetilde{X_k}}=-(q-q^{-1}{)}^4\cdot \frac{[r+1{]}^2[r-1]!}{[r-k]![k-2]!}\cdot \frac{D_{r+k-2}!D_{k-3}!}{D_{r-1}!}\cdot D_{-2}$$

Full text

MITP/TH-04/19

ITEP/TH-04/19

IITP/TH-04/19

Extension of KNTZ trick to non-rectangular representations

A.Morozov

MIPT, ITEP, & IITP, Moscow, Russia

ABSTRACT

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.

Spectacular success [1, 2] of the lasting program [3]-​​[9] to calculate colored knot polynomials [10] for antiparallel double braids (double twist knots) and Racah matrices [11] in all rectangular representations $R$ from the evolution properties [12]-​​[16] of their differential expansions [17, 13, 14, 18], opens a way to attack the main problem of arborescent calculus [19, 20]: understanding of non-rectangular representations. The main difference from rectangular case is that multiplicities occur in the product of representations, and this makes the notion of Racah matrices ambiguous. In the language of [20] this is described as the new gauge invariance and one of the problems is to define gauge-invariant arborescent vertices. However, before that there is a problem to calculate the Racah matrices $\bar{S}$ and $S$, which enter the definition of ”fingers” and ”propagators”, connected by these vertices. These problems, are not fully unrelated, because $\bar{S}$ and $S$ in non-rectangular case are not gauge invariant – still one can ask what they are in a particular gauge. As suggested in [4], the key to evaluation of $\bar{S}$ is differential expansion (DE) for twist knots [14] – which, once known, straightforwardly produces $\bar{S}$ for rectangular $R$, because of spectacular (and still unexplained!) factorization property of the DE coefficients for double braids. $S$ are then easily extractable as a diagonalizing matrix for $\bar{S}$ – it is enough to solve a system of linear equations. However, for non-rectangular $R$ the situation is worse: differential expansion for double braids includes not $\bar{S}$ itself, but some gauge-invariant combination of its matrix elements, and also the linear system for $S$ is degenerate and again provides only the information about gauge-invariant quantities. The problem therefore is to extract at this stages exactly the combinations, needed for arborescent calculus – and we do not yet know what they are. In other words, for non-rectangular $R$ we face a whole complex of related problems, which is partly surveyed in [20], [7] and, especially, [8]. Whatever the resolution will be, the first step is going to be the differential expansion for twist knots – and it is still not fully known for non-rectangular $R$. It is the goal of the present paper to suggest a mixture of the results of [7] and [1, 2] to advance in this direction.

We avoid repeating the whole story and refer to [2] for the latest summary and references. The crucial facts are the observation of [4] for the antiparallel double braid in Fig.1:

[TABLE]

and the second observation of [1, 2], that for rectangular $R$:

[TABLE]

where ${\cal B}$ is a universal triangular ”embedding” matrix with $Y\triangleleft\,X\subset R\otimes\bar{R}$. In this paper we consider the possibility for (5) 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 (4) are composite, see Fig.2. For rectangular representations $R=[r^{s}]=\overbrace{[r,\ldots,r]}^{s\ {\rm times}}$ only very special diagonal composites $(\lambda,\lambda)$ contribute to $R\otimes\bar{R}$ – and they are in one-to-one correspondence with the Young sub-diagrams of $\lambda\subset R$, and ”embedding” for diagonal composites is understood as embedding of the corresponding $\lambda$:

[TABLE]

The entries of the matrix ${\cal B}$ in (5) are expressed through the skew Schur functions:

[TABLE]

where $\vee$ stands for transposition of the Young diagram, and $\Lambda_{\mu}$ are the eigenvalues of ${\cal R}$-matrix in the channel

[TABLE]

best expressed through the hook parameters of $\lambda=(a_{1},b_{1}|a_{2},b_{2}|,\ldots)$:

[TABLE]

Index $\circ$ means that Schur functions are evaluated at the ”unit” locus in the space of time-variables,

[TABLE]

At $q=1$ this is equivalent to putting $p_{k}=\delta_{k,1}$, and there is even a a special notation for the result: $\chi_{\lambda}\{\delta_{k,1}\}=d_{\lambda}$. The value of skew Schur at the unit locus at $q=1$ can be also expressed through shifted Schur functions [21]

[TABLE]

evaluated at

[TABLE]

where $\lambda_{i}$ denotes the lengths of $l_{\lambda}$ lines of the Young diagram $\lambda$. According to this definition, the shifted $\bar{\chi}_{\mu}\{p^{\lambda}\}$ vanishes at the $\lambda$-locus (12) whenever $\mu$ is not a sub-diagram of $\lambda$. Since shifted Macdonald functions can be defined in just the same way as Schurs [22], eq.(11) can be immediately used to define a ”refined” matrix ${\cal B}$ and thus, through (5), 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 [23, 18] 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 (5) of $F_{X}$ through an auxiliary matrix ${\cal B}$ once again leads to drastic simplification. As we will see, this is indeed the case. Note that of the three properties

[TABLE]

for the figure-eight knot, unknot and trefoil respectively, the first one is automatic in (5), the second one requires that sum of the entries of ${\cal B}$ is zero along each line, $\sum_{Y}{\cal B}_{XY}=0$ $\ \forall X$, and the third one then says that $\sum_{Y}({\cal B}^{2})_{XY}={\cal B}_{X\emptyset}$ is a monomial $F_{X}^{(1)}$.

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 $X=(\lambda,\lambda)$ with $\lambda\subset R=[r,1]$, i.e. $\lambda=\emptyset,[i],[i,1]$, $i=1,\ldots,r$ there are $r-1$ additional composite pairs $\boxed{\tilde{X}_{i}=([i-1,1],[i])\oplus([i],[i-1,1])}$ with the same dimensions and eigenvalues

[TABLE]

each contributing once to the differential expansion. These $\tilde{X}_{i}$ contribute $r-1$ additional lines to the matrix ${\cal B}$, which thus becomes of the size $2r+1+r-1=3r$. Remarkably, ${\cal B}$ remains triangular, though a notion of embedding for generic composites $X$ gets somewhat more subtle than (8). The first $2r+1$ lines remain as they were in (7). The new entries in the new $r-1$ lines $\tilde{X}_{i}$ with $i=2,\ldots,r$ are:

.

[TABLE]

In particular,

[TABLE]

As we see, these entries essentially depend on $A$, and are therefore sensitive to characters (or something else) beyond the unit locus (10).

In the simplest case of $R=[2,1]$ the matrix is

[TABLE]

The new one – revealed by consideration of the non-rectangular $R$ – is the last line.

For $R=[3,1]$ the line $\tilde{X}_{2}$ remains the same – this is the universality property of ${\cal B}$ – and there is one more new, as compared to (7), line for $\tilde{X}_{3}$:

\begin{array}[]{c||ccccccccc}&\emptyset&[1]&[1,1]&[2]&[2,1]&[3]&[3,1]&\tilde{X}_{2}&\tilde{X}_{3}\\ &&&&&&&&&\\ \hline\cr&&&&&&&&&\\ \tilde{X}_{3}&q^{4}A^{8}&-[4]q^{5}A^{8}&-\frac{q^{2}(A^{2}-q^{6})A^{6}}{q^{4}-1}&\frac{q^{4}\big{(}A^{2}(q^{10}+q^{8}-1)-1\big{)}A^{6}}{q^{4}-1}&\frac{q^{4}(A^{2}-q^{2})(A^{2}-q^{6})A^{6}}{(q^{6}-1)(A^{2}-1)}&-\frac{q^{6}(A^{2}q^{10}-1)A^{6}}{q^{6}-1}&0&-\frac{q^{4}(A^{2}q^{2}-1)A^{6}}{A^{2}-1}&q^{6}A^{6}\end{array}

One can compare with the original formulas for $F_{X}$ in [7] to appreciate the simplification.

To make the story about $R=[r,1]$ complete, we need also explicit formula for the $Z$-factors. They are made from the differentials $D_{n}:=Aq^{N}-A^{-1}q^{-N}$, for example for $R=[2,1]$

[TABLE]

We see that one of the $Z$-factors is not fully factorized – this is the one, associated with the diagram $\lambda=[1]$ which has non-trivial multiplicity. Since multiplicity is two, it is naturally decomposed into sum of two factorized items. For a full understanding of how this works we need a more sophisticated theory, involving the analogue of $U$-matrices from [2] for non-rectangular $R=[2,1]$ and reduction from the full-fledged $10\times 10$ matrix representation to the $6\times 6$ one. This is a difficult and still not fully-developed subject beyond the scope of the present paper.

Known at this stage are all the $Z$-factors in the case of $R=[r,1]$, see [7]. They are not-quite-factorized for all single-line $\lambda=[k]$ with, which enter with multiplicities two:

[TABLE]

and are factorized for the other two series:

[TABLE]

[TABLE]

In fact, (34) is also factorized (the first term in the second line vanishes) at $k=r$.

For generic $R$ we should consider all composite $\boxed{X=\oplus_{|\lambda|=|\lambda^{\prime}|}(\lambda,\lambda^{\prime})}$ with all pairs of the same-size sub-diagrams of $R$: $\lambda,\lambda^{\prime}\subset R$, $|\lambda|=|\lambda^{\prime}|$. For example, for $R=[3,2]$ there will be three non-diagonal ($\lambda^{\prime}\neq\lambda$) pairs: the two already familiar $([2],[1,1])\oplus([1,1],[2])$, $([3],[2,1])\oplus([2,1],[3])$ and a new one: $([3,1],[2,2])\oplus([2,2],[3,1])$. For the psychologically important $R=[4,2]$ we encounter the first triple $\{[4],[3,1],[2,2]\}$, giving rise to three pairs $([4],[3,1])\oplus([3,1],[4])$, $([3,1],[2,2])\oplus([2,2],[3,1])$ and $([4],[2,2])\oplus([2,2],[4])$. Formulas (16) should be straightforwardly extendable to this general case – but it remains to be done, and it remains to be seen if triangular shape of ${\cal B}$ persists. Hopefully straightforward are also their Macdonald deformations – and it is interesting to see if this leads to hyper-polynomials, but not fully positive – as currently suspected for non-rectangular representations.

Acknowledgements

My work is partly supported by the grant of the Foundation for the Advancement of Theoretical Physics BASIS, by RFBR grant 19-02-00815 and by the joint grants 17-51-50051-YaF, 18-51-05015-Arm, 18-51-45010-Ind, RFBR-GFEN 19-51-53014.