Cauchy formula and the character ring
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Abstract
Cauchy summation formula plays a central role in application of character calculus to many problems, from AGTimplied Nekrasov decomposition of conformal blocks to topologicalvertex decompositions of link invariants. We briefly review the equivalence between Cauchy formula and expressibility of skew characters through the Littlewood–Richardson coefficients. As notquiteatrivial illustration we consider how this equivalence works in the case of plane partitions – at the simplest truly interesting level of just four boxes.
1 Introduction

in formal representation theory – as characters of \(Sl_N\) representations \(S_R[X]=\mathrm{Tr}_R \mathcal{X}^{(R)}\), and thus as the building blocks for integrable taufunctions through the general construction, reviewed in [33]
 in decomposition formulas for the integrands of freefield screening correlators like$$\begin{aligned} \left<\prod _i e^{\phi (x_i)} \prod _j e^{\phi (y_j)}\right>_{\phi }\sim & {} \prod _{i,j} (x_iy_j)^{1} \nonumber \\\sim & {} \sum _R S_R[X]S_{R^\vee }[Y^{1}] \end{aligned}$$(1)
 and as preserved quantities in Selberg–Kadelltype integrals [34, 35, 36, 37, 38, 39, 40], which stand behind the basic superintegrability/localization property [41]actually serving as a selection rule for “good” theories, which provide “matrixmodel \(\tau \)functions” [1, 2, 3, 4, 5, 6] after (functional) integration over fields.$$\begin{aligned} \Big < S_R[X] \Big >_X \sim S_R[X^*] \end{aligned}$$(2)
In this short note we consider two of such properties: Cauchy decomposition formula which stands behind (1) and the skewcharacter decomposition, which plays the central role in technical applications of Schur functions to representation theory. These two properties are in fact intimately related: imposing one implies another. This is a simple but important remark, it considerably weakens their impact on generalizations: one restriction (to keep these properties) is much less than two. Also, it reduces the number of “miracles” and thus the attractiveness of particular generalization attempts. After a brief presentation of the formal relation we present an explicit example – of the problems with the building project of the 3Schur functions, encountered at the level of the sizefour plain partitions, where the (general) relation between Cauchy and skew decompositions shows up in a somewhat unusual way.
2 Cauchy vs skew
This statement can be partly inverted: if the skew functions in (8) possess the expansion (9) with the same structure constants as in (3), this implies some version of Cauchy summation formula (4) with a bilinear exponent – but, strictly speaking, with some unspecified coefficients at the place of \(p_k^{2}\).
To avoid possible confusion, (3) and (10) are not the statements – these are just the definitions of the structure constants N and \(\mathbf{N}\) for a given set of functions \(S_\sigma \{p\}\). Of course, one can instead use (9) as a definition of \({\bar{N}}\), like it was done in [31, 32], – then the statement will be that (10) depends on validity of some version of (4).
3 Particular cases
However, already for the bynowconventional applications, restriction to \(\Sigma =\{partitions\}\) is insufficient. Nekrasov calculus for generic \(\Omega \)backgrounds (for \(c\ne 1\), i.e. \(\epsilon _1\ne \epsilon _2\)) requires “generalized” Macdonald functions [10, 11, 12, 13, 14, 15, 16], depending on collections (strings) of Young diagrams. This, however, is not a very big problem – it is enough just to consider several copies of time variables, though the scalar product can require nontrivial modification [97]. More challenging are the ordered sequences of Young diagrams (forming the plane partitions), which are needed in generic network models and representation theory of DIMalgebras. The corresponding “tripleMacdonald polynomials”, though constructible in terms of the ordinary ones [98], should depend on a very different set K of timevariables and be described by a morefirstprinciple theory.
An ever furthergoing challenge is adequate description of tensormodel characters, where some “nonabelization” looks unavoidable already at the level of (2) – straightforward lifting of Schur functions to these theories does not seem to provide a full basis in the operator space [17]. In this note we do not go as far as fullfledged tensormodel considerations, but provide just a simple example of difficulties, encountered at the plainpartition stage. We demonstrate that, conversely to possible expectations, Cauchy formula is considerably easier to satisfy than building a true collection of 3Schur functions.
4 The 3Schur attempt
5 Expectation at level four
6 The situation at level four
Coming back to multiplication rules, the differences between expected and actual formulas are marked by boxes. The main of them is the absence of any contribution from \(S_{[4]}\) – but, according to the argument in Sect. 2, this absence in both multiplication and Cauchy formulas is not independent. Thus it is enough to explain it in just one of these cases. The simplest is the first line in the multiplication list: there it is sufficient to look only at the terms \(p_3p_1\) and \(p_1^4\). The fact is that the ratio of coefficients in front of these structures is exactly the same in the sum \(\mathcal{S}^{\rho '\rho ''}_{[3,1]}+\mathcal{S}^{\rho ''\rho '}_{[3,1]}\) and \(\mathcal{S}^\rho _{[3]}\). Indeed, in the latter case the ratio is \(\frac{\vec \alpha ^\rho _3\vec p_3}{3}\left( \frac{p_1^3}{6}\right) ^{1} = \frac{2\vec \alpha ^\rho _3\vec p_3}{p_1^3}\), while in the former case it is rather \(\ \frac{(2\vec \alpha _3^\rho +\vec \beta _3^{\rho '}+\vec \beta _3^{\rho ''})\vec p_3p_1}{3} \left( 2\frac{p_1^4}{8}\right) ^{1} = \frac{4(2\vec \alpha _3^\rho +\vec \beta _3^{\rho '}+\vec \beta _3^{\rho ''})\vec p_3}{3p_1^3}\ \) – but since \(\ \vec \beta _3^{\rho '}+\vec \beta _3^{\rho ''}=\frac{1}{2}\vec \alpha _3^\rho \ \) this is actually the same. At the same time the same ratio for \(\mathcal{S}_{[4]}^\rho \) is four times bigger: \(\frac{\vec \alpha _3^\rho \vec p_3 p_1}{3}\left( \frac{p_1^4}{24}\right) ^{1} = \frac{8\vec \alpha _3^\rho \vec p_3}{p_1^4}\), thus already for these two items one has no chances to add \(\mathcal{S}_{[4]}^\rho \) with any nonvanishing coefficient.
Equally interesting can be the emerging additional terms in the multiplication rule. We remind that the product of representations \([2]\otimes [1,1] = [3,1]\oplus [2,1,1]\), i.e. this is the first example when the product does not contain the intermediate diagram [2, 2], which lies between \([2]+[1,1]=[2,1,1]\) and \([2]\cup [1,1]=[3,1]\) in the lexicographical ordering. This is exactly the situation reflected in (15), i.e. the [2, 2] contribution should vanish for Schur and Macdonald functions, but show up in the generic Kerov case. In fact, the Kerov function \(\mathrm{Kerov}_{[2,2]}\) appears in the product \(\mathrm{Kerov}_{[2]}\cdot \mathrm{Kerov}_{[1,1]}\) with a peculiar coefficient \(g_4g_1^53g_4g_2^2g_1+2g_4g_3g_1^2+2g_2^3g_1^33g_3g_2g_1^4+g_3g_2^3\) which is the simplest combination of gvariables, vanishing at the Macdonald locus (14). The fact that a boxed item appears in the product of the corresponding 3Schur functions \((\vec \alpha _2\vec p_2)^2 \in \mathcal{S}_{[2]}^\rho \cdot \mathcal{S}_{[2]}^{\rho ''}\) can be a signal that they know about the violation (15) of the representationproduct selection rule – and have a potential of describing the generic situation, including the Kerov functions.
7 Anomaly in the Cauchy formula
Since the true multiplication formulas at level 4 are different from the expectation, i.e. do not fully match the decomposition formulas (24), we should observe the violation of Cauchy formula. Indeed, this is what immediately observes in (28). This formula does not contain any reference to \(\mathcal{S}_{[4]}\) – and this is in accordance with the multiplication rule, where this function also does not appear, thus this not a violation. However, instead it contains an anomalous term \(X\{p,p'\}\), reflecting the true difference between multiplication and decomposition, which we now analyze in a little more detail.
8 Conclusion
In this note we explained the nearly rigid relation between Cauchy summation formula (4) and the equivalence of the structure constants in multiplication and skewdecomposition formulas (9) and (10). We illustrated this fact by an important example of the wouldbe 3Schur functions for 4box plane partitions: mismatches/anomailes are simultaneously present and well correlated in expressions of both kinds. Thus it is sufficient to cure just one of them – the other will be automatically fixed. This, however, remains to be done. More generally, beyond the 3Schur topic, this paper can help to understand the abundance of Cauchy formula, i.e. why it appears in one and the same form for a broad variety of special functions and why it may not be obligatory restricted to the case of Young diagrams.
Notes
Acknowledgements
My work is partly supported by the grant of the Foundation for the Advancement of Theoretical Physics BASIS, by RFBR Grant 190200815 and by the joint Grants 185105015Arm, 185145010Ind, 175150051YaF, RFBRGFEN 195153014. I also acknowledge the hospitality of KITP and partial support by the National Science Foundation under Grant no. NSF PHY1748958 at the final stage of this project.
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