aa r X i v : . [ h e p - t h ] N ov Kerov functions revisited
A.Mironov a,b,c ∗ and A.Morozov b,c † FIAN/TD-21/18IITP/TH-19/18ITEP/TH-33/18 a Lebedev Physics Institute, Moscow 119991, Russia b ITEP, Moscow 117218, Russia c Institute for Information Transmission Problems, Moscow 127994, Russia
Abstract
The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots andof topological vertices in DIM-based network theories, which could merge into a unified theory still to bedeveloped. The Macdonald functions do the same for hyperpolynomials and refined vertices, but mergingappears to be more problematic. For a detailed study of this problem, more knowledge is needed about theMacdonald polynomials than is usually available. As a preparation for the discussion of the knot/verticesrelation, we summarize the relevant facts and open problems about the Macdonald and, more generally,Kerov functions. Like Macdonald polynomials, they are triangular combinations of Schur functions, butorthogonal in a more general scalar product. We explain that parameters of the measure can be consideredas a set of new time variables, and the Kerov functions are actually expressed through the Schur functionsof these variables as well. Despite they provide an infinite-parametric extension of the Schur and Macdonaldpolynomials, the Kerov functions, and even the skew Kerov functions continue to satisfy the most importantrelations, like Cauchy summation formula, the transposition identity for reflection of the Young diagram andexpression of the skew functions through the generalized Littlewood-Richardson structure constants. Since these are the properties important in most applications, one can expect that the Kerov extension exists formost of them, from the superintegrable matrix and tensor models to knot theory.
Recently in [1] a program was originated to construct various (not obligatory torus) link polynomials [2]from topological vertices [3]. Already in the simplest example of the link L n considered in that paper,one encounters composite representations, associated ”uniform” Schur functions and their decompositions intobilinear combinations of the more conventional skew Schur functions, and the skew Schur is what is actuallyassociated with the topological vertex.This technique is however not easily lifted from the HOMFLY polynomials to the level of hyperpolynomials(the algebraically defined counterparts of Khovanov-Rozansky and superpolynomials), largely because of ourinsufficient knowledge about the Macdonald polynomials, which substitute the Schur functions in this lifting.This paper is a preparation for developing a theory of composite Macdonald functions and its application tostudy of the hyperpolynomial/refined-vertex relation, which will be presented elsewhere [4]. Needed for thistheory are the basic facts about the Macdonald functions in the form which allows one to easily calculate themfor arbitrary representations R at sufficiently high levels | R | .In this paper, we make such a list of properties, moreover, we present it for more general Kerov functions [5],because in applications it is important to understand what in these functions is depending on the obstacles forthe Schur −→ Macdonald −→ Kerov extension, what are the ways to overcome these obstacles and what oneneeds to sacrifice when doing this.We begin with the most general Kerov functions and then descend to the Macdonald and Schur functions,which have some additional properties absent in the Kerov case. The very few properties, which are true onlyfor the Schur functions and are destroyed already by the Macdonald deformation are underlined, those whichare true for the Macdonald, but not for generic Kerov functions are underlined twice. The Macdonald functionshave closer relation to group theory but many of their properties are actually true in the more general Kerovframework.The Kerov function should not be confused with much better known ”Kerov character polynomials”, alsoassociated with the name of S.Kerov [7]: to avoid the confusion, we use the term functions throughout thepresent text. ∗ [email protected]; [email protected] † [email protected] Kerov functions and their properties • The pair of Kerov functions Ker ( g ) R { p } and d Ker ( g ) R { p } depends on the Young diagram (representation) R = [ r ≥ r ≥ . . . ≥ r l R > p k homogeneousw.r.t. the grading grad( p k ) = k . Thus, a particular Ker R { p } depends only on | R | times, the level | R | being the size of R (number of boxes). The two functions Ker ( g ) R { p } , d Ker ( g ) R { p } differ by the choice ofordering in the sums in (1) below. • The best for practical purposes is to define the Kerov and Macdonald functions by a triangular transformfrom the Schur functions,
Ker ( g ) R { p } = Schur R { p } + P R ′ 1] = [4 , , 1] + [5 , b K -Kerov functions, the same relation contains notthree, but four terms:Ker [4] · Ker [1 , = ˆ α · d Ker [4 , , + ˆ β · Ker [4 , + ˆ δ · d Ker [3 , + ˆ γ · Ker [5 , (32)because there are two diagrams [2 , , , 1] = [4 , ∨ and [2 , , 2] = [3 , ∨ between [2 , , , , 1] = [5 , ∨ and [3 , , , 1] = [4 , , ∨ . We made it explicit that only two of the four emerging b K -Kerov functions aredifferent from the K -ones. 5 In the Macdonald case, however, (30) is believed to survive. In particular, the coefficient β in (31), vanishesin the Schur and Macdonald cases κ = 0 , 1, but not for κ > g ( k ) = x k + x k + x k β ∼ x x x ( x − x x )( x − x x )( x − x x ) · B ( x ) (33)with a complicated irreducible polynomial B ( x ). There is also a relatively simple denominator. d Ker and Ker in Macdonald case Actually only half of the Macdonald functions needs to be calculated directly, the other half can be obtainedfrom (13). This is a big calculational advantage, because the computer time needed for orthogonalization growsdramatically with the number of terms in the sum (1), and reducing the problem to sums of twice smaller lengthallows one to calculate the Kerov/Macdonald functions for higher levels | R | .At the same, for the Kerov functions this is not quite correct because of the two types of the Kerov functionsrelated by (13). In other words, (13) is a duality, and not self-duality condition in the generic Kerov case. • It is remarkable that (13) is a self-duality condition not only in the Schur, but also in the Macdonald caseof κ = 1 when it is sufficient to define d Mac = Mac so that (13) becomesMac R { q, t, p k } = ( − ) | R | · || Mac R || · Mac R ∨ (cid:26) t, q, − { t k }{ q k } p k (cid:27) (34)In the Schur case, this further simplifies to justSchur R ∨ { p } = ( − ) | R | · Schur R {− p } (35) • The reason for this simplification is that, at κ = 1, all the problematic Macdonald-Kostka coefficients, i.e.those for the pairs of diagrams, when both R > R ′ and R ∨ > R ′ , are vanishing, for example K (Mac)[3 , , , , [2 , , = 0 (36)This vanishing condition imposes severe restrictions on the function g , which are satisfied for κ = 1, i.e.for the Macdonald deformation, but are violated for κ > 1, i.e. for the generic Kerov functions.The analogue of (33) is also true: on the 3-dimensional locus g ( k ) = x k + x k + x k this Macdonald-Kostkacoefficient reduces to K [3 , , , , [2 , , ∼ x x x ( x − x x )( x − x x )( x − x x ) · Schur [7] { g } (37)but the substitute of the polynomial B ( x ) in this case is actually much simpler. Surprisingly, one againarrives at the Macdonald properties of the group theory expansion (30) and self-duality (34) on the two-dimensional locus, where both β in (33) and K [3 , , , , [2 , , in (37) vanish. It can be chosen, for instance,at x = x x . • Actually, in the Schur and Macdonald cases, one can use in (1) any partial ordering of Young diagramswhich satisfies the dominance rule : R ≥ R ′ if i X a =1 r a ≥ i X a =1 r ′ a for all i (38)since, for all pairs which remain unordered, the corresponding Kostka numbers vanish, together with theirinverse, e.g. k ± , , , , [2 , , = 0 (39)(inversion here, as in similar cases before, is inversion of the entire triangular matrix, not of the particularentry).To finish this piece of the story, we emphasize once again that the duality (13) is rather non-trivial, andit guarantees that the Kerov/Macdonald functions for symmetric representations and ”nearly-symmetric” areactually as simple as those for the antisymmetric, i.e. equivalence of the symmetric and antisymmetric ”ends” ofthe partition list, which is far from obvious from the definition (1), where the antisymmetric functions coincidewith the antisymmetric Schur polynomials, while symmetric are combinations of all Schur functions at a givenlevel. 6 Properties peculiar for the Schur functions • Specific for the Schur functions per se is determinant expression through those in symmetric representations S R = det l R i,j =1 S [ r i − i + j ] , S [ < = 0 (40)Deformations of this formula exist, for example, one can treat in this way the determinant formula for theMacdonald-Kostka coefficients. • The Schur functions are expanded in symmetric-group characters ψ R (∆):Schur R { p } = X ∆ ψ R (∆) z ∆ · p ∆ (41)and their orthogonality (7) is related to the orthogonality of characters, X ∆ ψ R (∆) ψ R ′ (∆) z ∆ = δ R,R ′ (42)In both these formulas, the sums are over Young diagrams of size | ∆ | = | R | = | R ′ | . Possible Kerov andMacdonald deformations of (41) would require a deep understanding of the Schur-Weyl-Howe duality [20]and is related [21] to deep and yet unanswered questions about the DIM algebra [22] and its furthergeneralizations. • As described in detail in [23], the Schur functions are the common eigenfunctions of the infinite set ofcommuting generalized cut-and-join operators ˆ W ∆ with eigenvalues made from symmetric group characters ψ R (∆): ˆ W ∆ (cid:26) p k , ∂∂p k (cid:27) Schur R { p } = ψ R (∆) d R z ∆ · Schur R { p } (43)where d R := Schur R { δ k, } . Existence of such operators at each level is a trivial linear-algebra propertyeasily generalizable to the Macdonald and Kerov functions (as well as far beyond them, see, for example,[9]). However, in the Schur (and, partly, Macdonald) case, they can be ”unified” into an operator madeby the Sugawara construction [24] from a linear Cherednik-Dunkl operator ˆ J (a counterpart of the Kac-Moody current in the WZW model [25]): ˆ W R ∼ Tr ˆ J R , which interrelates naive operators at differentlevels. Getting a deeper understanding of these relations and their proper extension beyond the Schurcase is one of the many open questions in the field. • The Gaussian averages of the Schur functions provide the same functions evaluated at the topologicallocus [26]: Z n × n Schur R [ X ] e − tr X dX ∼ Schur R { p k = N } = dim N ( R ) Z n × n Schur R [ e X ] e − ~ tr X dX ∼ Schur R (cid:26) p k = { e ~ N }{ e ~ } (cid:27) = D N ( R ) (44)which are ordinary and quantum dimensions of the representation R of SL n and U e ~ ( SL n ) correspond-ingly. This property is responsible for the superintegrability of matrix models [27], it has importantgeneralizations in many directions, from knot theory to a non-trivial extension to tensor models [28]. ItsMacdonald generalization is straightforward [29], but it involves the Jackson integrals, which have no yeta straightforward generalization to the generic Kerov deformations. • One could think that (40)-(44) are the only properties of the Schur functions which get significantlychanged (into much ”heavier” formulas) under the Kerov/Macdonald deformation, but we have foundanother one. As already mentioned at the beginning of this paper, the definition of Schur functions inthe composite representations also appears to be deformed not straightforwardly. An explanationof this fact and the ways to overcome the problem will be a starting point of our far-more-speculativeconsideration in the next paper [30]. 7 Measure g -variables as the new times As will be illustrated by examples in the Appendix, values of the function g n can actually be treated asnew time-variables so that the Kerov functions can be efficiently expressed through the Schur functions of thesevariables. Moreover, this expansion is somewhat different for the diagrams, which appear at the beginning andat the end of the lexicographical ordering, and the relation between the two expansions is not fully exhaustedby (13): there are additional symmetries between the structures for the k -th diagram at the beginning andthe k − p k and g k appear inone expansion, they become functions of p ∨ k = p k /g k in another, and it appears that g k is also substituted by g − k . The peculiarity of this part of the story is that the expansion in p ∨ k is much simpler than the one in p k ,which is originally used in (1). This expansion is upper instead of lower triangle and, in the Macdonald case,its coefficients are known to be rather simple minors [31]. It looks like this result is extendable to the Kerovfunctions: the coefficients can be expressed through the Schur functions at p k = g − k .Note that, while g k can clearly be treated as new (additional) time-variables, they participate in additionaloperations like inversion and multiplication, which are unusual for time-variables. In particular, these operationsare not seen on the Miwa loci, and thus are not captured by the usual symmetric functions technique; this can bethe reason why these interesting structures were not noticed, and thus theory of the Kerov functions remainedunderestimated and nearly undeveloped. • The Kerov functions are polynomials in p , but as functions of g they are rational, with g -dependentdenominators, which are related through (13) to the norms of Kerov functions for transposed diagrams,Ker ( g ) R { p } = pol R ( p, g )∆ R { g } (45) • For the first diagram in the lexicographical ordering, the Kerov function is independent of g , and thedenominator is unity: Ker ( g )[1 r ] { p } = Schur [1 r ] { p } , ∆ (1) r := ∆ [1 r ] = 1 (46) • However, already for the second diagram [2 , r − ] (associated with the adjoint representation of SL r )there is a non-trivial Macdonald-Kostka coefficient, which is actually the ratio of two Schur functions of g -variables: K ( g )[2 , r − ] , [1 r ] = − Schur [ r − , { g } Schur [ r ] { g } (47)This implies that Ker ( g )[2 , r − ] { p } = Schur [2 , r − ] { p } − Schur [ r − , { g } Schur [ r ] { g } · Schur [1 r ] { p } (48)Thus, the next denominator is ∆ (2) r := ∆ [2 , r − ] = Schur [ r ] { g } (49)If multiplied by r ! it is a polynomial with integer coefficients. • In the case of the third diagram [2 , , r − ], the denominator is described by a somewhat strange 3-termexpression: ∆ (3) r := ∆ [2 , , r − ] = det Schur [ r ] [1] Schur [ r ] Schur [ r − [ r − Schur [ r − (50)8his time it becomes an integer polynomial after multiplication by r !( r − K ( g )[2 , , r − ] , [1 r ] = − Schur [ r − , · Schur [ r − − Schur [ r − , · Schur [ r − · Schur [2] +Schur [ r − , · Schur [ r − · Schur [1] ∆ (3) r K ( g )[2 , , r − ] , [2 , r − ] = − Schur [ r − , · Schur [ r − − Schur [ r − , · Schur [ r − · Schur [2] +Schur [ r − , · Schur [ r − · Schur [2] ∆ (3) r (51)and, as a corollary, for the third Kerov function Ker ( g )[2 , , r − ] . Note that the two functions in (51) differonly in their last terms, the first two are the same. • The truly interesting is the fourth diagram, [2 , , , r − ], because it is the first one where the two orderingsbecome different: [2 , , , r − ] < [3 , r − ] but also [2 , , , r − ] ∨ = [ r − , < [ r − , , 1] = [3 , r − ] ∨ ;in other words, [2 , , , r − ] is the fourth in the lexicographical ordering, but in transposed ordering thefourth diagram is [3 , r − ]. To get an insight of what to expect from ∆ (4) relevant in this case, it ispractical to consider a complementary part of the story, what we will do next. We return to ∆ (4) in (58)below. • As one can anticipate from (13), the Kerov function in symmetric representation, which is the last in thelexicographical ordering, is almost as simple as in the antisymmetric one, which is the first, despite thisis not easy to see directly from the definition (1). In fact, (13) implies that Ker [ r ] is a Schur polynomialof p k /g k , but the g -dependence of the prefactor still needs to be found. In this case, it is easy. Accordingto (13), what is needed is the norm of the antisymmetric Schur function:Ker ( g )[ r ] { p } ( ) = ( − ) | R | Ker ( g − )[1 r ] {− p k /g k }|| Ker ( g − )[1 r ] || = ( − ) | R | Schur [1 r ] {− p k /g k }|| Ker ( g − )[1 r ] || = Schur [ r ] { p k /g k }|| Ker ( g − )[1 r ] || = ⇒ ∆ ∨ r ∼ || Ker ( g − )[1 r ] || = D Schur [1 r ] (cid:12)(cid:12)(cid:12) Schur [1 r ] E ( g − ) ( )+( ) = X ∆ ⊢ r ψ [1 r ] (∆) z ∆ g ∆ = Schur [ r ] { g − k } (52)where g ∆ = Q g δ i , and, at the last stage, we used peculiarities of antisymmetric and symmetric represen-tations: ψ [1 r ] (∆) = ( − ) l R + | R | and ψ [ r ] (∆) = 1 for any ∆ of the size r . l R Thus, for the Kerov functionsin symmetric representations, we get a universal expressionKer ( g )[ r ] { p } = Schur [ r ] n p k g k o Schur [ r ] n g k o (53)explicitly realizing their relation (13) to the g -independent Kerov functions in antisymmetric representa-tions Ker ( g )[1 r ] { p } = Schur [1 r ] { p k } (54) • Remarkably, the denominator in (53), which has no counterpart in (54), is related to the one in (48):∆ [ r ] { g k } = ∆ (2) r { g − k } (55) • Moreover, this property persists further: the denominator in the case of penultimate (second from theend) diagram [ r − , 1] is obtained by g -inversion from that for the third one, [2 , , r − ]:∆ [ r − , { g k } = ∆ (3) r { g − k } (56)Given (50), this is an absolutely explicit formula, and it enters the equally explicit expression for thecorresponding Kerov function:Ker ( g )[ r − , { p } = Schur [ r ] n g k o · Schur [ r − , n p k g k o − Schur [ r − , n g k o · Schur [ r ] n p k g k o(cid:16) Schur [1] · ∆ (3) r (cid:17) n g k o (57)9ote that, while the denominator is related to that for Ker ( g )[2 , , r − ] (this is not quite the relation (13),the two diagrams are not transposed, but additionally shifted by one in the lexicographical ordering!),this is not immediately so for the numerators. Instead, the numerators of the Kerov functions in thevicinity of symmetric case demonstrate a structure of minor expansion, which should be a straightforwardgeneralization of the one found in [31] for the Macdonald functions. • Since for the particular value of r = 3, the first diagram from the end coincides with the third one fromthe beginning, we can expect that, in this case, ∆ (2) { g − } when expanded in g will have the same shapeas ∆ (3) , i.e. the Schur function of inverted g will be a three-term determinant. This is indeed the case, andit implies that the shape of ∆ [ r − , can actually reveal the structure of ∆ (4) { g − } and thus of ∆ (4) { g } .Already the simplest ∆ [2 , is given by a somewhat sophisticated 19-term formula (see the very end of thenext section), and this illustrates the type of problems one should deal with and resolve. • The full expression for ∆ (4) is rather lengthy, for illustrative purposes, we present just the first threeterms, which fully capture the contribution with the product g [ r ] g [ r − :∆ (4) r := ∆ [2 , , , r − ] b ∆ (4) r := b ∆ [3 , r − ] == Schur [ r ] · Schur [ r − · (cid:16) Schur [ r − Schur [2] + Schur [ r − Schur + Schur [ r − Schur (cid:17) + . . . (58)(for example of the full expression in a particular case, see the last formula in (71) in the Appendix). Thistime the factorial would be (( r − ( r − (4) r at even r are divisible by 2 (i.e. areeven integer polynomials). What is important, the hatted and ordinary quantities are the same,∆ (4) = d ∆ (4) (59)despite they appear in the denominators of very different functions, Ker [2 , , , r − ] and d Ker [3 , r − ] . It isinteresting, if this property (transposition independence) persists for all higher ∆ ( m ) . • We emphasize that all these strangely looking combinations of the Schur functions in ∆’s are factorized atthe topological locus, which explains factorization of the associated Macdonald quantities. The Macdonaldfactorizations begin to attract interest in various contexts, see, for example, [12, 32, 33], and our observa-tions suggest that these studies can have relation to formulas for the Kerov functions (which themselvesare not factorized). Factorizations are also crucial for knot theory applications, and further extension of,say, the Rosso-Jones formula, which defines the Macdonald based torus knot/link hyperpolynomials, tothe Kerov functions depends on understanding of what substitutes the factorization in the Kerov case. To conclude, we presented a list of properties of the Kerov functions. Our goal was to illustrate thattheory of these functions is very rich, and they rather have more interesting properties that the Schur andMacdonald functions, in contrary to the common belief. Lost in the Kerov case are very special features likevanishing the multiplication structure constants for diagrams which are not in the product of representationsand similar rigid (precise) links with representation theory, as well as a Sugawara-like construction of thegeneralized cut-and-join operators through the U (1)-like Cherednik-Dunkl operators. Though important, theseproperties are not truly crucial for many applications, especially because the other ones, which are really neededfor practical calculations, like the Cauchy summation formula (19) for the skew Kerov functions and even thetransposition rule (13) nicely persist. Instead of the lost properties, an entire new world of additional time-variables g k emerges. Moreover, a role appears for non-trivial operations over the time-variable like inversionand multiplication, and these unexpected operations do act on the Kerov functions, quite non-trivially, butnicely and explicitly.We hope that our modest review will attract an attention to this interesting field. Particular applicationswere mentioned in the introduction, they will be addressed elsewhere. A special role among them is played byknot theory applications, also because this theory provides one of the most promising approaches to quantumprogramming [34, 35]. 10 cknowledgements Our original interest to Kerov functions was largely inspired by Anton Zabrodin many years ago. We areindebted to Anton Khoroshkin for fresh comments on existing folklore about the role and (in)significance ofdominance rule and other peculiarities of Macdonald theory. We highly appreciate our constant discussions onrelated subjects with Hidetoshi Awata, Hiroaki Kanno, Andrey Morozov, Alexei Sleptsov and Yegor Zenkevich.This work was supported by Russian Science Foundation grant No 18-71-10073. Appendix. Examples Kerov functions d Ker and Ker begin to deviate from each other only at level | R | = 6, which is beyond thisexample section, thus we do not distinguish between them here. At various smaller levels, we illustrate otherimportant phenomena. Like Ker ∅ = 1 at level | R | = 0, the polynomial at the first level is also universal, but already its norm is g -dependent:Ker ( g )[1] = p , || Ker ( g )[1] || = g , Ker [1] { p } = p ) = || Ker ( g )[1] || · Ker [1] (cid:26) − p k g k (cid:27) (60)In what follows, we omit the index ( g ) to simplify the formulas. At level 2, there are just two functions Ker [1 , = Schur [1 , = − p + p [2] = Schur [2] + K · Schur [1 , = (1 − K ) p + (1 + K ) p g p + g p g + g = 2 g g g + g · Schur [2] (cid:26) − p k g k (cid:27) (61)with the norms || Ker [1 , || = g + g and || Ker [2] || = g g g + g . Conditions (13) imply thatKer [2] { p } = || Ker [2] || · Ker ∨ [1 , {− p ∨ } ⇐⇒ g p + g p g + g = 2 g g g + g · p ∨ + p ∨ [1 , { p } = || Ker [1 , || · Ker ∨ [2] {− p ∨ } ⇐⇒ − p + p − g ∨ p ∨ + g ∨ p ∨ · g + g g ∨ + g ∨ (62)which can be easily solved: p ∨ = p g , p ∨ = p g , g ∨ = 1 g , g ∨ = 1 g (63)The product (10)Ker = Ker [2] + 2 g g + g · Ker [1 , = ⇒ N [2][1] , [1] ( g ) = 1 , N [1 , , [1] ( g ) = 2 g g + g = S S [2] (64)is related to the shape of the skew Kerov functions:Ker [2] / [1] = 2 g g + g · Ker [1] = N [2] ∨ [1] , [1] ( g − ) · Ker [1] , Ker [11] / [1] = Ker [1] = N [1 , ∨ [1] , [1] · Ker [1] (65)11 .3 Level 3 This is the first level where different g -dependent denominators emerge in the formulas, but they are stillrelated by the g -inversion:Ker [3] = 2 g g p + 3 g g p p + g g p g g + 3 g g + g g = pol( p, g )∆ ∨ = Schur [3] n p k g k o Schur [3] n g k o Ker [2 , = − g ( g + g ) p + ( g − g ) p p + ( g + g g ) p g + 3 g g + g = pol( p, g )∆ Ker [1 , , = Schur [1 , , = p − p p p g : ∆ ∨ { g } := 2 g g + 3 g g + g g = g g g · ∆ { g − } with ∆ { g } = 2 g + 3 g g + g = 6 Schur [3] { g , g , g } .The products and skew Kerov functions areKer [2] · Ker [1] = Ker [3] ++ g g (2 g +3 g g + g )( g + g )( g g +3 g g +2 g g ) · Ker [2 , Ker [1 , · Ker [1] = Ker [2 , + g ( g + g )2 g +3 g g + g · Ker [1 , , = ⇒ N [3][2] , [1] ( g ) = 1 N [2 , , [1] ( g ) = g g (2 g +3 g g + g )( g + g )( g g +3 g g +2 g g ) N [2 , , , [1] ( g ) = 1 N [1 , , , , [1] ( g ) = g ( g + g )2 g +3 g g + g = Schur [2] { g }· Schur [1] { g } Schur [3] { g } (67)Ker [3] / [2] = N [1 , , , , [1] ( g − ) · Ker [1] , Ker [3] / [1] = N [1 , , , , [1] ( g − ) · Ker [2] Ker [2 , / [2] = N [2 , , , [1] ( g − ) · Ker [1] , Ker [2 , / [1 , = N [2 , , [1] ( g − ) · Ker [1] = Ker [1] , Ker [2 , / [1] = N [2 , , , [1] ( g − ) · Ker [2] + N [2 , , [1] ( g − ) · Ker [1 , = Ker [2] + N [2 , , [1] ( g − ) · Ker [1 , Ker [1 , , / [1 , = N [3][2] , [1] ( g − ) · Ker [1] = Ker [1] , Ker [1 , , / [1] = N [3][2] , [1] ( g − ) · Ker [1 , = Ker [2] (68)Note also that ∆ · Ker [2 , = − g p · ∆ + g p · ∆ · Ker [2] + 2 g p · Ker [1 , (69) Expressions for various denominators ∆ { g } will be provided in the next subsection. Other quantities areKer [4] = 6 g g g · p + 8 g g g · p p + 3 g g g · p + 6 g g g g · p p + g g g · p ∆ ∨ = Schur [4] n p k g k o Schur [4] n g k o Ker [3 , = pol( p, g )2∆ ′∨ , Ker [2 , = pol( p, g )2∆ ′ , Ker [1 , , = pol( p, g )2∆ Ker [1 , , , = Schur [1 , , , { p k } (70)The simplest of the three polynomials in the numerators is in Ker [1 , , = g z }| { (2 g + 3 g g + g ) · (2 p − p ) + 2(2 g + g − g g − g ) · p p − (6 g + 4 g g + 3 g − g ) · p p + (2 g + 2 g g + g + g g ) · p but they are hardly useful, if presented in this form. Remarkably, this complicated expression is nothing but avery simple (48), which is a direct generalization of (69).12 .5 Examples of the emerging structure We can now list the emerging denominators and observe that they are actually the Schur functions with g k playing the role of the time variables:∆ = g + g = 2 Schur [2] { g , g } ∆ = 2 g + 3 g g + g = 6 Schur [3] { g , g , g } , ∆ ∨ = g g + 3 g g + 2 g g = 6 g g g · Schur [3] ( g − , g − , g − ) == 6 (cid:16) Schur [3] · Schur [2] + Schur [3] · Schur − Schur · Schur [1] (cid:17) { g } ∼∼ (cid:16) [32] + [311] − [221] (cid:17) = det Schur [3] [1] Schur [3] Schur [2] [2] Schur [1] ∆ = 6 g + 8 g g + 3 g + 6 g g + g = 24 Schur [4] { g k } ∆ ∨ = g g g + 6 g g g g + 8 g g g + 3 g g g + 6 g g g ∼ Schur [4] { g − k } ∆ ′ = 2 (cid:16) g g + g g + 6 g g g + 3 g g + 2 g g g + 4 g g + 2 g g + 3 g g + g g (cid:17) == 48 (cid:16) Schur [4] · Schur [3] + Schur [4] · Schur [2] · Schur [1] − Schur · Schur [1] (cid:17) { g } ∼∼ (cid:16) [43] + [421] − [331] (cid:17) = det Schur [4] [1] Schur [4] Schur [3] [3] Schur [2] ∆ ′∨ = ∆ ′ { g − k } ∼ (cid:16) − −− − [3311111] − [32222] + 4[322211] − [222221] (cid:17) (71)Explicitly written in (58) is the lifting to arbitrary r of just the two underlined terms from the last formula.Note that the degeneracy is lifted for r > 4, and they become three independent structures. The same is goingto happen to the other items, thus, in general, it is a 19-term expression.This form of ∆ r can explain why the Macdonald choice of g k is distinguished: it is an exact counterpart ofthe topological locus, which converts the Schur functions into the quantum dimensions g k = { A k }{ t k } . However,the factorization of other denominators is far from obvious. What if we further change the Schur functions of g k for the Macdonald ones and then to the Kerov functions?Since this Appendix is devoted to honest examples, which illustrate and underlie general theory rather thanfollow from it, we do not interpret the denominators ∆ in the terms of sec.7, instead we present them as theyare. Actually, in general terms (i.e. for all r ),second diagram ∆ r = ∆ [2 , r − ] = ∆ (2) r { g k } third diagram ∆ ′ r = ∆ [2 , , r − ] = ∆ (3) r { g k } fourth diagram ∆ ′′ r = ∆ [2 , , , r − ] = b ∆ [3 , r − ] = ∆ (4) r { g k } . . . third-from-the end diagram ∆ ′′∨ r = ∆ [ r − , = ∆ ′′ r { g − k } penultimate diagram ∆ ′∨ r = ∆ [ r − , = ∆ ′ r { g − k } last diagram ∆ ∨ r = ∆ [ r ] = ∆ r { g − k } (72)13ote that the lexicographically-last diagram is related by the g -inversion to the second diagram, not to the firstone, and so on: there is a shift-by-one in these formulas as compared to a more naive expectation from (13).In result, while the ”true” fourth diagram appears only at level 6, its counterpart is the third diagram from theend, and it is non-trivial already at level 4, which allows one to learn something about ∆ (4) from the exampleat this level. The structure which is revealed in this way is indeed true in general, see (58) in sec.7. 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