On Factorization of Generalized Macdonald Polynomials

A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to {\it generalized} Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a codimension-one slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.

Generalized Macdonald polynomials (GMP) [1] play a constantly increasing role in modern studies of 6d version [2,3] of AGT relations [4,5] and spectral dualities [6]. At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [7]. Even the simplest questions about them are yet unanswered. In this letter we address one of them -what happens to the hook formulas for classical, quantum and Macdonald dimensions at the level of GMP. We find that they survive, but only partly -on a one-dimensional line in the space of time-variables. Lifting to the entire 2-dimensional topological locus remains to be found.
We begin by reminding that the Schur functions χ R {p}, depend on representation (Young diagram) R and on infinitely many time-variables p k (actually, particular χ R depends only on p k with k ≤ |R| = # boxes in R). They get nicely factorized on peculiar two-dimensional topological locus 6 -? a l a l Coarm a and coleg l are the ordinary coordinates of the box in the diagram. To keep notation continuous throughout the text, in (1) we call the relevant parameter t, not q, from the very beginning.
It is often convenient to ignore the simple overall coefficient and substitute the product formulas like (2) by a polynomial expression for plethystic logarithm ∈R t l ( ) where we use the definition for A = t 0 and t = t 1 . Plethystic exponential S • (f ) is the character of the symmetric algebra of f as a representation of C * t0 × ... × C * tn . By factorization of S • (f ) we actually mean that its plethystic logarithm f is a polynomial or, more generally, a rational function, with integer coefficients. The celebrated Gopakumar-Ooguri-Vafa hypothesis [8] is that such are not only the quantum dimensions, but also the HOMFLY polynomials of arbitrary knots (in this context the quantum dimensions are associated with the unknots).
Since there is a double product/sum in (2) and (3), it is natural to consider their refinement, where a second tparameter is introduced, usually called q (for knots this means going from HOMFLY to super-and hyper-polynomials [9]). The refined version of Schur is Macdonald polynomial [10] M q,t R {p}, which -unlike Schur functions -explicitly depends on q and t. If t is the same as in (1), then (2) lifts to We introduced a special notation for the character of the Young diagram and which fully describe the plethystic logarithm of Macdonald dimension M * R . Additional star in the label of µ and ν emphasizes that they do not depend on A.
The quantities χ * R and M * R are deformations of dimensions of representation R of SL N -algebras, which factorize due to Weyl formulas, and are therefore called quantum and Macdonald dimensions. In the former case they can be considered as graded dimension of representation R of U t (SL N ), while in the latter case there is still no commonly-accepted group-theory interpretation. In the absence of such interpretation Macdonald polynomials are defined not as characters, but by other less straightforward methods -of which generalized to GMP is currently only the definition as eigenvectors of Calogero-like Hamiltonians. Factorization in these terms is not straightforward and is in fact a separation-of-variables phenomenon, more-or-less equivalent to integrability of the associated theory. Its actual derivation on these lines is usually quite tedious. However, the very fact that factorization occurs can be observed experimentally, far before the proofs. derivations and real understanding. Our goal in this letter is to search for such evidence in the case of GMP.
GMP depends on set of Young diagrams and a set of time variables -in what follows we consider the simplest non-trivial case, when there are two: two Young diagrams and two sets of times. Then just one more deformation parameter adds to t and q, we call it Q. Thus the GMP in question will be denoted by M q,t,Q A,B {p,p} and, in full analogy with the Schur functions [11], they are eigenfunctions of the quantum cut-and-join operator, which is the Hamiltonian ∆ DIM (E(z)) of the DIM algebra (see [2,3] for details of the definition): Somewhat remarkably, just a single Hamiltonian is needed to describe the whole set of GMP -no higher Hamiltonians are needed, because all its eigenvalues are non-degenerate. As to label 1 in H 1 , it refers to the first coproduct in DIM, higher coproducts provide Hamiltonians for GMP,depending on more time variables.
Explicitly, the simplest GMP in a "natural" normalization -which will appear consistent with the factorization property -are: When all p k = 0, GMP become just ordinary Macdonald polynomials ofp k : and thus factorize at the topological locus (1) ofp, i.e. at However, forp k = 0, they remain quite complicated functions of p k : [1,1] (13) which do not look much simpler than the general expressions forp k = 0. Already from these examples it is clear that GMP are non-trivial functions of Q. Moreover, this complexity can seem to persist in restriction to topological locus. However, this is not quite the case if one looks at the right quantities -our claim is that at A = 0 andp = 0 plethystic logarithm is just linear in Q with a further factorized coefficient: see also (32) below for a more explicit expression. Here * * denotes the special locus and Υ is made from the dual characterν * * (q, t) = ν * * (q −1 , t −1 ): Note that the switch p k ↔p k is a non-trivial operation for GMP, described by the action of the DIM-algebra R-matrix [12]. This implies an amusing interplay between R-matrix structure and the two very different branches of factorization locus (12) and (14).
We checked the factorization conjecture (14) up to level five, i.e. for |Y 1 | + |Y 2 | ≤ 5. Here are particular examples: are just the ordinary Macdonald polynomials, factorized according to (5), and they are omitted from the table.
Despite (15) is a relatively small slice of what one could expect from refinement of the topological locus, the property (14) looks quite spectacular and mysterious. Its understanding can provide new insights about both the GMP and integrality conjectures. GMP are clearly simpler and closer to conventional group theory than knot (super)polynomials -and (14) is naturally more structured than one can expect for generic knots, however, its continuation to A = 0 can already be of more general type.

Factorization does not survive neither the deformation to non-vanishingp
while at A = 0 we return to a nicely factorized As an A-series, plethystic logarithm of the ratio M * [1], [1] /M * * [1], [1] is: The first term at the r.h.s. resembles the Q-linear term in (14). An additional surprise is that the second term also factorizes, and really bad things happen only in the order A 3 .
Along with factorization, Schur functions satisfy the Cauchy identity: Already in the simplest case of Schur polynomials (where only single-line Young diagrams R = [n], i.e. pure symmetric representations, contribute), combination of this identity and factorization provide a remarkable product formula, identifying the t-exponent and Pochhammer symbol: This can be also considered as a formula for the Euler characteristic of the structure sheaf of Hilb (C, n). In Miwa coordinates p k = i x k i Cauchy identity (20) can be considered as following from the decomposition If we restrict both p i andp i to the topological locus, For Macdonald polynomials it deserves taking the proper form of the Cauchy identity: It is related to a more complicated but better known version through the transposition identity [13,1] M q,t It actually simplifies greatly at the topological locus: This formula can be seen as equivariant Euler characteristic of a certain tautological bundle over the Hilbert scheme.
Refs. [14] provide factorization formulas for the equivariant Euler characteristic of the sheaf of differential forms: In the case A = B = 0 and M = 0 both (28) and (29) represent the equivariant Euler characteristic of the structure sheaf, and the formulas become identical. For GMP the Cauchy identity is [1] Y1,Y2 and our factorization conjecture (14) implies the following analogue of (21): and Eq.(31) is a new identity, which is a generalization of (21) with 2 additional parameters. This formula expresses the Euler characteristic of a certain tautological bundle on the moduli space of framed sheaves on P 2 . The left hand side is the sum of localization contribution of the fixed points (which are parameterized by pairs of Young diagrams) due to the K-theoretic Lefschetz fixed point formula.
To conclude, we investigated the possibility that generalized Macdonald polynomials satisfy some kind of a hook factorization formula on some kind of a topological locus -and discovered that this is indeed the case. The main observation (conjecture) is a rather beautiful (14), where Q enters only linearly -like A in (3), and the coefficient in front of it is further decomposes into Y 1 -and Y 2 -dependent factors. This result is twice surprising: it is remarkable that factorization at all persists for GMP, but, if it does, why only on the restricted one-parametric subspace (15) and not on the naive three-parametric extension of (1), Hopefully, this small discovery will cause interest to the problem and this will help to clarify the structure of Weyl formulas for DIM and double Hekke algebras, which stand behind the generalized Macdonald polynomials and their properties. The first step in this direction is already made in [15]: as one could anticipate from (12), whenp is non-vanishing, the factorization locus increases, to but still remains of codimension one w.r.t. (33). Despite the sign difference with (15), which makes the loci, big and small, not even intersecting, restrictions of generalized Macdonalds are nearly the same: Proportionality factor is a power of t. The next immediate questions concern extension to GMP, depending on many time variables (eigenfunctions of the higher coproducts of DIM), and extension to more general representations of DIM, labeled by 3d (plane) partitions.