A commutative algebra on degenerate CP1 and Macdonald polynomials

Abstract

Abstract. We introduce a unital associative algebra A over degenerate CP 1 . We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii [FO]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors [S2]. It is found that the Ding-Iohara algebra [DI]

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…(1.1), the job of the R-matrix is to permute the components in the tensor product of representations of the algebra G. This is the property we will use in refined topological strings. The representations in question are going to be Fock modules [37][38][39] and their permutation exchanges the legs of the toric diagram corresponding to a DIM intertwiner [40,41]. The permutation of the legs performed by the R-matrix has a simple interpretation in terms of the corresponding conformal blocks of the q-Virasoro or qW N -algebras.…”

“…The generators, x + n , ψ ± n , x − n and their commutators form a lattice, which is sketched in figure 1. The exact definition of the DIM algebra can be found in [41,[63][64][65][66] (see also [67][68][69][70][71] for elliptic DIM algebra).…”

“…a bosonization of the DIM generators, which are expressed through exponentials of the free bosons (for concrete expressions see [37][38][39][40]). The second central charge of this representation is trivial, C 2 = 1, while the first one is given by C 1 = (t/q) 1/2 .…”

Macdonald operators at infinity

“…However, in the remarkable work of Shiraishi [18] the limits of certain linear combinations of all the coefficients were expressed in terms of the vertex operators associated with an infinite dimensional Heisenberg Lie algebra, see also [5,20]. In a more recent work [1] this result of [18] has been reformulated by using the well-known correspondence [7] between the vertex operators and the HallLittlewood symmetric functions, which are specializations of the Macdonald symmetric functions at q = 0.…”

“…In the present article we consider the limits at N → ∞ of linear combinations of all the coefficients of D N (u), different from those in [1,5,18,20]. Our linear combinations arise naturally from the theory of the double affine Hecke algebras, see for instance [17].…”