Free field approach to the Macdonald process
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Free field approach to the Macdonald process Shinji Koshida1 Received: 1 November 2019 / Accepted: 28 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Macdonald process is a stochastic process on the collection of partitions that is a (q, t)-deformed generalization of the Schur process. In this paper, we approach the Macdonald process identifying the space of symmetric functions with a Fock representation of a Heisenberg algebra. By using the free field realization of operators diagonalized by the Macdonald symmetric functions, we propose a method of computing several correlation functions with respect to the Macdonald process. It is well known that expectation value of several observables for the Macdonald process admits determinantal expression. We find that this determinantal structure is apparent in free field realization of the corresponding operators and, furthermore, it has a natural interpretation in the language of free fermions at the Schur limit. We also propose a generalized Macdonald measure motivated by recent studies on generalized Macdonald functions whose existence relies on the Hopf algebra structure of the Ding–Iohara–Miki algebra. Keywords Macdonald process · Macdonald symmetric function · Ding–Iohara–Miki algebra · Generalized Macdonald functions · Generalized Macdonald measure Mathematics Subject Classification 05E05 · 33D52 · 16T05
1 Introduction 1.1 Backgrounds Let Yn be the collection of partitions of n ∈ Z≥1 , and set Y := ∞ n=0 Yn , where Y0 = {∅}. The Macdonald measure MMq,t is a probability measure on Y defined by [8,14]
B 1
Shinji Koshida [email protected] Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo, Tokyo 112-8551, Japan
123
Journal of Algebraic Combinatorics
MMq,t (λ) =
1 Pλ (X ; q, t)Q λ (Y ; q, t), λ ∈ Y. (X , Y ; q, t)
Here, Pλ (X ; q, t) is the Macdonald symmetric function of X = (x1 , x2 , . . . ) for a partition λ and Q λ (Y ; q, t) is its dual symmetric function of Y = (y1 , y2 , . . . ) (see Sect. 2 for definition). From the Cauchy-type identity, the normalization factor is computed as (X , Y ; q, t) =
Pλ (X ; q, t)Q λ (Y ; q, t) =
λ∈Y
(t xi y j ; q)∞ , (xi y j ; q)∞
i, j≥1
n where (a; q)∞ = ∞ n=0 (1 − aq ). In the following, we suppress the parameters q and t if there is no ambiguity, for instance, by writing Pλ (X ) = Pλ (X ; q, t). Precisely speaking, to obtain a genuine probability measure, we have to adopt a nonnegative specialization of the Macdonald symmetric functions whose classification was conjectured in [38] and recently proved in [41]. As a generalization of the Macdonald measure, the N -step Macdonald process N on Y N defined so that the probability [8,14] for N ≥ 1 is a probability measure MPq,t (1) (N ) N for a sequence (λ , . . . , λ ) ∈ Y of partitions is given by N MPq,t (λ(1) , . . . , λ(N ) )
:=
Pλ(1) (X (1) )λ(1) ,λ(2) (Y (1) , X (2) ) · · · λ(N −1) ,λ(N ) (Y (N −1) , X (N ) )Q λ(N ) (Y (N ) ) . (i) ( j) ) 1≤i≤ j≤N (X , Y (
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