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Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure

Gary Greaves, Naihuan Jing, Haoran Zhu

Abstract

We study the $t$-Schur measure on partitions, defined by $ \mathbb{P}(λ)=Z^{-1}S_λ(x;t)s_λ(y) $, where $S_λ(x;t)$ denotes the $t$-Schur symmetric functions and $s_λ(y)$ the ordinary Schur functions, and $Z$ is the normalising constant. Using vertex operator calculus, we realise $S_λ(x;t)$ in the charged free-fermion Fock space, yielding a $t$-deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the $t$-Cauchy identities and $t$-Gessel identity. Building on this framework, we compute the correlation functions of the $t$-Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised $t$-Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at $t=0$, connect to the Schur-$Q$ theory at $t=-1$, and provide a probabilistic interpretation of a natural $t$-refinement of increasing subsequences via a generalised RSK correspondence.

Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure

Abstract

We study the -Schur measure on partitions, defined by , where denotes the -Schur symmetric functions and the ordinary Schur functions, and is the normalising constant. Using vertex operator calculus, we realise in the charged free-fermion Fock space, yielding a -deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the -Cauchy identities and -Gessel identity. Building on this framework, we compute the correlation functions of the -Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised -Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at , connect to the Schur- theory at , and provide a probabilistic interpretation of a natural -refinement of increasing subsequences via a generalised RSK correspondence.
Paper Structure (27 sections, 34 theorems, 173 equations)

This paper contains 27 sections, 34 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. A probabilistic model
  3. Symmetric functions and their $t$-analogue
  4. Generalised RSK over the alphabet $\mathcal{A}$
  5. The $t$-Schur measure and organisation of the paper
  6. Vertex operators realisations
  7. Vertex operators arising from $t$-Schur functions
  8. Realisations of the $t$-Schur function
  9. Generalised Cauchy identities and Gessel identity
  10. Infinite wedge representations
  11. Fermionic Fock space
  12. Free fermions
  13. Bosons and $t$-half vertex operators
  14. Correlation functions of $t$-Schur measure
  15. Coordinates on partitions
  16. ...and 12 more sections

Key Result

Theorem 1.1

The above procedure is a bijection between matrices $A=(a_{ij})\in\mathcal{A}_{m,n}$ and pairs $(S,U)$ consisting of a marked tableau $S$ and a semistandard tableau $U$ of the same shape $\lambda\in\mathbb Y$. Moreover, and $\mathrm{mark}(S)=\mathrm{mark}(A)$.

Theorems & Definitions (71)

  • Theorem 1.1: Mat1
  • Remark 1.2
  • Lemma 1.3: Mat1
  • Theorem 1.4: Mat1
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Proposition 2.1: J1
  • Remark 2.2
  • Proposition 2.3
  • ...and 61 more