[Paper Review] Matrix Integrals, Symmetric Functions theory and matrix integrals

A. Yu. Orlov|ArXiv.org|Jul 17, 2002

Advanced Topics in Algebra22 references4 citations

TL;DR

This paper establishes a deep connection between symmetric functions, soliton theory, and random matrix models by identifying scalar products in symmetric function theory with vacuum expectations in fermionic Fock space, leading to integral representations of tau-functions of hypergeometric type. The key contribution is showing that such scalar products correspond to matrix integrals—particularly in normal and multi-matrix models—thereby unifying combinatorial, integrable systems, and random matrix perspectives.

ABSTRACT

We consider certain scalar product of symmetric functions which is parameterized by a function $r$ and an integer $n$. One the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of this product with the help of multi-integrals. This gives links between a theory of symmetric functions, soliton theory and models of random matrices (such as a model of normal matrices).

Motivation & Objective

• To unify three major mathematical frameworks: symmetric functions, soliton theory (KP and Toda hierarchies), and random matrix models.
• To establish a fermionic representation of scalar products in the ring of symmetric functions parameterized by a function $ r $ and integer $ n $.
• To derive integral representations of these scalar products using matrix integrals, particularly in the context of normal, Hermitian, and two-matrix models.
• To demonstrate that the scalar product of two tau-functions of hypergeometric type is itself a tau-function, extending integrability to bilinear structures.
• To generalize results to Gelfand-Graev hypergeometric functions and multi-matrix integrals, linking them to lattice-based tau-functions.

Proposed method

• Utilizes fermionic Fock space formalism with vacuum expectation values to represent scalar products of symmetric functions, particularly Schur functions.
• Employs Hirota-Miwa variables $ \mathbf{x}, \mathbf{y} $ to express tau-functions as double series in Schur functions, connecting to power sums and KP/Toda hierarchies.
• Derives integral representations of scalar products via matrix integrals: normal matrix model, Hermitian one-matrix model, and two-matrix model.
• Applies bosonization rules and vertex operator techniques to relate fermionic fields to bosonic tau-functions in the Sato Grassmannian framework.
• Constructs multi-matrix integrals and Gelfand-Graev hypergeometric series using lattice-based vector $ \upsilon $ and generalized parameters $ a^{(i)}, b^{(i)} $, linking to tau-function identities.
• Uses determinant and bilinear identities (e.g., Hirota-Miwa equations) to verify consistency and derive closed-form expressions for the integral representations.

Experimental results

Research questions

• RQ1 How can scalar products in the ring of symmetric functions be represented both via fermionic vacuum expectations and via matrix integrals?
• RQ2 What is the precise connection between tau-functions of hypergeometric type and matrix models such as the normal matrix model or two-matrix model?
• RQ3 Can the scalar product of two such tau-functions also be a tau-function, and if so, under what conditions?
• RQ4 How do Gelfand-Graev hypergeometric functions arise from multi-matrix integrals and lattice-based fermionic constructions?
• RQ5 What role do $ \Psi DO $ on a circle and additional symmetries play in the structure of these integrals and tau-functions?

Key findings

• The scalar product of Schur functions in the standard inner product is realized as a vacuum expectation value in the fermionic Fock space, providing a physical interpretation.
• A deformation of the scalar product leads to hypergeometric-type series, which are shown to be equivalent to certain matrix integrals in the normal matrix model.
• The two-matrix model and Hermitian one-matrix model yield integral representations that match the fermionic scalar product, confirming the link between random matrix ensembles and symmetric function theory.
• The scalar product of two $ \tau $-functions of hypergeometric type is itself a $ \tau $-function, suggesting a closed algebraic structure under bilinear operations.
• Gelfand-Graev hypergeometric series are constructed as multi-matrix integrals with specific lattice vectors $ \upsilon $, and their generating functions are shown to be proportional to $ \tau $-functions via scaling factors $ c_i(a,b) $.
• Explicit integral representations are derived using Hirota-Miwa variables, with $ z $-dependent factors $ g_i(\mathbf{z}) $ and normalization constants $ c_i^{-1}(a,b) $, linking the series to tau-functions through multiplicative factors.