[Paper Review] A description of characters on the infinite wreath product

Artem Dudko, N. I. Nessonov|ArXiv.org|Oct 27, 2005

Random Matrices and Applications9 references3 citations

TL;DR

This paper provides a complete classification of indecomposable characters on the infinite wreath product $\Gamma\wr\mathfrak{S}_{\infty}$ using a generalization of Okounkov's method for $\mathfrak{S}_{\infty}$-representations. It shows that such characters are fully determined by finite characters of $\Gamma$, with the classification relying on spectral projections and asymptotic operators linked to the Tomita-Takesaki modular operator in type $III$ representations. The key result is a structural description of $II_1$-factor representations in terms of $\Gamma$-characters and invariant projections.

ABSTRACT

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $Γ$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $Γ^\infty$ by automorphisms, so one can form a semidirect product $Γ^\infty times \mathfrak{S}_\infty$, known as the {\it wreath} product $Γ\wr\mathfrak{S}_\infty$ of $Γ$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $II_1-$factor-representations of $Γ\wr\mathfrak{S}_\infty$ in terms of finite characters of $Γ$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty imes\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $II_1$, where the {\it modular operator} of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.

Motivation & Objective

• To classify all indecomposable characters on the infinite wreath product $\Gamma\wr\mathfrak{S}_{\infty}$, which are associated with factor-representations of type $II_1$.
• To extend Okounkov’s classification method for $\mathfrak{S}_{\infty}\times\mathfrak{S}_{\infty}$-representations to the wreath product setting.
• To analyze the connection between asymptotic operators in characters-theory of $\mathfrak{S}_{\infty}$ and the modular operator in type $III$ representations.
• To establish a correspondence between spectral projections of certain operators and irreducible components of $\Gamma$-representations in the GNS construction.

Proposed method

• The authors use the semigroup method of Olshanski and extend Okounkov’s approach to analyze representations of $\Gamma\wr\mathfrak{S}_{\infty}$ via asymptotic operators and double coset structures.
• They define the wreath product $\Gamma\wr\mathfrak{S}_{\infty} = \Gamma^\infty_e \rtimes \mathfrak{S}_{\infty}$ with the inductive limit topology on $\Gamma^\infty_e$ and discrete topology on $\mathfrak{S}_{\infty}$.
• The key technical tool is the spectral decomposition of the operator $\mathcal{O}_k$ defined via the GNS construction, leading to projections $E^{(k)}(r)$ associated with eigenvalues $r \in \{\alpha_i, \beta_i\} \subset (0,1)$.
• They use the projection $e_m(r)$ constructed from $E^{(k)}(r)$ and the action of $\mathfrak{S}_m$ to define a character $\phi_r(s)$ on $\mathfrak{S}_m$, which is shown to be indecomposable.
• The proof relies on the identity $\sum_{s \in \mathfrak{S}_m} \text{sgn}(s) t^{|\mathbb{P}_m(s)|} = t(t-1)\cdots(t-m+1)$, used to derive a contradiction unless $\nu(r) \in \mathbb{Z}$.
• The classification is completed by showing that for $r \neq 0$, the $W^*$-algebra $E_k(r)\mathfrak{A}_k$ is finite-dimensional and that the $\Gamma$-representation on $E_k(r)\mathcal{H}_\phi$ decomposes into irreducible components corresponding to the characters of $\Gamma$.

Experimental results

Research questions

• RQ1 How can the indecomposable characters on $\Gamma\wr\mathfrak{S}_{\infty}$ be fully classified in terms of finite characters of $\Gamma$?
• RQ2 What is the role of asymptotic operators in the characters-theory of $\mathfrak{S}_{\infty}$ and how are they related to the modular operator in type $III$ representations?
• RQ3 Under what conditions does a positive definite function on $\Gamma\wr\mathfrak{S}_{\infty}$ satisfying $\varphi(sg) = \varphi(gs)$ for all $g \in \Gamma\wr\mathfrak{S}_{\infty}$, $s \in \mathfrak{S}_{\infty}$, give rise to a factor-representation?
• RQ4 How do spectral projections of the operator $\mathcal{O}_k$ relate to the irreducible components of the $\Gamma$-action in the GNS representation?

Key findings

• The paper establishes a complete classification of indecomposable characters on $\Gamma\wr\mathfrak{S}_{\infty}$, showing they are parametrized by finite characters of $\Gamma$ and spectral data from the asymptotic operators.
• For each $r \in \{\alpha_i, \beta_i\} \subset (0,1)$, the spectral projection $E^{(k)}(r)$ of $\mathcal{O}_k$ gives rise to a finite-dimensional $W^*$-algebra $E_k(r)\mathfrak{A}_k$, ensuring the representation is of type $II_1$.
• The value $\nu(r)$ associated with each eigenvalue $r$ must be an integer, as shown by contradiction using the identity $\sum_{s \in \mathfrak{S}_m} \text{sgn}(s) t^{|\mathbb{P}_m(s)|} = t(t-1)\cdots(t-m+1)$.
• The irreducible components of the $\Gamma$-representation on $E_k(r)\mathcal{H}_\phi$ are precisely the $\varrho^r$ representations described in Theorem 7.
• The modular operator in type $III$ representations is naturally expressed via the asymptotic operators, confirming a deep link between characters-theory and Tomita-Takesaki theory.
• The construction of the GNS representation for each character is explicitly realized in Section 2, showing the correspondence between the spectral data and the factor-representation structure.