[Paper Review] Algebras of conjugacy classes in symmetric groups

Yury A. Neretin|arXiv (Cornell University)|Apr 19, 2016

Finite Group Theory Research10 references3 citations

TL;DR

This paper extends Ivanov and Kerov's stabilization phenomenon of structure constants in symmetric group conjugacy class algebras to a broader class of group pairs $G \supset K$, focusing on $G = S_n \times S_n$ and $K = S_n$ diagonally embedded. It constructs an infinite-dimensional algebra of conjugacy classes $G_\infty // K_\infty$, introduces a filtration, and shows its associated graded algebra forms a commutative semigroup algebra under a natural product, with a compatible Poisson bracket structure arising from commutators.

ABSTRACT

In 1999 V. Ivanov and S. Kerov observed that structure constants of algebras of conjugacy classes of symmetric groups $S_n$ admit a stabilization (in a non-obvious sense) as $n o \infty$. We extend their construction to a class of pairs of groups $G\supset K$ and algebras of conjugacy classes of $G$ with respect to $K$. In our basic example $G$ is a product of symmetric groups, $G=S_n imes S_n$, $K$ is the diagonal subgroup $S_n$.

Motivation & Objective

• To generalize Ivanov and Kerov's stabilization of structure constants in symmetric group conjugacy class algebras to a broader class of group pairs $G \supset K$.
• To study the infinite-dimensional limit of conjugacy class algebras $G_n // K_n$ for $G_n = S_n \times S_n$ and $K_n = S_n$ diagonally embedded.
• To construct a filtration on the algebra of conjugacy classes of the infinite product group $G_\infty = S_\infty \times S_\infty$ with respect to the diagonal subgroup $K_\infty = S_\infty$, and analyze its associated graded algebra.
• To identify the structure of the associated graded algebra as a semigroup algebra via a natural product on $\coprod_{j=0}^\infty G_j // S_j$, and to define a Poisson bracket on this graded algebra.

Proposed method

• Define a filtration $\mathcal{B}_k$ on the algebra $\mathcal{B}[G_\infty // K_\infty]$ based on the rank of partial bijections in the conjugacy classes.
• Construct the associated graded algebra $\mathop{\mathrm{gr}}\nolimits\mathcal{B}[G_\infty // K_\infty] = \bigoplus_{k=0}^\infty \mathcal{B}_k / \mathcal{B}_{k-1}$, where each component is spanned by basis elements indexed by $G_k // S_k$.
• Define a product $\diamond$ on the graded algebra using the partial bijection $\theta_{n,k}$, mapping $({\,{\overline{\overline{g}}\,}}, {\,{\overline{\overline{h}}\,}}) \mapsto \overline{\overline{\theta_{n,k}(g,V_n)\theta_{n,k}^{-1}(h,V_k)}}}$, which corresponds to the disjoint union of checkerboard triangulated surfaces.
• Define a Lie bracket on the graded algebra via the commutator $[V,W] = V*W - W*V$, inducing a Poisson structure on $\mathop{\mathrm{gr}}\nolimits\mathcal{B}[G_\infty // K_\infty]$.
• Show that when $X = \varnothing$, the graded algebra is commutative and the Poisson bracket is given by a sum over rank-1 partial bijections $\lambda$, with terms $B[\overline{\overline{g \circledast_\lambda h}}] - B[\overline{\overline{h \circledast_{\lambda^{-1}} g}}]$.
• Establish a canonical identification $\coprod_{j=0}^\infty G_j // S_j \simeq (G_\infty // K_\infty) \times \mathbb{Z}_+$, linking the semigroup structure to the infinite conjugacy class space.

Experimental results

Research questions

• RQ1 Can the stabilization of structure constants in symmetric group conjugacy class algebras be generalized beyond $S_n$ to other group pairs $G \supset K$?
• RQ2 What is the structure of the infinite-dimensional limit algebra $\mathcal{B}[G_\infty // K_\infty]$ for $G_\infty = S_\infty \times S_\infty$ and $K_\infty = S_\infty$?
• RQ3 Does the associated graded algebra of this limit algebra admit a natural semigroup structure, and if so, how is it realized?
• RQ4 Can a Poisson bracket be defined on the associated graded algebra, and how does it relate to the commutator in the original algebra?
• RQ5 How do the resulting algebraic structures (semigroup, Poisson algebra) compare to those in previous works on double cosets, particularly in terms of identifications of combinatorial objects?

Key findings

• The associated graded algebra $\mathop{\mathrm{gr}}\nolimits\mathcal{B}[G_\infty // K_\infty]$ is isomorphic to the semigroup algebra of the semigroup $\coprod_{j=0}^\infty G_j // S_j$ under the product $\bullet$ defined via partial bijections $\theta_{n,k}$.
• The product in the graded algebra is given explicitly by $B[\overline{\overline{g}}]\diamond B[\overline{\overline{h}}] = B[\overline{\overline{\theta_{n,k}(g,V_n)\theta_{n,k}^{-1}(h,V_k)}}]$, which corresponds to the disjoint union of checkerboard triangulated surfaces in the case $G_\infty = S_\infty \times S_\infty$.
• When $X = \varnothing$, the graded algebra is commutative, and the Poisson bracket is defined by $[B[\overline{\overline{g}}], B[\overline{\overline{h}}]]_{\mathrm{gr}} = \sum_{\lambda \in \mathrm{PB}(J_k, J_n), \mathop{\mathrm{rk}}\nolimits\lambda=1} \left( B[\overline{\overline{g \circledast_\lambda h}}] - B[\overline{\overline{h \circledast_{\lambda^{-1}} g}}] \right)$.
• The semigroup structure on $\coprod_{j=0}^\infty G_j // S_j$ is isomorphic to $G_\infty // K_\infty$ via a canonical identification with $(G_\infty // K_\infty) \times \mathbb{Z}_+$, linking the finite and infinite levels.
• The construction generalizes the known stabilization of structure constants in symmetric group algebras to a broader class of group pairs, with the infinite limit algebra capturing combinatorial data through a natural filtration and graded structure.
• The Poisson bracket on the graded algebra arises naturally from the commutator in the original algebra, and its explicit formula involves only rank-1 partial bijections, highlighting a deep combinatorial structure.