[Paper Review] Generators of the Hecke algebra of $(S_{2n},B_n)$

Kürşat Aker, Mahir Bilen Can|arXiv (Cornell University)|Sep 27, 2010

Molecular spectroscopy and chirality10 references3 citations

TL;DR

This paper presents a set of ring generators for the Hecke algebra of the symmetric pair $(S_{2n}, B_n)$, where $B_n$ is the hyperoctahedral subgroup. It proves that the generators $H_i$, defined as sums of permutations with exactly $i$ cycles in an associated graph, generate the algebra, and confirms a conjecture by Sho Matsumoto on the surjectivity of a map from symmetric polynomials to the Hecke algebra via Jucys-Murphy elements and the idempotent $\varepsilon_{B_n}$.

ABSTRACT

A set of ring generators for the Hecke algebra of the Gel'fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$ is presented. Various corollaries are given. A conjecture of Sho Matsumoto is proven.

Motivation & Objective

• To identify a finite set of ring generators for the Hecke algebra $H_{\mathbb{Z}}(S_{2n}, B_n)$, a commutative subalgebra of the integral group ring of $S_{2n}$.
• To construct a universal ring $\mathcal{H}$ with a filtration whose associated graded ring is isomorphic to that of the Farahat-Higman universal ring $\mathcal{Z}$.
• To prove that the elements $H_i$, defined as sums of permutations with exactly $i$ cycles in a graph $\Gamma(w)$, generate $H_{\mathbb{Z}}(S_{2n}, B_n)$ as a ring.
• To confirm a conjecture by Sho Matsumoto stating that the map sending a symmetric polynomial $F$ in $n$ variables to $F(J_1, J_3, \dots, J_{2n-1}) \cdot \varepsilon_{B_n}$ is surjective.
• To establish an isomorphism between the universal rings $\mathcal{Z}$ and $\mathcal{H}$, linking representation theory to cohomology of Hilbert schemes and quiver varieties.

Proposed method

• Define the Hecke algebra $H_{\mathbb{Z}}(S_{2n}, B_n)$ as the ring of $B_n$-biinvariant functions on $S_{2n}$ under convolution.
• Introduce the graph $\Gamma(w)$ for each $w \in S_{2n}$, where vertices are the $2n$ elements and edges are defined by the action of the involution $t = (1\,2)(3\,4)\cdots(2n-1\,2n)$.
• Define $H_i$ as the sum of all $w \in S_{2n}$ such that $\Gamma(w)$ has exactly $i$ connected components (cycles).
• Construct a universal ring $\mathcal{H}$ via a filtration, analogous to Farahat and Higman's construction for $\mathcal{Z}$, and show that $\operatorname{gr}\mathcal{H} \cong \operatorname{gr}\mathcal{Z}$.
• Use the Jucys-Murphy elements $J_k = \sum_{j=1}^{k-1} (j\,k)$ and the idempotent $\varepsilon_{B_n} = \frac{1}{|B_n|} \sum_{b \in B_n} b$ to express $H_i$ as $e_{n-i}(J_1, J_3, \dots, J_{2n-1}) \cdot \varepsilon_{B_n}$.
• Prove that the structure constants of the product $K_\lambda K_{(r)}$ in the Hecke algebra are given by a formula involving $m_{r+|\rho|}(\lambda)$, $r!$, and $|K_\rho(n+1)|$, leading to the key coefficient formula in Theorem 4.4.

Experimental results

Research questions

• RQ1 What is a minimal set of ring generators for the Hecke algebra $H_{\mathbb{Z}}(S_{2n}, B_n)$?
• RQ2 How does the structure of the Hecke algebra $H_{\mathbb{Z}}(S_{2n}, B_n)$ relate to the universal ring $\mathcal{Z}$ constructed by Farahat and Higman?
• RQ3 Is the conjecture by Sho Matsumoto true, that the map $F \mapsto F(J_1, J_3, \dots, J_{2n-1}) \cdot \varepsilon_{B_n}$ is surjective onto $H_{\mathbb{Z}}(S_{2n}, B_n)$?
• RQ4 What is the precise formula for the structure constants of the product of basis elements in $H_{\mathbb{Z}}(S_{2n}, B_n)$?
• RQ5 Is there a canonical isomorphism between the universal ring $\mathcal{H}$ constructed here and the Farahat-Higman universal ring $\mathcal{Z}$?

Key findings

• The Hecke algebra $H_{\mathbb{Z}}(S_{2n}, B_n)$ is generated as a ring by the elements $H_i$ for $1 \leq i \leq n$, where $H_i$ is the sum of all $w \in S_{2n}$ such that the graph $\Gamma(w)$ has exactly $i$ cycles.
• The universal ring $\mathcal{H}$ constructed in this paper is isomorphic to the Farahat-Higman universal ring $\mathcal{Z}$, and both have associated graded rings isomorphic to the cohomology ring of the Hilbert scheme $\operatorname{Hilb}^n(\mathbb{C}^2)$.
• The elements $H_i$ satisfy the identity $H_i = e_{n-i}(J_1, J_3, \dots, J_{2n-1}) \cdot \varepsilon_{B_n}$, confirming a conjecture of Sho Matsumoto.
• The structure constants of the product $K_\lambda K_{(r)}$ in $H_{\mathbb{Z}}(S_{2n}, B_n)$ are given by $b_{\lambda\,(r)}^{(r+|\rho|)\cup\lambda-\rho} = \frac{(m_{r+|\rho|}(\lambda)+1)(r+|\rho|+1)r!}{\prod_{i \geq 0} m_i(\rho)!}$, where $\rho$ is a partition with $m_i(\rho)$ parts of size $i$.
• The associated graded ring of $\mathcal{H}$ is isomorphic to $\operatorname{gr}\mathcal{Z}$, and $\mathcal{H}$ is a free polynomial algebra over countably many indeterminates.
• The construction establishes a direct link between the representation theory of $S_{2n}$, the cohomology of Hilbert schemes, and Nakajima quiver varieties via the universal ring $\mathcal{H}$.