在 \cite{RS} 中，Rosso-Savage 提出了一類名叫 quantum affine wreath algebras 的代數，這類代數可視為是A型仿射Hecke代數的推廣。他們表明，這類代數的中心總是由對稱多項式給出。我們將這一結果擴展到 Lai-Nakano和Xiang在 \cite{LNX} 中的量子花環積上。我們提供了這些量子花環積的中心是某些代數中的對稱多項式的充分條件。我們進一步表明，中心可以包含不僅僅是對稱多項式。具體來說，我們通過Hu代數（即 $S_m \wr S_2$ 的 Hecke代數）來提供一個反例。特別地，我們通過在 Heisenberg categorifications中出現的 symmetric intertwiners 來描述Hu代數的中心。此外，我們使用 weak Bruhat orders 來研究由 symmetric intertwiners 所組成的子空間，並證明其維度等於正整數$m$的分割數。

In \cite{RS}, Rosso-Savage proved a center theorem for the quantum affine wreath algebras, which generalizes the notion of the affine Hecke algebras of type A. They show that the centers are always given by the symmetric polynomials in symmetric Frobenius algebras and the Laurent polynomials. We extend this result to the quantum wreath products introduced recently by Lai-Nakano and Xiang in \cite{LNX}. We provide sufficient conditions for these quantum wreath products whose centers are symmetric polynomials in certain algebras. We further show that the center can consist of more than just symmetric polynomials. To be precise, we provide a counterexample via the Hu algebra, i.e., the Hecke algebra for the wreath product $S_m \wr S_2$. In particular, we describe the center of the Hu algebra via symmetric intertwiners which appear in Heisenberg categorifications. Furthermore, we investigate its subspace of symmetric intertwiners using weak Bruhat orders and show that its dimension equals the number of partitions of m.

1 Introduction
1.1 Motivation and our work . . . . . . . . . . . . . . . . . . . . 1
1.2 The main theorem . . . . . . . . . .. . . . . . . . . . . . . . 3

2 Preliminaries
2.1 The Hecke algebras of type A. . . . . . . . . . . . . . . . . . 6
2.2 The affine Hecke algebras of type A . . . . . . . . . . . . . . 8
2.3 The degenerate affine Hecke algebras of type A .. . . . . . . . 9
2.4 The Hu algebras . . . . . . . . . . . . . . . . . . . . . . . 11
3 The centers of generalization of Rosso-Savage’s center theorem
3.1 A motivation of Rosso-Savage’s quantum affine wreath algebra. 12
3.2 The centers of generalization of Rosso-Savage’s center theorem 14

4 The Hu algebras
4.1 The Hu algebras . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 The centers of the Hu algebras . . . . . . . . . . . . . . . 17

A Appendix
A.1 Geck-Rouquier basis . . . . . . . . . . . . . . . . . . . . . 27
A.2 Jucys-Murphy elements . . . . . . . . . . . . . . . . . . . . 28

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