In this paper we show that the center of the partition algebra $\mathcal{A}_{2k}(\delta)$, in the semisimple case, is given by the subalgebra of supersymmetric polynomials in the normalised Jucys-Murphy elements. For the non-semisimple case, such a subalgebra is shown to be central, and in particular it is large enough to recognise the block structure of $\mathcal{A}_{2k}(\delta)$. This allows one to give an alternative description for when two simple $\mathcal{A}_{2k}(\delta)$-modules belong to the same block.

中文翻译：

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划分代数的中心

在本文中，我们证明了划分代数 $\mathcal{A}_{2k}(\delta)$ 的中心，在半简单情况下，由归一化 Jucys-Murphy 元素中的超对称多项式的子代数给出。对于非半简单的情况，这样的子代数被证明是中心的，特别是它大到足以识别 $\mathcal{A}_{2k}(\delta)$ 的块结构。这允许当两个简单的 $\mathcal{A}_{2k}(\delta)$-modules 属于同一个块时给出替代描述。