Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra ℂRn and the subalgebra ℂIk of the partition algebra ℂAk(n) acting on (ℂn)⊗k. In this paper, we consider a subalgebra ℂIk+1 2 of ℂIk+1 such that there is Schur–Weyl duality between the actions of ℂRn−1 and ℂIk+1 2 on (ℂn)⊗k. This paper studies the representation theory of partition algebras ℂIk and ℂIk+1 2 for rook monoids inductively by considering the multiplicity free tower ℂI1 ⊂ ℂI3 2 ⊂ ℂI2 ⊂⋯ ⊂ ℂIk ⊂ ℂIk+1 2 ⊂⋯. Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of ℂIk and ℂIk+1 2. Also, we describe the Jucys–Murphy elements of ℂRn which play a central role in the demonstration of the actions of Jucys–Murphy elements of ℂIk and ℂIk+1 2.

中文翻译：

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rook monoid 分区代数的 Jucys-Murphy 元素

Kudryavtseva 和 Mazorcuk 在 rook monoid 代数之间展示了 Schur-Weyl 对偶性ℂRn和子代数ℂ一世ķ分区代数的ℂ一种ķ(n)作用于(ℂn)⊗ķ. 在本文中，我们考虑一个子代数ℂ一世ķ+1 2的ℂ一世ķ+1使得在行动之间存在 Schur-Weyl 对偶ℂRn-1和ℂ一世ķ+1 2在(ℂn)⊗ķ. 本文研究了分区代数的表示论ℂ一世ķ和ℂ一世ķ+1 2通过考虑多重性自由塔来归纳地计算车子幺半群 ℂ一世1 ⊂ ℂ一世3 2 ⊂ ℂ一世2 ⊂⋯ ⊂ ℂ一世ķ ⊂ ℂ一世ķ+1 2 ⊂⋯. 此外，通过描述 Jucys-Murphy 元素及其在由上述多重自由塔确定的规范 Gelfand-Tsetlin 碱基上的不可约表示ℂ一世ķ和ℂ一世ķ+1 2. 此外，我们描述了 Jucys-Murphy 元素ℂRn在演示 Jucys-Murphy 元素的动作中起着核心作用ℂ一世ķ和ℂ一世ķ+1 2.