Tanabe代数的上同调
Cohomology of Tanabe algebras
AI总结：
本文证明Tanabe代数的（上）同调与对称群的（上）同调同构，且与参数和索引奇偶性无关，首次展示此类结果。
摘要：
在本文中，我们研究了Tanabe代数的（上）同调，这些代数是分区代数的子代数，表现出与某些复反射群的Schur-Weyl对偶性。分区代数的同调已被Boyd-Hepworth-Patzt证明与对称群的同调相关，他们获得的结果依赖于一个参数。在所有已知的结果中，图代数的同调依赖于两种情况之一：地面环中参数的可逆性或正整数索引的奇偶性。我们证明Tanabe代数的（上）同调同构于对称群的（上）同调，并且与参数和索引的奇偶性无关。据我们所知，这是此类结果的第一个例子。在过程中，我们还将研究均匀块置换代数和完全传播分区代数的（上）同调，并收集分区代数和Jones环形代数已知同调结果的上同调类比。
英文摘要：
In this paper we study the (co)homology of Tanabe algebras, which are a family of subalgebras of the partition algebras exhibiting a Schur-Weyl duality with certain complex reflection groups. The homology of the partition algebras has been shown to be related to the homology of the symmetric groups by Boyd-Hepworth-Patzt and the results they obtain depend on a parameter. In all known results, the homology of a diagram algebra is dependent on one of two things: the invertibility of a parameter in the ground ring or the parity of the positive integer indexing the number of pairs of vertices. We show that the (co)homology of Tanabe algebras is isomorphic to the (co)homology of the symmetric groups and that this is independent of both the parameter and the parity of the index. To the best of our knowledge, this is the first example of a result of this sort. Along the way we will also study the (co)homology of the uniform block permutation algebra and the totally propagating partition algebra as well collecting cohomological analogues of known results for the homology of partition algebras and Jones annular algebras.
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