A remark on Leclerc's Frobenius categories

Graded Frobenius cluster categories

Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in p…

相对簇范畴与Higgs范畴

Relative cluster categories and Higgs categories

本文将簇范畴构造推广至相对语境，证明了由左(n+1)-Calabi--Yau态射产生的Higgs范畴中存在稳定n-Calabi--Yau的n-cluster tilting对象，并应用于相对Ginzburg dg代数与高维Auslander代数。

具有无限维态射空间的相对丛范畴与希格斯范畴

Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

该研究在相对雅可比无限设定下构建了具有外三角结构的希格斯范畴与相对丛范畴，涵盖格拉斯曼流形丛范畴情形，还构建了改进的丛特征标，附录中完成了准丛态射的范畴化并被用于证明相关猜想。

A remark on Frobenius characters for set representations of symmetric groups

For any set representation (permutation representation) of the symmetric group $S_n$, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

On cluster algebras with coefficients and 2-Calabi-Yau categories

Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not al…

关于微分拓扑弗罗贝尼乌斯互易性的remarks

Remarks on Diffeological Frobenius Reciprocity

本文研究了微分拓扑中弗罗贝尼乌斯互易性的性质，证明了相关猜想并给出了约简形式存在的新条件。