More Like this

1. Generalized cluster structures related to the Drinfeld double of 𝑮𝑳_𝒏

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (October 2021, Journal of the London Mathematical Society)

   We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿𝑛 constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties. 

   more » « less
2. Periodic Staircase Matrices and Generalized Cluster Structures

   https://doi.org/10.1093/imrn/rnaa148  

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (July 2022, International mathematics research notices)

   As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot– Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators. 

   more » « less
3. Generalized cluster trees and singular measures

   https://doi.org/10.1214/18-AOS1744  

   Chen, Yen-Chi (August 2019, The Annals of Statistics)
4. Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras

   https://doi.org/10.1007/s00026-022-00623-1  

   Cheung, Man-Wai; Kelley, Elizabeth; Musiker, Gregg (September 2023, Annals of Combinatorics)
5. Integral quantum cluster structures

   https://doi.org/10.1215/00127094-2020-0061  

   Goodearl, K. R.; Yakimov, M. T. (April 2021, Duke Mathematical Journal)

   null (Ed.)