We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov’s plabic graphs. This description is a natural generalization of the description given by (Scott 2006) for the Grassmannian and has been believed by experts essentially since (Scott 2006), though the statement was not formally written down until (Muller–Speyer 2016). To prove this conjecture we use a result of (Leclerc 2016), who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety admit cluster structures. We also adapt a construction of (Karpman 2016) to build cluster seeds associated to reduced expressions. Further, we explicitly describe cluster structures in skew Schubert varieties using plabic graphs whose boundary vertices need not be labeled in cyclic order.
|Journal||Seminaire Lotharingien de Combinatoire|
|State||Published - 2019|
Bibliographical noteFunding Information:
We are grateful to Bernard Leclerc for numerous helpful discussions. K.S. and M.S.B. acknowledge support from National Science Foundation Postdoctoral Fellowship MSPRF-1502881 and NSF Graduate Research Fellowship No. DGE-1752814, respectively. L. W. was partially supported by the NSF grant DMS-1600447. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
© 2019,Seminaire Lotharingien de Combinatoire. All Rights Reserved.
- cluster algebra
- plabic graph
- Schubert variety
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics