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Mathematics > Representation Theory

Title: Cyclic Sieving and Cluster Duality for Grassmannian

Abstract: We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.
Comments: 40 pages
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Combinatorics (math.CO)
Cite as: arXiv:1803.06901 [math.RT]
  (or arXiv:1803.06901v1 [math.RT] for this version)

Submission history

From: Linhui Shen [view email]
[v1] Mon, 19 Mar 2018 13:14:43 GMT (55kb)