2024/04/22 by Bernt Tore Jensen, Alastair King, Jensen, Bernt Tore +3
Mathematics · #FOS: Mathematics #Mathematics and Applications #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.2404.14572
Minor changes. 85 pages
openalex publication_date 2024/04/22 · openalex created_date 2025/10/10 · arxiv created 2026/07/30 · arxiv updated 2026/07/31 · openalex updated_date 2026/08/02
The homogeneous coordinate ring ℂ[Gr(k,n)] of the Grassmannian is a cluster algebra, with an additive categorification CMC. Thus every M\inCMC has a cluster character ΨM∈ℂ[Gr(k,n)]. For any cluster tilting object T, with A=End(T)op, we define two new cluster characters, a generalised partition function PTM∈ℂ[K(CMA)], whose leading exponent is g-vector/index of M, and a generalised flow polynomial FTM∈ℂ[K(fdA)], whose leading exponent is \boldsymbolκ(T,M), an invariant introduced in earlier paper. These (formal) polynomials are related by applying a map wt\colon K(CMA)→ K(fdA) to their exponents. In the \mathbbX-cluster chart corresponding to T, the function ΨM becomes FTM. Further more when T mutates, FTM undergoes \mathbbX-mutation and \boldsymbolκ(T,M) undergoes tropical \mathbbA-mutation. We show that the monoid of g-vectors is given by a rational polyhedral cone, which can be described, following Rietsch-Williams' mirror symmetry strategy, by tropicalisation of the Marsh-Reitsch superpotential~W and, from that, by module-theoretic inequalities. In the process, the NO-body of Rietsch--Williams can be described in terms of \boldsymbolκ(T,M). This leads to a categorical incarnation of Grassmannian mirror symmetry, in the sense of Rietsch-Williams. Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of Gr(k,n).