## Abstract

The homogeneous coordinate ring C[Gr(k,n)] of the Grassmannian is a cluster algebra, with an additive categorification CMC. Thus every M∈CMC has a cluster character ΨM∈C[Gr(k,n)].

The aim is to use the categorification to enrich Rietsch-Williams' mirror symmetry result that the Newton-Okounkov (NO) body/cone, made from leading exponents of functions in C[Gr(k,n)] in an X-cluster chart, can also be described by tropicalisation of the Marsh-Reitsch superpotential~W.

For any cluster tilting object T, with endomorphism algebra A, we define two new cluster characters, a generalised partition function PTM∈C[K(CMA)] and a generalised flow polynomial FTM∈C[K(fdA)], related by a `dehomogenising' map wt:K(CMA)→K(fdA).

In the X-cluster chart corresponding to T, the function ΨM becomes FTM and thus its leading exponent is κ(T,M), an invariant introduced in earlier paper (and the image of the g-vector of M under wt). When T mutates, FTM undergoes X-mutation and κ(T,M) undergoes tropical A-mutation.

We then show that the monoid of g-vectors is saturated, and that this cone can be identified with the NO-cone, so the NO-body of Rietsch--Williams can be described in terms of κ(T,M). Furthermore, we adapt Rietsch-Williams' mirror symmetry strategy to find module-theoretic inequalities that determine the cone of g-vectors.

Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of Gr(k,n).

The aim is to use the categorification to enrich Rietsch-Williams' mirror symmetry result that the Newton-Okounkov (NO) body/cone, made from leading exponents of functions in C[Gr(k,n)] in an X-cluster chart, can also be described by tropicalisation of the Marsh-Reitsch superpotential~W.

For any cluster tilting object T, with endomorphism algebra A, we define two new cluster characters, a generalised partition function PTM∈C[K(CMA)] and a generalised flow polynomial FTM∈C[K(fdA)], related by a `dehomogenising' map wt:K(CMA)→K(fdA).

In the X-cluster chart corresponding to T, the function ΨM becomes FTM and thus its leading exponent is κ(T,M), an invariant introduced in earlier paper (and the image of the g-vector of M under wt). When T mutates, FTM undergoes X-mutation and κ(T,M) undergoes tropical A-mutation.

We then show that the monoid of g-vectors is saturated, and that this cone can be identified with the NO-cone, so the NO-body of Rietsch--Williams can be described in terms of κ(T,M). Furthermore, we adapt Rietsch-Williams' mirror symmetry strategy to find module-theoretic inequalities that determine the cone of g-vectors.

Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of Gr(k,n).

Original language | English |
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Publisher | arXiv |

Number of pages | 78 |

Publication status | Published - 22 Apr 2024 |