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SUMMARY:Jiarui Fei (Shanghai Jiao Tong)
DTSTART:20210401T120000Z
DTEND:20210401T130000Z
DTSTAMP:20260423T005752Z
UID:notts_ag/51
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/notts_ag/51/
 ">Tropical $F$-polynomials and Cluster Algebras</a>\nby Jiarui Fei (Shangh
 ai Jiao Tong) as part of Online Nottingham algebraic geometry seminar\n\n\
 nAbstract\nThe representation-theoretic interpretations of $g$-vectors and
  $F$-polynomials are two fundamental ingredients in the (additive) categor
 ification of cluster algebras. We knew that the $g$-vectors are related to
  the presentation spaces. We introduce the tropical $F$-polynomial $f_M$ o
 f a quiver representation $M$\, and explain its interplay with the general
  presentation for any finite-dimensional basic algebra. As a consequence\,
  we give a presentation of the Newton polytope $N(M)$ of $M$. We propose a
 n algorithm to determine the generic Newton polytopes\, and show it works 
 for path algebras. As an application\, we give a representation-theoretic 
 interpretation of Fock-Goncharov's cluster duality pairing. We also study 
 many combinatorial aspects of $N(M)$\, such as faces\, the dual fan and $1
 $-skeleton. We conjecture that the coefficients of a cluster monomial corr
 esponding to vertices are all $1$\, and the coefficients inside the Newton
  polytope are saturated. We show the conjecture holds for acyclic cluster 
 algebras. We specialize the above general results to the cluster-finite al
 gebras and the preprojective algebras of Dynkin type.\n
LOCATION:https://researchseminars.org/talk/notts_ag/51/
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