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CATEGORIES:Geometry and Mathematical Physics seminar
SUMMARY:Compactifications of cluster varieties and convexi
ty - Alfredo Nájera Chávez\, UNAM Oaxaca
DTSTART:20200211T160000Z
DTEND:20200211T170000Z
UID:TALK4145AT
URL:/talk/index/4145
DESCRIPTION:In 2014\, Gross\, Hacking\, Keel and Kontsevich (G
HKK) introduced the so-called theta basis for a cl
uster algebra. The elements of the theta basis -th
eta functions- are a very special class of functio
ns on a cluster variety. In order to describe thet
a functions one considers sophisticated objects ca
lled scattering diagrams and broken lines inside t
hem. A key insight of the work of GHKK is that the
ta functions on a cluster varieties play a similar
role to the role played by the characters of an a
lgebraic torus in toric geometry. In particular\,
they showed that one can consider "positive subset
s" inside the scattering diagram to compactify a c
luster variety. Various projective varieties arisi
ng in representation theory (such as flag varietie
s\, Grassmannians\, certain Schubert varieties\, e
tc.) fit this framework. \nThe definition of a pos
itive set is of algebraic nature and is a notion d
irectly linked to the positivity property of clust
er algebras. The purpose of this talk is to presen
t a geometric/combinatorial interpretation of posi
tive sets. The main result I will talk about is th
e following: a set is positive if and only if it i
s "broken line convex".\n\nThis is joint work with
Man-Wai Cheung and Timothy Magee
LOCATION:Physics West 115
CONTACT:Timothy Magee
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