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The g-fan and the Gröbner fan of Gr(2,n) and Gr(3,6)

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CARW03 - Interdisciplinary applications of cluster algebras

In this talk I will discuss recent results of joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez. We show that for the Grassmannains Gr(2,n) and Gr(3,6) there exist a unique maximal simplicial cone in the Gröbner fan (of a suitable ideal) whose intersection with the tropical Grassmannian is the totally positive part of the tropicalization. The initial monomial ideal of the maximal cone is the Stanley-Reisner ideal of the g-fan (or cluster complex) associated with the corresponding cluster algebra. Moreover, there is family of Gröbner degenerations associated with the maximal cone that is canonically isomorphic to the cluster algebra with universal coefficients. This isomorphism yields an identification of the rays of the totally positive tropicalization with the cluster variables.

This talk is part of the Isaac Newton Institute Seminar Series series.

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