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Francisco Santos:Multitriangulations and tropical Pfaffians

Time: Wed 2022-05-04 10.15 - 11.15

Location: Zoom meeting ID: 654 5562 3260

Participating: Francisco Santos (Universidad de Cantabria)

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Let $V=\binom{[n]}{2}$ label the possible diagonals among the vertices of a convex $n$-gon. A subset of size $k+1$ is called a $(k+1)$-crossing if all elements mutually cross, and a general subset is called $(k+1)$-crossing free if it does not contain a $k$-crossing. $(k+1)$-crossing free subsets form a simplicial complex that we call the $k$-associahedron and denote $Ass_k{n}$, since for $k=1$ it (essentially) equals the simplicial associahedron. The $k$-associahedron on the $n$-gon is known to be a shellable sphere of dimension $k(n-2k-1)$ and conjectured to be polytopal (Jonsson 2003). It is also a subword complex in the root system of the $A$.

The Pfaffian of an anti-symmetric matrix of size $2k+2$ is the square root of its determinant, and it is a homogeneous polynomial of degree $k+1$ with one monomial for each possible complete matching among $2k+2$ nodes representing the rows and columns. Thus, monomials correspond to certain $(k+1)$-subsets of $V$ and among them there is a unique $(k+1)$-crossing. Calling $I_k(n)$ the ideal of all principal Pfaffians of degree $k+1$ in an antisymmetric matrix of size $n$, it is known (Jonsson and Welker 2007) that for certain term orders the corresponding initial ideal equals the Stanley-Reisner ideal of the $k$-associahedron.

In this talk we explore the relation between Pfaffians and the $k$-associahedron from the tropical perspective. We show that the part of the tropical Pfaffian variety $trop(I_k(n))$ lying in the ``four-point positive orthant’’ realises the $k$-associahedron as a fan, and that this intersection is contained in (but is not equal to, except for $k=1$) the totally positive tropical Pfaffian variety $trop^+(I_k(n))$. We hope this to be a step towards realising the $k$-associahedron as a complete fan, but have only attained this for $k=1$: we show that for any seed triangulation $T$, the projection of $trop^+(I_1(n))$ to the coordinates corresponding to diagonals in $T$ produces a complete polytopal simplicial fan, that is, the normal fan of an associahedron. In fact, the fans we obtain are linearly isomorphic to the $g$-vector fans in cluster algebras of type $A$, as realized by Hoheweg, Pilaud and Stella (2018).

Zoom meeting ID: 654 5562 3260

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