# Francisco Santos:Multitriangulations and tropical Pfaffians

Time: Wed 2022-05-04 10.15 - 11.15

Location: Zoom meeting ID: 654 5562 3260

Lecturer: Francisco Santos (Universidad de Cantabria)

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