## ISM Discovery School:

Advances in Geometric Combinatorics and Representation Theory: Flow Polytopes, Gentle Algebras, and Associated Posets

Geometric combinatorics and representation theory are two important areas of mathematics that are increasingly intertwined. This summer school will highlight new developments connecting the representation theory of *gentle algebras* and the geometry and combinatorics of *flow polytopes* and *permutree lattices*. We will hold intertwined minicourses in these subjects followed by a 1-day conference.

The minicourses are aimed at graduate students specializing in combinatorics and representation theory. The goal of the workshop is to bring participants quickly up to speed on the side of the topic with which they are less familiar, so that all are able to appreciate the problems and research directions discussed at the conference.

Some financial support will be available to help cover the travel expenses of students coming from outside Montreal. To apply please send a CV and a letter of recommendation by April 1st to haedrich.alexandra@uqam.ca.

Please note that registration is now closed and any new registrations will be put on our waiting list.

### Description

The underlying motivation of the summer school is that triangulations of flow polytopes of directed graphs are deeply related to partial orders coming from both gentle algebras and the unified model of permutree lattices. These connections have already shed light on unsolved problems in these areas, such as computing the h^{*}-polynomial of flow polytopes, characterizing tau-tilting posets of gentle algebras, the geometric realization of the s-permutahedron, and realizing geometrically all permutree lattices.

A *flow* in a directed acyclic graph G is an assignment of a non-negative real number to each edge so that there is conservation of flow at each internal vertex. The *flow polytope* F_{G} consists of the set of flows of G for which the total netflow from the source vertices is 1. The lattice points of F_{G} are of interest in representation theory since they are related to Kostant’s vector partition function. Surprisingly, the volume of F_{G} is related to the number of integer flows of a related flow polytope by the work of Baldoni-Vergne [1], and Postnikov-Stanley [unpublished].

In 2012, motivated by cluster algebras, Danilov–Karzanov–Koshevoy (DKK) [7] introduced a family of regular unimodular triangulations of F_{G}, that depend on an ordering of the incoming and outgoing edges of the graph G (also known as framing), in terms of certain sets of pairwise coherent routes in the directed graph called cliques. The latter are in bijection with the set of integer flows of F_{G} whose cardinality gives its normalized volume [9]. The dual graphs of these triangulations include geometric realizations of posets that are fundamental in combinatorics; for instance, they have been used to geometrically realize the lattices that are given by the edge graphs of generalized associahedra [3] and permutahedra [6]. There are open questions related to the poset structure of these dual graphs and how to compute the h^{*}-polynomials of families of flow polytopes that are generalizations of F_{G} with various types of netflow vectors .

*Gentle algebras* are finite dimensional algebras introduced by Assem and Skowroński in 1987 and are receiving renewed interest due to combinatorial aspects of their module categories and connections to homological mirror symmetry. They have rich combinatorics [5] related to surfaces with marked points and *non-kissing complexes*. An important invariant in gentle algebras are tau-tilting posets and in 2022, a connection was found between DKK triangulations of flow polytopes of certain graphs and tau-tilting posets of gentle algebras [4]. Among other things, this connection allowed them to prove the lattice structure of the dual graph of these particular triangulations and develop Gorenstein and unimodality results for the h-polynomials of the triangulations.

*Permutree lattices* are a family of posets introduced in 2016 [10] to unify several important posets — such as boolean algebras, the weak order of the symmetric group, the Tamari lattice — in an object that has combinatorial, geometric, and algebraic information. Other special cases of permutree lattices are the Cambrian lattices of Reading [11] related to Catalan-Coxeter combinatorics. Some of these posets are known to be the dual graphs of triangulations of flow polytopes [6], and this connection is part of a more general phenomenon.

### References:

[1] Welleda Baldoni and Michele Vergne. “Kostant partitions functions and flow polytopes”. In: Transform. Groups 13.3-4 (2008), pp. 447–469.issn:1083-4362.doi:10.1007/s00031-008-9019-8

https://doi.org/10.1007/s00031-008-9019-8

[2] Welleda Baldoni-Silva, Jesus A. De Loera, and Michele Vergne. “Counting integer flows in networks”. In: Found. Comput. Math. 4.3 (2004), pp. 277–314. issn: 1615-3375. doi:10.1007/s10208-003-0088-8

https://doi.org/10.1007/s10208-003-0088-8

[3] Matias von Bell et al. “A unifying framework for the ν-Tamari lattice and principal order ideals in Young’s lattice”. In:Combinatorica 43.3 (2023), pp. 479–504. issn: 0209-9683.doi:10.1007/s00493-023-00022-x

https://doi.org/10.1007/s00493-023-00022-x

[4] Matias von Bell et al. Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras. 2022.

arXiv: 2203.01896 [math.CO]

[5] Thomas Brustle et al. “On the combinatorics of gentle algebras”. In: Canad. J. Math. 72.6 (2020), pp. 1551–1580.issn: 0008-414X. doi:10.4153/s0008414x19000397.

https://doi.org/10.4153/s0008414x19000397

[6] Rafael S. González D’León et al. Realizing the s-permutahedron via flow polytopes. 2023.

arXiv:2307.03474 [math.CO]

[7] Vladimir I. Danilov, Alexander V. Karzanov, and Gleb A. Koshevoy. “Coherent fans in the space of flows in framed graphs”. In: 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012). pp. 481–490.

[8] Karola Mészáros and Alejandro H. Morales. “Volumes and Ehrhart polynomials of flow poly-topes”. In: Math. Z. 293.3-4 (2019), pp. 1369–1401.
issn: 0025-5874.doi:10.1007/s00209-019-02283-z.

https://doi.org/10.1007/s00209-019-02283-z.

[9] Karola Mészáros, Alejandro H. Morales, and Jessica Striker. “On flow polytopes, order poly-topes, and certain faces of the alternating sign matrix polytope”. In: Discrete Comput. Geom.62.1 (2019), pp. 128–163. issn: 0179-5376.doi:10.1007/s00454-019-00073-2

https://doi.org/10.1007/s00454-019-00073-2.

[10] Vincent Pilaud and Viviane Pons. “Permutrees”. In: Algebr. Comb. 1.2 (2018), pp. 173–224.doi:10.5802/alco,

https://doi.org/10.5802/alco.

[11] Nathan Reading. “Cambrian lattices”. In: Adv. Math.205.2 (2006), pp. 313–353. issn: 0001-8708.doi:10.1016/j.aim.2005.07.010

https://doi.org/10.1016/j.aim.2005.07.010

[12] Christian Stump, Hugh Thomas, and Nathan Williams. Cataland: Why the Fuss? 2018.

arXiv:1503.00710