Séminaire Lotharingien de Combinatoire (2021)
Sém. Lothar. Combin. 85B (2021), Art. 14, 12 pp.
Proceedings of the 33rd Conference on Formal Power
Series and Algebraic Combinatorics
Ben Drucker, Eli Garcia, Emily Gunawan, and Rose Silver
A box-ball system is a collection of discrete time states representing a permutation,
on which there is an action called a BBS move. After a finite number of BBS moves
the system decomposes into a collection of soliton states; these are weakly
increasing and invariant under BBS moves. The students proved that when this
collection of soliton states is a Young tableau or coincides with a partition of a type
described by Robinson-Schensted (RS), then it is an RS insertion tableau. They also
studied the number of steps required to reach this state.
Emily Cormier, Peter Dillery, Jill Resh, John Whelan
Maximal green sequences (MGS’s) are combinatorial objects that involve local transformations of directed graphs, also called quivers. We studied minimal length MGS’s for quivers of type A. It is know that such quivers are in bijection with triangulations of polygons. Moreover, these local transformations of quivers behave well under this correspondence, and there is a related notion of mutation of triangulations. This enabled us to study MGS’s both in terms of quivers and in terms of triangulations. We showed that any minimal length MGS has length n+t, where n is the number of vertices in the quiver and t is the number of 3-cycles. We also developed an algorithm that yields such sequences of minimal length.
Journal of Algebraic Combinatorics volume 44, pages 905–930 (2016)
Minimal Length Maximal Green Sequences
Jed Chou and Ben Whitney
A toric variety is an algebraic variety containing the algebraic torus (C*)n as an open dense subset such that the action of the torus extends to the whole variety. Every n-dimensional toric variety can be associated to a fan, which can be given as a set of primitive vectors in an n-dimensional lattice N. Because of this association, many properties of toric varieties can be studied using combinatorial methods. This group was interested in determining which projective toric varieties are Frobenius split. If a projective toric variety is Frobenius split, it has many nice properties. For example, it can be given as the solution set to homogeneous degree two polynomials.
Sam Payne proved in 2008 that a toric variety is Frobenius split if and only if an associated polytope called the splitting polytope contains representatives of every residue class of (1/q)M/M where M is the dual lattice to N. The group’s goal was to use this characterization of Frobenius splitting to classify the Frobenius split projective toric varieties in n dimensions. Given a trivalent tree where edges are labeled with variables and integers, it’s possible to construct the fan of a toric variety. They determined which toric varieties arising in this way are Frobenius split for certain classes of edge labelings.
Participating Schools: Amherst, Colgate, Smith, UConn, UMass, Williams
(photo courtesy Megan Brunner)