Orbits of Groups Generated by Transvections over F2

Abstract

Let V be a finite dimensional vector space over the two element field. We compute orbits for the linear action of groups generated by transvections with respect to a certain class of bilinear forms on V . In particular, we compute orbits that are in bijection with connected components of real double Bruhat cells in semisimple groups, extending results of M. Gekhtman, B.

Recognizing Cluster Algebras of Finite Type

“…Our basic idea to develop such a method is to view the underlying graph of a diagram as an alternating bilinear form on a vector space over the 2-element field, and characterize an arbitrary (simply-laced) 2finite diagram using algebraic invariants of the corresponding bilinear form. A nice combinatorial set-up to carry out this idea is provided by a class of (undirected) graph transformations called basic moves, which were introduced and studied in [2,11]. A basic move changes a graph in a way similar to a mutation does modulo 2: it introduces or deletes edges containing a fixed vertex connected to a given vertex (thus a basic move is assigned to a pair of vertices connected to each other, for a precise description see Definition 5.1).…”

“…We prove the converse of this statement for 2-finite diagrams: any simply-laced diagram that does not contain any non-oriented cycle is 2-finite if and only if its underlying graph can be obtained from a Dynkin graph using basic moves (Theorem 5.3). The advantage of characterizing 2-finite diagrams using basic moves is that a basic move is a simpler operation than a mutation; there is also a classification of graphs under basic moves using algebraic and combinatorial invariants which can be easily implemented [9,11]. In Proposition 5.7, we give such a characterization for graphs that can be obtained from Dynkin graphs with 6,7 or 8 vertices using basic moves.…”

“…is provided by a class of (undirected) graph transformations called basic moves, which were introduced and studied in [2,11]. A basic move is a simpler operation than a mutation; there is also a classification of graphs under basic moves using algebraic and combinatorial invariants which can be easily implemented [9,11]. We take advantage of this classification thanks to our following characterization: a simply-laced diagram that does not contain any non-oriented cycle is 2-finite if and only if its underlying graph can be obtained from a Dynkin graph using basic moves (Theorem 5.3).…”

Cluster algebras: Notes for the CDM-03 conference

“…The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26,39].) The solution in [51] utilized the following general approach, which goes back to [41]: try to find a "simple" Zariski open subvariety U ⊂ G u,v such that the codimension in G u,v of the complement of U is greater than 1.…”

The flipping puzzle on a graph

“…A survey of related work, a brief discussion of Humphries results, and a discussion of the isomorphism types of groups occurring are given by Hall [6]. More recent results are in [14,16].…”