arXiv subjects
Publications and source records attributed to Travis Scrimshaw.
We describe a combinatorial realization of the crystals $B(\infty)$ and $B(λ)$ using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced finite and affine types.
In his PhD thesis, Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n+1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n+1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n-1.
The FindStat project at www.FindStat.org provides an online platform for mathematicians, particularly for combinatorialists, to gather information about combinatorial statistics and their relations. This outline provides an overview over the project.
We explicitly construct an embedding of a right-angled Artin group into a classical pure braid group. Using this we obtain a number of corollaries describing embeddings of arbitrary Artin groups into right-angled Artin groups and linearly independent subgroups of a right-angled Artin group.