Travis Scrimshaw

A crystal to rigged configuration bijection and the filling map for type D4 (3)

Abstract We give a bijection Φ from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals of the form B r , 1 and B 1 , s in type D 4 ( 3 ) . We show that the cocharge statistic is sent to the energy statistic for tensor products ⨂ i = 1 N B r i , 1 and ⨂ i = 1 N B 1 , s i . We extend this bijection to a single B r , s , show that it preserves statistics, and obtain the so-called Kirillov–Reshetikhin tableaux model for B r , s . Additionally, we show that Φ commutes with the virtualization map and that B 1 , s is naturally a virtual crystal in type D 4 ( 1 ) , thus defining an affine crystal structure on rigged configurations corresponding to B 1 , s .

A rigged configuration model for B (

We describe a combinatorial realization of the crystals B ( ∞ ) 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.

Abrams's stable equivalence for graph braid groups

Abstract In his PhD thesis [1] , Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G, denoted C n ( 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 Formans 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 .

VIRTUALIZATION MAP FOR THE LITTELMANN PATH MODEL

We show the natural embedding of weight lattices from a diagram folding is a virtualization map for the Littelmann path model, which recovers a result of Kashiwara. As an application, we give a type-independent proof that certain Kirillov-Reshetikhin crystals respect diagram foldings, which is a known result on a special case of a conjecture given by Okado, Schilling, and Shimozono.

Rigged configurations and the \(*\)-involution

We give an explicit description of the

Type

*

rigged configuration bijection

∗-involution on the rigged configuration model for