Rhombic tilings and Bott-Samelson varieties

TL;DR

This paper explores the connection between rhombic tilings and Bott-Samelson varieties, proposing a geometric perspective and extending tiling constructions to desingularizations, linking combinatorics with algebraic geometry.

Contribution

It establishes a natural connection between rhombic tilings and Bott-Samelson varieties, introducing a new geometric interpretation and extending tiling methods to desingularizations.

Findings

Connected tilings with Bott-Samelson data in type A

Provided a geometric perspective on Elnitsky's combinatorics

Extended tiling constructions to include desingularizations

Abstract

S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P.~Magyar (1998) found an important construction of the Bott-Samelson varieties introduced by H.C.~Hansen (1973) and M.~Demazure (1974). We explain a natural connection between S.~Elnitsky's and P.~Magyar's results. This suggests using tilings to encapsulate Bott-Samelson data (in type $A$). It also indicates a geometric perspective on S.~Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by B.~Tenner (2006).