Rhombic tilings and Bott–Samelson varieties

Abstract

S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of 2 n 2n -gons and commutation classes of reduced words in the symmetric group on n 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 A ). It also indicates a geometric perspective on S. Elnitsky’s bijection. We also extend this construction by assigning desingularizations of Schubert varieties to the zonotopal tilings considered by B. Tenner (2006).