On 321-Avoiding Permutations in Affine Weyl Groups

• Authors: R. M. Green
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2002

We introduce the notion of 321-avoiding permutations in the affine Weyl group iW of type iAin − 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in iW coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of iW (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group iW, we use our main result to show that the fully commutative elements of iW form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to {\it GL}_n({\bb C}).