TY - JOUR

T1 - On 321-avoiding permutations in affine Weyl groups.

AU - Green, Richard

PY - 2002/5

Y1 - 2002/5

N2 - We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A n – 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 W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (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 W, we use our main result to show that the fully commutative elements of W 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 GL n ()

AB - We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A n – 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 W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (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 W, we use our main result to show that the fully commutative elements of W 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 GL n ()

KW - pattern avoidance - Kazhdan-Lusztig cells

U2 - 10.1023/A:1015012524524

DO - 10.1023/A:1015012524524

M3 - Journal article

VL - 15

SP - 241

EP - 252

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 3

ER -