Fully commutative elements in finite and affine Coxeter groups

Abstract

Let W be an irreducible finite or affine Coxeter group and let W c be the set of fully commutative elements in W. We prove that the set W c is closed under the Kazhdan-Lusztig preorder LR if and only if W c is a union of two-sided cells of W .

Depth in classical Coxeter groups

“…The depth of each reflection is given below the corresponding arrow. Then we further apply Step 1, first to the entry 8 and then to 7: 3,2,4,1,5,7,8,9]. Now, none of the positive entries are located to the left of its natural position, so we proceed with Step 2 to unsign the two most negative entries, which are6 and4: [6,3,2,4,1,5,7,8,9] We apply again Step 1 to push 6 and then 4 forward to their natural positions:…”

“…On the other hand, w = [8,1,9,3,5,2,6,4,7] is indecomposable with o B (w ) = 0. The negative identity [1, .…”

“…For example, if we let w = [4,3,1,2,7,5,6,9,8], then w is decomposable with w = w 1 ⊕ w 2 ⊕ w 3 , where the blocks are w 1 = [4,3, 1,2], w 2 = [3, 1,2], and w 3 = [2,1]; moreover, o B (w) = 2. On the other hand, w = [8,1,9,3,5,2,6,4,7] is indecomposable with o B (w ) = 0.…”

“…The sum of the depths of these reflections is 9 > 8. One can also write w as the product of t1 4 = [4, 2, 3,1], t2 3 = [1,3,2, 4], and t 23 = [1,3,2,4] in some order. The sum of the depths of these reflections is also 9 > 8.…”

“…However, T (w) = 3, and there are essentially only two ways to write w as the product of 3 reflections. One is to write w as the product of t1 4 = [4, 2, 3,1], t2 2 = [1,2,3,4], and t3 3 = [1,2,3,4] in some order. (These reflections pairwise commute.)…”