A classification of affine fully commutative elements

Fully commutative elements in finite and affine Coxeter groups

“…Given any subset I ⊂ S, we will note H I the subposet induced by all elements of H with labels in I. In particular H {s} is the chain H s = s (1) < s (2) < • • • < s (k) where k = |H s | is its cardinality. If s, t are two labels such that m st ≥ 3, note that H {s,t} is also a chain.…”

“…In the recent work [1] by Al Harbat, the author classifies FC elements by indicating a normal form for each of them. That is, the main theorem exhibits a family of reduced FC expressions where each FC element is represented exactly once.…”

“…Then the knowledge of all terms up to 6 Proof. Let H be a FC heap with k ≥ 0 t-elements t (1) < • • • < t (k) . By Proposition 1.7, there are at least two with labels in {s 1 , s 2 , s 3 } between t (i) and t (i+1) for any i < k. But by Lemma 5.1, there is at most one s j -element between t (i) and t (i+1) for j = 1, 2, 3.…”

“…Now we assume k ≥ 4, so that case (a) holds: between two successive t-elements, there are two elements with distinct labels in {s 1 , s 2 , s 3 }. Consider first t (1) , t (2) and assume without loss of generality that these labels are {s 1 , s 2 }. The possible labels in ]t (2) , t (3) [ are then either {s 1 , s 3 } or {s 2 , s 3 }; indeed {s 1 , s 2 } cannot occur since the same argument as in the previous paragraph shows that this forces k = 3.…”