A Subalgebra of 0-hecke Algebra

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [BK] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type. 1.1. Let W be a Coxeter group generated by the simply reflections si (for i ∈ I). Let H be the Iwahori-Hecke algebra associated to W with parameter q = 0, i.e., H is a Q-algebra generated by Tsi for si ∈ I with relations T 2 si = −Tsi and the braid relations. The algebra H is called 0Hecke algebra. It was introduced by Norton in [No]. Representations of H were later studied in the work of Carter [Ca], Hivert-Novelli-Thibon [HNT] and etc. More recently, Stembridge [St] used the 0-Hecke algebra to obtain a new proof for the Möbius function of the Bruhat order of W . 1.2. Set T ′ si = −Tsi . For w ∈ W , we define T ′ w = T ′ si1 · · ·T ′ si k , where w = si1 · · · sik is a reduced expression of w. Tit’s theorem implies that T ′ w is well defined. Moreover, we have a binary operation ∗ : W ×W → W such that T ′ xT ′ y = T ′ x∗y for any x, y ∈ W . It is easy to see that (W, ∗) is a monoid with unit element 1. Now we state our main theorem. Theorem 1. Let W be a finite Coxeter group. For any subset J ⊂ I, let w 0 be the maximal element in the subgroup generated by sj (for j ∈ J). Then {w 0w I 0; J ⊂ I} is a commutative submonoid of (W, ∗). In other words, there exists a commutative monoid structure ⋆I on the set of subsets of I, such that T ′ w J1 0 w 0 T ′ w J2 0 w 0 = T ′ w J1⋆IJ2 0 w 0 . Remark. In the case where W is a Weyl group, this result was discovered by Berenstein and Kazhdan in [BK, Proposition 2.30]. Their approach was based on unipotent χ-linear bicrystals. The proof below is more elementary. It is based on some combinatorial properties of The author is partially supported by (USA) NSF grant DMS 0700589 and (HK) RGC grant DAG08/09.SC03. 1