Joint Relations on Elements of the Symmetric Group

We introduce two methods which reduce the checking of the joint relations on element pairs x, y (i.e., checking if μ(x, y) 6= 0, see 1.2) inside a two-sided cell of the symmetric group Sn to some special cases up to star operations. One involves the involutions in Sn, and the other involves the set Σλ for λ ∈ Λn (see 3.1 and 5.1). We give a detailed investigation for the properties of the set Σλ in terms of standard tableaux. Also, we study the sets U(S∞) and U′(S∞) and give an affirmative answer to a question of N. H. Xi (see Proposition 2.9). §0. Introduction. 0.1. Let W be a Coxeter group with S its distinguished Coxeter generator set. Let H be the Hecke algebra associated to (W,S). In their construction for the representations of W and H, Kazhdan and Lusztig introduced the concept of left, right and two-sided cells of W (see [8]). The joint relations w—y (i.e., μ(w, y) 6= 0) on the pairs of elements w, y ∈ W (call {w, y} a joint pair in this case) play an important role in the representation of W and H afforded by these cells (see [8]). However, checking the joint relation on a pair of elements w, y ∈ W is usually a difficult task, which involves the complicated computation of the related Kazhdan–Lusztig polynomial Pw,y or Py,w (see [8]). Thus it is desirable to reduce the amount of such work to be as simple as possible. This is just the aim of our work. In the present paper, we shall mainly deal with the case where W is the symmetric group Sn, i.e., the Coxeter group of type An−1 for n > 1.