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)≠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).