Positivity results for Stanley's character polynomials

Abstract

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (B50d) (2003) 11 pp.] the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results in Goulden and Rattan [I.P. Goulden, A. Rattan, An explicit form for Kerov's character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005].We shall see that the generating series C(t) introduced in [I.P. Goulden, A. Rattan, An explicit form for Kerov's character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005] is critical to our discussion.

Stanley's character polynomials and coloured factorisations in the symmetric group

“…Thus, I p;q (x) − 1 and −G p;−q (−x) both satisfy (5). Combining this with (9) and Proposition 2.1 gives the result.…”

“…This conjecture has only been proved in the case m = 1 for general µ (see [8,Theorem 1.1]); in particular, the conjecture is not known to be true for m > 1 even when µ has one part; that is, the conjecture is unknown even for (−1) k F k (p; −q). Some partial results showing positivity of the coefficients of (−1) k F k (p; −q) were given in Rattan [5], but otherwise little is known about these polynomials. Recently, Stanley [9] has a conjectured combinatorial interpretation for F µ (p; q), which we now explain.…”

“…2 [6,Theorem 4.9] and [7, Theorem 9] and references therein. To see how Kerov polynomials are used to obtain characters of the symmetric group and Stanley's polynomials, see Rattan [5]). Assuming that αβ = γ in S k where γ has cycle type μ and κ(α) + κ(β) = k + κ(γ ) then the product αβ = γ necessarily decomposes into κ(γ ) products of the form …”

Stanley’s Formula for Characters of the Symmetric Group

“…As suggested by Rattan in [6] where he gives a new easier proof of the case m = 1, we will use the following theorem:…”

A positivity conjecture for Jack polynomials

“…In this situation, we prove that (−1) k z µ θ λ µ,1 n−k (α) is a polynomial in (p, −q, β), with nonnegative rational coefficients. The proof is much more cumbersome and lenghty than in the case α = 1, studied in [15,13].…”