A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group

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1 A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group Richard P. Stanley Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 019, USA version of 6 July 006 The irreducible characters χ λ of the symmetric group S n are indexed by partitions λ of n (denoted λ n or λ = n), as discussed e.g. in [, 1.7] or [4, 7.18]. If w S n has cycle type ν n then we write χ λ (ν) for χ λ (w). Let µ be a partition of n, and let (µ, 1 n ) be the partition obtained by adding n 1 s to µ. Thus (µ, 1 n ) n. Regarding as given, define the normalized character χ λ (µ, 1 n ) by χ λ (µ, 1 n ) = (n) χ λ (µ, 1 n ), χ λ (1 n ) where χ λ (1 n ) denotes the dimension of the character χ λ and (n) = n(n 1) (n + 1). Thus [, (7.6)(ii)][4, p. 49] χ λ (1 n ) is the number f λ of standard Young tableaux of shape λ. Suppose that (the diagram of) the partition λ is a union of m rectangles of sizes p i q i, where q 1 q q m, as shown in Figure 1. The following result was proved in [5, Prop. 1] for µ = () and attributed to J. Katriel (private communication) for arbitrary µ. Proposition 1. Let λ be the shape in Figure 1, and fix 1. Let µ. Set n = λ and F µ (p; q) = F µ (p 1,...,p m ; q 1,..., q m ) = χ λ (µ, 1 n ). Then F µ (p; q) is a polynomial function of the p i s and q i s with integer coefficients, satisfying ( 1) F µ (1,...,1; 1,..., 1) = ( + m 1).

2 q 1 p 1. q p q m p m Figure 1: A union of m rectangles Note. When µ = (), the partition with a single part, we write F for F (). A formula was given in [5, (9)] for F (p; q), viz., F (p; q) = 1 [x 1 ] m (x) (x (q i + p i + p i p m )) i=1, m (x (q i + p i+1 + p i+ + + p m )) i=1 where [x 1 ]f(x) denotes the coefficient of x 1 in the expansion of f(x) in descending powers of x (i.e., as a Taylor series at x = ). It was conjectured in [5] that the coefficients of the polynomial ( 1) F µ (p; q) are nonnegative, where q = ( q 1,..., q m ). This conjecture was proved in [5] for the case m = 1, i.e., when λ is a p q rectangle, denoted λ = p q. For w S n let κ(w) denote the number of cycles of w (in the disjoint cycle decomposition of w). The main result of [5] was the following (stated slightly differently but clearly equivalent).

3 Theorem. Let µ and fix a permutation w µ S of cycle type µ. Then F µ (p; q) = ( 1) p κ(u) ( q) κ(v), uw µ=v where the sum ranges over all! pairs (u, v) S S satisfying uw µ = v. To state our conjectured generalization of Theorem, let S (m) denote the set of permutations u S whose cycles are colored with 1,,..., m. More formally, if C(u) denotes the set of cycles of u, then an element of S (m) is a pair (u, ϕ), where u S and ϕ : C(u) [m]. (We use the standard notation [m] = {1,,..., m}.) If α = (u, ϕ) S (m) and v S, then define a product αv = (w, ψ) S (m) as follows. First let w = uv. Let τ = (a 1, a,...,a j ) be a cycle of w, and let ρ i be the cycle of u containing a i. Set ψ(τ) = max{ϕ(ρ 1 ),...,ϕ(ρ j )}. For instance (multiplying permutations from left to right), 1 {}}{{}}{{}}{{}}{{}}{{}}{{}}{ ( 1,, )( 4, 5 )( 6, 7 )( 8 ) (1, 7)(, 4, 8, 5)(, 5) = ( 1, 4,, 6)(, 7 )( 5, 8 ). Note that it an immediate consequence of the well-nown formula w S x κ(w) = x(x + 1) (x + 1) that #S (m) = ( + m 1). Note. The product αv does not seem to have nice algebraic properties. In particular, it does not define an action of S on S (m), i.e., it is not necessarily true that (αu)v = α(uv). For instance (denoting a cycle colored 1 by leaving it as it is, and a cycle colored by an overbar), we have [(1)() (1, )] (1, ) = (1)() (1)() [(1, ) (1, )] = (1)().

4 Given α = (u, ϕ) S (m), let p κ(α) = i pκ i(α) i, where κ i (α) denotes the number of cycles of u colored i, and similarly q κ(β), so ( q) κ(β) = i ( q i) κ i(β) We can now state our conjecture. Conjecture. Let λ be the partition of n given by Figure 1. Let µ and fix a permutation w µ S of cycle type µ. Then F µ (p; q) = ( 1) αw µ=β p κ(α) ( q) κ(β), where the sum ranges over all (+m 1) pairs (α, β) S (m) satisfying αw µ = β. S (m) Example 1. Let m = and µ = (), so w µ = (1, ). There are six pairs (α, β) S () n for which α(1, ) = β, viz. (where as in the above Note an unmared cycle is colored 1 and a barred cycle ), α β p κ(α) q κ(β) (1)() (1, ) p 1q 1 (1)() (1, ) p 1 p q (1)() (1, ) p 1 p q (1)() (1, ) (1, ) (1)() p q p 1 q1 (1, ) (1)() p q. It follows (since the conjecture is true in this case) that F (p 1, p ; q 1, q ) = p 1q 1 p 1 p q p q + p 1 q 1 + p q. We can reduce Conjecture to the case p 1 = = p m = 1; i.e., λ = (q 1,, q,..., q m ). Let G µ (p, q) = p κ(α) q κ(β), αw µ=β so that Conjecture asserts that F µ (p; q) = ( 1) G µ (p, q). 4

5 Proposition 4. We have G µ (p, q) qi+1 =q i = G µ (p 1,...,p i 1, p i + p i+1, p i+,...,p m ; q 1,...,q i 1, q i, q i+,...,q m ). (1) Proof. Let αw µ = β, where α, β S (m) and µ. If τ is a cycle of β colored i + 1 then change the color to i, giving a new colored permutation β. We can also get the pair (α, β ) by changing all the cycles in α colored i + 1 to i, producing a new colored permutation α for which α w µ = β, and then changing bac the colors of the recolored cycles of α to i + 1. Equation (1) is simply a restatement of this result in terms of generating functions. It is clear, on the other hand, that F µ (p, q) qi+1 =q i = F µ (p 1,...,p i 1, p i + p i+1, p i+,...,p m ; q 1,...,q i 1, q i, q i+,...,q m ), because the parameters p 1,...,p m ; q 1,...,q i 1, q i, q i, q i+,...,q m and p 1,...,p i 1, p i + p i+1, p i+,...,p m ; q 1,...,q i 1, q i, q i+,...,q m specify the same shape λ. (Note that Proposition 1 requires only q 1 q q m, not q 1 > q > > q m.) Hence if Conjecture is true when p 1 = = p m = 1, then it is true in general by iteration of equation (1). Remars. 1. Conjecture has been proved by Amarpreet Rattan [] for the terms of highest degree of F, i.e., the terms of F (p; q) of total degree Kerov s character polynomials (e.g., [1]) are related to F (p; q) and are also conjectured to have nonnegative (integral) coefficients. Is there a combinatorial interpretation of the coefficients similar to that of Conjecture? 5

6 References [1] I. P. Goulden and A. Rattan, An explicit form for Kerov s character polynomials, Trans. Amer. Math. Soc., to appear; math.co/ [] I. G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, [] A. Rattan, in preparation. [4] R. Stanley, Enumerative Combinatorics, vol., Cambridge University Press, New Yor/Cambridge, [5] R. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lotharingien de Combinatoire (electronic) 50 (00), B50d. 6