Sminaire Lotharingien de Combinatoire 72 (2015), Article B72c

COMPARING ESTIMATES ON THE NUMBER OF ZEROS OF IRREDUCIBLE CHARACTERS IN SYMMETRIC GROUPS

F. PETÉNYI

To the memory of Professor David Chillag

Abstract. We compare lower estimates of Gallagher, Chillag and Wilde on the number of zeros of irreducible characters of finite groups. We formulate each estimate both for group elements and for conjugacy classes. We compare for each pair of estimates, if there are infinitely many symmetric groups where one of them is better than the other.

1. Introduction

In the literature zeros of characters are widely investigated, see e.g. [2], [3], [4], [5], [6], [7], [8], [9], [12], [14], [17], [18], [19] [21], [25], [26], [27], and [28].

In this paper we investigate lower estimates on the number of zeros of irreducible complex characters of finite groups.

The paper consists of two parts. In the first part we reformulate the lower estimate of Gallagher for the case of conjugacy classes, and the lower estimate of Chillag for the case of group elements. We also formulate estimates of Wilde.

We introduce for these estimates the following notation: the upper index (G or C) indicates if the estimate is for group elements or for conjugacy classes, respectively. The lower index Ga, Ch, W refers to the name of the person whose estimate is used.

We give the smallest finite groups for each ordered pair of the above mentioned estimates where the first one is better than the second for some irreducible character.

In the second part of the paper we consider symmetric groups. We examine, which estimate is better than the other for irreducible, non-linear characters of symmetric groups, namely which of them is bigger than the other for all but finitely many characters. In the paper we use standard notations; for example, Cl(G) denotes the set of conjugacy classes of G, and C_{G}(g) denotes the centralizer of g in G.

2. Some lower estimates for the number of zeros of irreducible characters of finite groups

Let G be a finite group and χ one of its complex irreducible characters. We say that an element g is vanishing with respect to χ, if χ(g) = 0. Our aim is to investigate and compare lower estimates for the number of vanishing conjugacy classes or elements.

2010 Mathematics Subject Classification. Primary 20C15; Secondary 20C30.

Key words and phrases. zeros of characters, estimates, partition, hook length formula, primes in interval.

The work reported in the paper has been developed in the framework of the project “Talent care and cultivation in the scientific workshops of BME" project. This project is supported by the grant TÁMOP -4.2.2.B-10/1-2010-0009.

The first well-known result is due to Burnside (see [11, Thm. 3.15]), which states that every nonlinear irreducible character of a finite group vanishes on some element of the group. We will use the results of Gallagher [8, Thm. 4], Chillag [4, Thm. 1.1], and Wilde [25, Thm. 2.1] to construct three different lower bounds.

We introduce the following notations. Denote by n^{G}(χ) and n^{C}(χ) the number of vanishing group elements and conjugacy classes of χ, respectively, i.e.,

n^{G}(χ) = |{g ∈ G | χ(g) = 0}| and n^{C}(χ) = |{C ∈ Cl(G) | χ|_{C} = 0}|.

Gallagher improved the result of Burnside as follows; see [11, Thm. 3.15].

Theorem 2.1 (Gallagher [8]). For every irreducible character χ of a finite group, we have

n^{G}(χ) ⩾ (χ(1)^{2} − 1)|Z(χ)|,

where Z(χ) is the center of χ. If |χ| takes only the values 0, 1, and χ(1), then equality holds.

Let us introduce the notation 𝔈_{Ga}^{G}(χ) = (χ(1)^{2} − 1)|Z(χ)|.

Remark 2.2. There are infinitely many groups where n^{G}(χ) = 𝔈_{Ga}^{G}(χ) for some nonlinear irreducible character χ. An example is the case of groups of nilpotency class 2, see [11, Cor. 2.30, Thm. 2.31].

Let us recall the following result.

Theorem 2.3 (Chillag [4]). Let G be a finite non-abelian group such that G ≠ G′ and χ a non-linear irreducible character of G. Then one of the following holds:

(i) G has an element x such that |C_{G}(x)| ⩽ 2n^{C}(χ). In fact, one can choose such an x from any conjugacy class of maximal size among the classes on which χ vanishes.

(ii) ϕ = χ_{G′} ∈ Irr(G′) and ϕ^{G} = χ ∑{λ | λ ∈ Irr(G), λ(1) = 1}. Furthermore the set {χλ | λ ∈ Lin(G)} consists of exactly |G : G′| extensions of ϕ, which are all the extensions of ϕ. In particular, ϕ is not linear and G′′ ≠ 1.

We can use this theorem to give a lower bound for n^{C}(χ). This lower bound will be denoted by 𝔈_{Ch}^{C}(χ), which is given by

𝔈_{Ch}^{C}(χ) = \begin{cases}0, & if χ linear, \\ min_{\{x∈G | χ(x)=0\}} ⌈|C_{G}(x)|/2⌉, & if G ≠ G′ and χ_{G′} ∉ Irr(G′), \\ 1, & otherwise. \end{cases}

Let us consider the case where 𝔈_{Ch}^{C}(χ) = 1.

Proposition 2.4. Let G be a finite group with G ≠ G′, and let χ ∈ Irr(G) be nonlinear and χ_{G′} ∉ Irr(G′). Let 𝔈_{Ch}^{C}(χ) = 1. Then n^{G}(χ) = \frac{|G|}{2} and n^{C}(χ) = 1. Moreover, G is a Frobenius group with a complement of order 2.

Proof. If 𝔈_{Ch}^{C}(χ) = 1, then min_{{g|χ(g)=0}} |C_{G}(g)| is 1 or 2. Suppose the minimum is obtained at some element g_{0}.

If |C_{G}(g_{0})| = 1, then the group is the one-element group, which is a contradiction. Otherwise, we have |C_{G}(g_{0})| = 2, so | Cl(g_{0})| = \frac{|G|}{2} ⩽ n^{G}(χ).

ESTIMATES ON ZEROS OF CHARACTERS IN S_{n} 3

Consider the subgroup H = {1, g_{0}}. This is a trivial intersection set, i.e., H ⋂ H^{x} = 1 for all x ∈ G \ H. Therefore G is a Frobenius group with H as complement. In G every irreducible character is either linear or is induced from the Frobenius kernel N. Since χ has at least one vanishing conjugacy class, it cannot be linear. Furthermore, χ = ν^{G}, where ν ∈ Irr(N). Since |H| is even, the kernel N is abelian, see e.g. [11, Thm. 7.21]. Therefore ν is linear. Let n ∈ N such that χ(n) = 0. Then

χ(n) = \frac{1}{|N|} ∑_{x∈G} ν^{∘}(xnx^{−1}) = \frac{1}{|N|} ∑_{x∈G} ν(xnx^{−1}) = ν(n) + \frac{1}{|N|} ∑_{x∈G\N} ν(xnx^{−1}).

However, every g ∈ G\N is inverting the elements of N. Hence χ(n) = ν(n)+\overline{ν(n)} = 0. This holds exactly when ν(n) = i or − i. Thus 4 divides the order of n, and also |N|. This is a contradiction. This means that exactly the elements in G \ N are the vanishing elements, and this is the only vanishing conjugacy class. Thus n^{C}(χ) = 1 and n^{G}(χ) = \frac{|G|}{2}. □

Suppose we know the sizes of the conjugacy classes of G. Then, using 𝔈_{Ga}^{G}(χ), we can construct another lower bound for n^{C}(χ). This lower bound will be denoted by

𝔈_{Ga}^{C}(χ) = min{|J| | J ⊆ I, ∑_{j∈J} |C_{j}| ⩾ 𝔈_{Ga}^{G}(χ)},

where Cl(G) = {C_{i}}_{i∈I}.

Similarly, from 𝔈_{Ch}^{C}(χ) we can construct a lower estimate 𝔈_{Ch}^{G}(χ) of n^{G}(χ). If we use the result of Proposition 2.4, as well as the fact that χ cannot vanish on Z(G), then we obtain

𝔈_{Ch}^{G}(χ) = \begin{cases}0, & if χ is linear, \\ \frac{|G|}{2}, & if 𝔈_{Ch}^{C}(χ) = 1, G ≠ G′, \\ & and χ_{G′} ∉ Irr(G′), \\ min_{J⊆I}\{∑_{j∈J} |C_{i}| | \\ C_{j} ∈ Cl(G), C_{j} ⊈ Z(G) and |J| = 𝔈_{Ch}^{C}(χ)\}, & otherwise. \end{cases}

We remark that, if χ_{G′} ∈ Irr(G′), then 𝔈_{Ch}^{G}(χ) is the size of the smallest non-central conjugacy class.

Before formulating the lower bound by Wilde, we mention the following fact.

Theorem 2.5 (Brauer, Nesbitt [11, Thm. 8.17]). Let χ ∈ Irr(G) and suppose p ∤ \frac{|G|}{χ(1)} for some prime p. Then χ(g) = 0 whenever p | o(g).

This is equivalent to saying that, whenever χ(g) ≠ 0, then the squarefree part o(g)_{0} of o(g) divides \frac{|G|}{χ(1)}.

In [25, Conj. 1.1], Wilde proposed the following conjecture.

Conjecture 2.6 (Wilde). Let G be a finite group. Furthermore, let χ ∈ Irr(G) and g ∈ G, and suppose that χ(g) ≠ 0. Then o(g) divides \frac{|G|}{χ(1)}.

Wilde proved in [25] that Conjecture 2.6 holds for solvable groups, and he mentioned that it is true also for S_{n}, which we prove later for the sake of selfcontainedness, see Lemma 3.10. In [25, Thm. 2.1] the following partial result was proved.

Theorem 2.7 (Wilde). Let χ be an irreducible character of G and g ∈ G a group element with χ(g) ≠ 0. Then o(g)o(g)_{0} | (\frac{|G|}{χ(1)})^{2} and o(g)^{3} | \frac{|G|^{3}}{χ(1)^{2}} .

We use Theorem 2.7 and Burnside’s theorem, see [11, Thm. 3.15], to give lower estimates for n^{G}(χ) and n^{C}(χ), which we will denote by 𝔈_{W}^{G} (χ), 𝔈_{W}^{C} (χ), respectively:

𝔈_{W}^{G} (χ) = max \{|\{g ∈ G | o(g)o(g)_{0} ∤ \frac{|G|^{2}}{χ(1)^{2}} or o(g)^{3} ∤ \frac{|G|^{3}|}{χ(1)^{2}} \}| , min_{C∈Cl(G),C⊈Z(G)} |C|\} ,
𝔈_{W}^{C} (χ) = max \{|\{C ∈ Cl(G) | o(g)o(g)_{0} ∤ \frac{|G|^{2}}{χ(1)^{2}} or o(g)^{3} ∤ \frac{|G|^{3}}{χ(1)^{2}} for all g ∈ C\}| , 1\} .

We know that Conjecture 2.6 is true for p-groups. In fact one can prove more.

Proposition 2.8. If G is a p-group, then for every g ∈ G and for every χ ∈ Irr(G), we have o(g) | \frac{|G|}{χ(1)}. In particular, for every χ ∈ Irr(G), we have 𝔈_{W}^{C} (χ) = 1 and 𝔈_{W}^{G} (χ) = min_{C∈Cl(G), C⊈Z(G)} |C|.

Proof. We use induction on the order of the group. We may suppose that χ(1) > 1. Then ⟨g⟩ ≠ G.

Let M be a maximal subgroup in G containing g. Then χ_{M} is either irreducible or it is the sum of p irreducible characters. If χ_{M} = ψ_{1} + ⋯ + ψ_{p}, where ψ_{i} ∈ Irr(M), then ψ_{i}^{G} = χ, and so χ(1) = pψ_{i}(1). Since M is a smaller p-group, by induction hypothesis we have o(g) | \frac{|M|}{ψ_{i}(1)} = \frac{|G|}{χ(1)}.

If χ_{M} = ψ, where ψ ∈ Irr(M), then by induction hypothesis we have

o(g) | \frac{|M|}{ψ(1)} = \frac{|M|}{χ(1)} | \frac{|G|}{χ(1)}.

□

For both n^{G}(χ) and n^{C}(χ) we have the following results on the above lower bounds. We checked the examples with the help of the GAP system [10], see the used programs in [22].

Theorem 2.9. For each ordered pair of the above estimates on conjugacy classes {𝔈_{Ch}^{C}( ), 𝔈_{Ga}^{C}( ), 𝔈_{W}^{C} ( )} and group elements {𝔈_{Ch}^{G}( ), 𝔈_{Ga}^{G}( ), 𝔈_{W}^{G} ( )}, there exists a group and an irreducible character of that group such that the first estimate is better than the second.

We demonstrate the result using the following diagrams:

[figure: 𝔈_{Ch}^{C}( ); D_{8} (2); C_{5}⋊C_{4}; (20,3) (1); 𝔈_{Ga}^{C}( ); C_{3}⋊Q_{8} (2); C_{7}⋊C_{4} (2); 𝔈_{W}^{C} ( ); D_{8} (2); D_{8} (2); 𝔈_{Ch}^{G}( ); D_{8} (2); C_{3}⋊C_{4} (2); 𝔈_{Ga}^{G}( ); D_{10} (1); D_{10} (1); 𝔈_{W}^{G} ( ); D_{8} (2); D_{8} (2)]

Here, an arrow points to the estimate which is better on some irreducible character of the group corresponding to it. We labelled the arrows with the smallest possible groups having suitable character. When the semidirect product is not unique, we included the group ID of the GAP system. In brackets you can find the size of the centre of the character. With this data, the group and set of suitable characters are uniquely determined.

3. Estimates for symmetric groups

Let consider the estimates of the previous section for symmetric groups.

Lemma 3.1. Let χ be an irreducible character of the symmetric group S_{n} for n ≥ 5. Then 𝔈_{Ga}^{G}(χ) = χ(1)^{2} − 1. Thus n^{G}(χ) ≥ χ(1)^{2} − 1.

Proof. If χ is linear, then the statement is true. Otherwise ker(χ) = 1, and Z(χ) = Z(S_{n}) = 1. □

Lemma 3.2. Let χ ∈ Irr(S_{n}) be the character corresponding to the partition λ. Then 𝔈_{Ch}^{C}(χ) = 1 if and only if λ is not symmetric and χ is not linear, or n = 3 and χ is corresponding to the partition (2, 1).

Proof. According to [20, Sec. 6.7], the character χ_{A_{n}} is an irreducible character in A_{n} if and only if the diagram corresponding to χ is not symmetric. We then use Proposition 2.4 to conclude. □

We will need the following well-known result.

Lemma 3.3. The largest conjugacy class in S_{n} corresponds to the partition (n − 1, 1), and the second largest is (n).

Proof. The idea of the proof is that, if we replace one of the largest cycles and another cycle by a cycle of length equal to the sum of lengths of the two cycles, then the corresponding conjugacy class is bigger, unless we had just a 1-cycle and an (n − 1)- cycle.

The size of the conjugacy class (1^{j_{1}}, 2^{j_{2}}, …, k^{j_{k}}), which contains exactly j_{l} cycles of length l, is

\frac{n!}{∏_{i=1,i≠l}^{k−1} i^{j_{i}}j_{i}! · l^{j_{l}}j_{l}! · k^{j_{k}}j_{k}!},

and the size of the conjugacy class (1^{j_{1}}, 2^{j_{2}}, …, l^{j_{l}−1}, …, k^{j_{k}−1}, (k + 1)^{0}, …, (k + l − 1)^{0}, (k + l)^{1}) is

\frac{n!}{∏_{i=1,i≠l}^{k−1} i^{j_{i}}j_{i}!·, l^{(j_{l}−1)}(j_{l} − 1)! · k^{(j_{k}−1)}(j_{k} − 1)! · (k + l)}.

The second one is always larger, unless j_{1} = j_{n−1} = 1, and for the other indices k we have j_{k} = 0 or j_{n} = 1. Thus the largest conjugacy class is (n − 1, 1), and the second largest is (n). □

Now we can compute 𝔈_{Ch}^{C}(χ_{λ}) for symmetric λ as well.

Lemma 3.4. If χ is an irreducible character of S_{n} corresponding to a symmetric partition, then 𝔈_{Ch}^{C}(χ) = ⌈\frac{n−1}{2}⌉.

Proof. Since λ is symmetric, χ is induced from an irreducible character of A_{n}. Thus χ vanishes on the conjugacy classes of odd permutations. If n is odd, then the largest conjugacy class Cl(n − 1, 1) is vanishing. Thus

𝔈_{Ch}^{C}(χ) = min_{χ(g)=0} ⌈\frac{|C_{G}(g)|}{2}⌉ = ⌈\frac{n − 1}{2}⌉ .

Let n be even. In Lemma 3.3 we have seen that the largest conjugacy class is the class Cl(n − 1, 1), and the second largest is Cl(n). Since n is even, the conjugacy class Cl(n) is vanishing. Therefore, we have

𝔈_{Ch}^{C}(χ) = min_{χ(g)=0} ⌈\frac{|C_{G}(g)|}{2}⌉ = \begin{cases}⌈\frac{n−1}{2}⌉ , & n odd, \\ ⌈\frac{n}{2}⌉ = ⌈\frac{n−1}{2}⌉ , & n even. \end{cases}

□

Lemma 3.5. If χ ∈ Irr(S_{n}) and n ≥ 851, then 𝔈_{Ga}^{C}(χ) ≤ 1.

Proof. If χ is an irreducible character of S_{n}, then we know by [24] that

χ(1) ≤ e^{−0,11565\sqrt{n}}\sqrt{n!}.

This implies that

χ^{2}(1) − 1 = 𝔈_{Ga}^{G}(χ) < e^{−0,2313\sqrt{n}}n!.

Let f(n) := e^{−0,2313\sqrt{n}}n!/(n(n−2)!). Since f(n) ≤ 1 for n = 851 and f(n) is a decreasing function for n ≥ 851, for n ≥ 851 the bound 𝔈_{Ga}^{G}(χ) is less than n(n − 2)!, which is the size of the largest conjugacy class of S_{n}. Thus 𝔈_{Ga}^{C}(χ) ≤ 1 for n ≥ 851. □

Moreover, we have the following conjecture.

Conjecture 3.6. Let χ be a non-linear irreducible character of a symmetric group. The above defined 𝔈_{Ga}^{C}(χ) is two exactly if χ corresponds to one of the following partitions:

(3, 1^{2}), (3, 2, 1), (3, 2, 1^{2}), (4, 2, 1), (4, 2, 1^{2}), (4, 2^{2}, 1), (4, 3, 1^{2}),

(4, 3, 2, 1), (4, 3, 2, 1^{2}), (5, 3, 2, 1), (5, 3, 2, 1, 1), (6, 4, 3, 2, 1, 1).

In all other cases, 𝔈_{Ga}^{C}(χ) is exactly 1.

Remark 3.7. Using results of McKay in [16], we can see that this conjecture is true if n ≤ 75. It is easy to check (for example with the help of GAP [10]) that the conjecture holds for 2-part and 3-part partitions. One can also prove by induction that the conjecture is true for hook-partitions. The conjecture shows for symmetric groups that Gallagher’s estimate for conjugacy classes (𝔈_{Ga}^{C}( )) is not really better than Burnside’s, see [11, Thm. 3.15].

Proposition 3.8. (i) The minimal degrees of non-linear irreducible characters of

S_{n} are n − 1, if n > 4. (ii) The minimal degrees of those non-linear irreducible characters of S_{n} which cor-

respond to symmetric partitions are at least 2^{n/4}.

Proof. (i). The proof can be found in [13, Thm. 4.10].

(ii). We can easily check the statement for n = 3, …, 7 directly. Let n be at least 8. We will prove by induction that, if λ = (λ_{1}, …, λ_{k}) is a partition not necessarily symmetric, and λ_{1} ≥ k ≥ 2, then χ_{λ}(1) ≥ λ_{1}2^{\frac{n−λ_{1}−1}{2}} . Since χ_{λ}(1) = χ_{λ^{T}} (1), in this way we can estimate the degree of every non-linear irreducible character. Every partition not corresponding to a linear character contains a (λ_{1}, 1) subdiagram. Let i be the number of the boxes outside of this subdiagram, and denote the n-partition λ by λ(n, i). Rephrasing the statement for a function of i, we have

χ_{λ}(1) ≥ (n − i − 1)2^{i/2}.  (3.1)

We will prove this by double induction on n and i. We checked the statement for n = 8, 9 by GAP [10]. For i = 0 and i = 1 it follows from this table that (3.1) holds.

i diagram of λ filled by hook-lengths  χ_{λ}(1)  (n − i − 1)2^{i/2}
0  \begin{matrix}n & n − 2 & n − 3 & … & 1 \\ 1 \end{matrix}  n − 1  n − 1
1  \begin{matrix}n − 1 & n − 2 & n − 4 & … & 1 \\ 2 & 1 \end{matrix}  n(n − 3)/2  (n − 2)\sqrt{2}
1  \begin{matrix}n & n − 3 & n − 4 & … & 1 \\ 2 \\ 1 \end{matrix}  (n − 1)(n − 2)/2 (n − 2)\sqrt{2}

Let us suppose n > 9, i > 1, and that we know the statement (3.1) for n_{1}, i_{1}, where either n_{1} = n and 0 ≤ i_{1} < i or 8 ≤ n_{1} < n. If λ ≠ (λ_{1}, …, λ_{1}), then there exist two rows each of them bigger than the one following it. Using the branching rule (see [23, Thm. 2.8.3]), χ_{λ}|_{S_{n−1}} is the sum of at least two irreducible characters corresponding to (n − 1)-partitions. At most one deleted box could be in the first row. Therefore we have to check both options: χ_{λ}(1) ≥ χ_{ν(n−1,i−1)}(1) + χ_{μ(n−1,i−1)}(1) or χ_{λ}(1) ≥ χ_{ν(n−1,i−1)}(1) + χ_{μ(n−1,i)}(1). We illustrate this by two examples:

(1) =  (1) +  (1),

(1) =  (1) +  (1).

By the induction hypothesis, we get

χ_{λ}(1) ≥ min{(n − i − 1)2^{(i−1)/2} · 2, (n − i − 1)2^{(i−1)/2} + (n − i − 2)2^{i/2}}.  (3.2)

In the other case, we have λ = (λ_{1}, …, λ_{1}). For computing χ_{λ}(1) we use the branching rule (see [23, Thm. 2.8.3]) twice. In the first step we cannot delete a box from the first row. In the second step, the integer i decreases by 2 or 1:

χ_{λ}(1) = χ_{θ(n−1,i−1)} ≥ χ_{ν(n−2,i−2)}(1) + χ_{μ(n−2,i−2)}(1)

or

χ_{λ}(1) = χ_{θ(n−1,i−1)}(1) ≥ χ_{ν(n−2,i−2)}(1) + χ_{μ(n−2,i−1)}(1),

respectively. Here are again two examples for the sake of illustration:

(1) =  (1) =  (1) +  (1),

(1) =  (1) =  (1) +  (1).

Thus we know by induction that

χ_{λ}(1) ≥ min{(n − i − 1)2^{(i−2)/2} · 2, (n − i − 1)2^{(i−2)/2} + (n − i − 2)2^{(i−1)/2}}. (3.3)

Since n > 9, by the assumption on k we have n−i−1 = λ_{1} > 3. Therefore \sqrt{2}(n−i−2) > (n − i − 1). Now the inequalities (3.2) and (3.3) imply that χ_{λ}(1) ≥ (n − i − 1)2^{i/2}. The minimum of the function f(i) = (n − i − 1)2^{i/2} on the interval [0, n − 4] is at i = 0. (In this way, we also have an alternative proof of the first part of the proposition.) Since the function f(i) is increasing on the interval [0, n − 4], if we want to compute a lower bound of the degrees of irreducible characters corresponding to symmetric partitions, then by inequality (3.1) it is enough to find the symmetric partition where i is minimal. This minimum is either (k, 1^{k−1}) if n is 2k − 1, or (k, 2, 1^{k−2}) if n is 2k. Consequently, min\{χ_{λ}(1)|λ symmetric\} ≥ \frac{n}{2}2^{(n−3)/4} ≥ 2^{n/4} for n ≥ 4. □

Corollary 3.9. If χ is a non-linear, irreducible character of S_{n} for n > 4, then 𝔈_{Ga}^{G}(χ) ≥ (n − 1)^{2} − 1 and, if χ corresponds to a symmetric partition, then 𝔈_{Ga}^{G}(χ_{λ}) ≥ 2^{n/2} − 1 for n > 2.

Here we give a proof of Conjecture 2.6 for S_{n}.

Lemma 3.10. If χ is an irreducible character of S_{n} and the order of g does not divide \frac{|S_{n}|}{χ(1)}, then χ(g) = 0.

Proof. Let λ be a partition of n and χ ∈ Irr(S_{n}) the character corresponding to λ. Consider a permutation g ∈ S_{n}. We know, by the hook-length formula (see [23, Thm. 3.10.2]), that \frac{|S_{n}|}{χ(1)} = ∏_{h∈H(λ)} h, where H(λ) is the multi-set of the hook-lengths of λ. Then there exists a prime p and an exponent α with p^{α} | o(g) and p^{α} ∤ ∏_{h∈H(λ)} h. Let us choose α to be maximal. Then g has a cycle of length divisible by p^{α}, however no hook-length in λ is divisible by p^{α}. Thus there is no rim-hook of length divisible by p^{α}, either. Hence, by the Murnaghan–Nakayama rule (see [23, Thm. 4.10.2]), we have χ(g) = 0. □

Consequently we can use Burnside’s Theorem (see [11, Thm. 3.15]) and Lemma 3.10 to give two other lower estimates for n^{G}(χ) and n^{C}(χ), for a nonlinear, irreducible character χ of S_{n}:

𝔈_{Co}^{G}(χ) = max \{|\{g ∈ G | o(g) ∤ \frac{|S_{n}|}{χ(1)}\}| , min_{C∈Cl(G),|C|≠1} |C|\} ,
𝔈_{Co}^{C}(χ) = max \{|\{C ∈ Cl(S_{n}) | o(g) ∤ \frac{|S_{n}|}{χ(1)} for all g ∈ C\}| , 1\} .

For linear characters, we define 𝔈_{Co}^{G}(χ) = 𝔈_{Co}^{C}(χ) = 0.

Of course, the inequalities 𝔈_{Co}^{G}(χ) ≥ 𝔈_{W}^{G} (χ) and 𝔈_{Co}^{C}(χ) ≥ 𝔈_{W}^{C} (χ) always hold. Thus we use 𝔈_{Co}^{C}( ), 𝔈_{Co}^{G}( ) instead of 𝔈_{W}^{C} ( ), 𝔈_{W}^{G} ( ) for symmetric groups.

4. Comparing estimates for symmetric groups

Theorem 4.1. For each ordered pair of elements of the set {𝔈_{Ch}^{C}, 𝔈_{Ga}^{C}, 𝔈_{Co}^{C}}, except for the pairs (𝔈_{Ga}^{C}, 𝔈_{Co}^{C}), (𝔈_{Ga}^{C}, 𝔈_{Ch}^{C}), and (𝔈_{Ch}^{C}, 𝔈_{Co}^{C}), there are infinitely many symmetric groups having a non-linear, irreducible character, with the property that the first estimate is better then the second. In the remaining cases, there are just finitely many symmetric groups with this property.

We illustrate the theorem in the following diagram. The broken arrow notation means that there are just finitely many symmetric groups where either 𝔈_{Ch}^{C}(χ) > 𝔈_{Co}^{C}(χ) or 𝔈_{Ga}^{C}(χ) > 𝔈_{Ch}^{C}(χ) or 𝔈_{Ga}^{C}(χ) > 𝔈_{Co}^{C}(χ) for some irreducible character χ. Each ordered pair of estimates is handled in a separate proposition below.

[figure: 𝔈_{Ch}^{C}( ); 4.2; 4.3; S_{p+2} (p,1^{2}); p>3 prime; 𝔈_{Ga}^{C}( ); 4.4 S_{2n+1 (n+1,1^{n})}, n>2; 4.5; S_{p+2} (p,1^{2}); p>3 p prime; 𝔈_{Co}^{C}( ); 4.7; 4.2]

We prove the theorem in Propositions 4.2–4.7.

Since for each nonlinear irreducible character χ ∈ Irr(S_{n}) the estimates 𝔈_{Co}^{C}(χ) and 𝔈_{Ch}^{C}(χ) are at least 1, by Lemma 3.5, we obtain the following fact.

Proposition 4.2. There are just finitely many symmetric groups having an irreducible character χ such that 𝔈_{Ga}^{C}(χ) > 𝔈_{Ch}^{C}(χ) or 𝔈_{Ga}^{C}(χ) > 𝔈_{Co}^{C}(χ).

Despite of this, we show that there are infinitely many symmetric groups having an irreducible character χ such that 𝔈_{Ch}^{C}(χ) = 𝔈_{Ga}^{C}(χ) = 𝔈_{Co}^{C}(χ). For example, if n > 5 with n − 1 being not prime, and, if in addition χ corresponds to the partition (n − 1, 1), then we get 𝔈_{Ch}^{C}(χ) = 𝔈_{Ga}^{C}(χ) = 𝔈_{Co}^{C}(χ) = 1 (see Lemma 3.2, Proposition 4.6). However this value is smaller than n^{C}(χ).

Proposition 4.3. Let p be a prime larger than 3. Then the symmetric group S_{p+2} possesses an irreducible character χ with 𝔈_{Co}^{C}(χ) > 𝔈_{Ch}^{C}(χ) and 𝔈_{Co}^{G}(χ) > 𝔈_{Ch}^{G}(χ).

Proof. Let χ ∈ Irr(S_{p+2}) be the irreducible character corresponding to the partition (p, 1^{2}). Since (p, 1^{2}) does not contain a hook of length divisible by p, we have p ∤ \frac{|S_{p+2}|}{χ(1)}. Thus

𝔈_{Co}^{C}(χ) ≥ |{C ∈ Cl(S_{p+2}) | p divides o(g) for all g ∈ C}|.

The set on the right-hand side contains the conjugacy classes corresponding to the partitions (p, 2) and (p, 1^{2}), respectively. Hence we have 𝔈_{Co}^{C}(χ) ≥ 2. Since (p, 1^{2}) is not symmetric, by Lemma 3.2 we have 𝔈_{Ch}^{C}(χ) = 1, and 𝔈_{Ch}^{G}(χ) is the size of a minimal noncentral conjugacy class. Thus the size of the conjugacy class (p, 1^{2}) is at least 𝔈_{Ch}^{G}(χ). Therefore the second statement is trivially true. □

Proposition 4.4. For n > 2, the symmetric group S_{2n+1} possesses an irreducible character χ with 𝔈_{Ga}^{C}(χ) < 𝔈_{Ch}^{C}(χ).

Proof. Let χ ∈ Irr(S_{2n+1}) be the character corresponding to the partition (n + 1, 1^{n}). Then we have 𝔈_{Ga}^{G}(χ) = \binom{2n}{n}^{2} − 1. By induction on n we can prove that, for n > 2, the latter quantity is at most (2n + 1)(2n − 1)!. Thus 𝔈_{Ga}^{C}(χ) = 1. By Lemma 3.4, we have 𝔈_{Ch}^{C}(χ) = n. So we are done. □

Proposition 4.5. Let p be a prime larger than 3. Then the symmetric group S_{p+2} possesses an irreducible character χ with 𝔈_{Co}^{C}(χ) > 𝔈_{Ga}^{C}(χ) and 𝔈_{Co}^{G}(χ) > 𝔈_{Ga}^{G}(χ).

Proof. Let χ ∈ Irr(S_{p+2}) correspond to the partition (p, 1^{2}). Since 𝔈_{Ga}^{G}(χ) = \binom{p+1}{2}^{2}−1 ≤ (p + 2)p! for p > 3, we have 𝔈_{Ga}^{C}(χ) = 1.

On the other hand, we have seen in Proposition 4.3 that

\{C ∈ Cl(S_{p+2}) | o(g) ∤ \frac{|G|}{χ(1)} for all g ∈ C\} ⊇ \{(p, 1^{2}), (p, 2)\} .

Thus we have 𝔈_{Co}^{C}(χ) ≥ 2. So we are done with the first part.

We can easily check that χ^{2}(1) − 1 = ⌊\frac{(p+1)p}{2}⌋^{2} − 1 < | Cl(p, 2)| + | Cl(p, 1^{2})| = \frac{(p+2)!}{p}. This implies that 𝔈_{Ga}^{G}(χ) < 𝔈_{Co}^{G}(χ). □

Proposition 4.6. Let n be an integer such that n − 1 is not a prime and n > 5. Then the symmetric group S_{n} possesses an irreducible character χ, with 𝔈_{Ga}^{C}(χ) = 𝔈_{Co}^{C}(χ) = 1 and 𝔈_{Ga}^{G}(χ) > 𝔈_{Co}^{G}(χ).

Proof. Let χ be the irreducible character of degree n − 1 corresponding to the partition (n − 1, 1). We will prove that, for every group element g, o(g) divides \frac{|G|}{χ(1)} = n · (n − 2)!. Let us fix a prime p, and compute the exponents of p in the prime factorizations of o(g) and n·(n−2)!. Let l denote the exponent of p in o(g). We have l ≤ k = max{j | p^{j} ⩽ n}. We will prove that k is at most the exponent of p in n · (n − 2)!. In order to that, we distinguish between three cases depending on the relative sizes of n and p^{k}.

If n = p^{k}, then the exponent of p in n · (n − 2)! is k + ⌊\frac{n−2}{p}⌋ + ⋯ + ⌊\frac{n−2}{p^{k−1}} ⌋, which is at least k.

If n − 1 = p^{k}, then k > 1 by our assumption, thus ⌊\frac{n−2}{p}⌋ + ⋯ + ⌊\frac{n−2}{p^{k−1}} ⌋ ⩾ (p^{k−1} − 1) + ⋯ + (p − 1), and we can easily see by induction that k ⩽ p^{k−1} − 1, since k = 2 and p = 2 cannot hold at the same time.

If n − 2 ≥ p^{k}, then the exponent of p in n · (n − 2)! is larger than k.

Therefore 𝔈_{Co}^{C}(χ) = 1 and 𝔈_{Co}^{G}(χ) = min_{g≠1} | Cl(g)| = \frac{n(n−1)}{2}.

Furthermore, we have 𝔈_{Ga}^{G}(χ) = χ^{2}(1) − 1 = (n − 1)^{2} − 1 = n(n − 2). Since the size of the largest conjugacy class is n(n − 2)! (see Lemma 3.3), we know that 𝔈_{Ga}^{C}(χ) = 1. This concludes the proof. □

Theorem 4.7. There exist only finitely many symmetric groups having a non-linear, irreducible character χ with 𝔈_{Ch}^{C}(χ) > 𝔈_{Co}^{C}(χ) or 𝔈_{Ch}^{G}(χ) > 𝔈_{Co}^{G}(χ).

Proof. Let us consider a symmetric group S_{n} with n > 3, and let χ be an irreducible character of S_{n}. If χ corresponds to a nonsymmetric partition, then 𝔈_{Ch}^{C}(χ) is 1 by Lemma 3.2 and 𝔈_{Co}^{C}(χ) ≥ 1. Therefore

𝔈_{Ch}^{C}(χ) ≤ 𝔈_{Co}^{C}(χ) and 𝔈_{Ch}^{G}(χ) ≤ 𝔈_{Co}^{G}(χ).

So at least one of 𝔈_{Ch}^{C}(χ) > 𝔈_{Co}^{C}(χ) and 𝔈_{Ch}^{G}(χ) > 𝔈_{Co}^{G}(χ) cannot hold.

Thus we may assume that the partition λ = (λ_{1}, …, λ_{n}) corresponding to χ is symmetric, so, by Lemma 3.4, we have 𝔈_{Ch}^{C}(χ) = ⌈\frac{n−1}{2}⌉. We prove the result in several steps.

Step 1. If there is a prime p with p ∤ \frac{|G|}{χ(1)}, then

𝔈_{Co}^{C}(χ) ≥ |{conjugacy classes containing a p-cycle}| = p(n − p).

(Here, p(n) is the number of n-partitions.)

Step 2. If n ≥ 16, then π(3n) − π(2n) ≥ 4. (Here, π(n) is the number of prime numbers, which are at most n.) Proof of Step 2. We used a GAP program to check π(3n)−π(2n) > 3 for 16 ≤ n ≤ 1216, see [22].

Using [1] we will now prove that, if n > 1216, then π(3n) − π(2n) ≥ 4.

Let T_{3} be the product of primes in the interval (2n, 3n]. By the proof of [1, Thm. 1.3], we have

T_{3} > (\frac{6, 5}{\sqrt{27}})^{n} \frac{1}{(3n)^{\sqrt{3n}/2}} =: t(n).

It is enough to prove that t(n) ≥ (3n)^{3}, since then we know that T_{3} has at least 4 prime divisors, which means that π(3n)−π(2n) ≥ 4. By taking the logarithm, we get what we

have to prove, namely that log \frac{6,5}{\sqrt{27}} ≥ ^{\frac{\frac{\sqrt{3n}}{2} +3}{n}} log(3n). Since ^{\frac{\frac{\sqrt{3n}}{2} +3}{n}} log(3n) is a decreasing function for n ≥ 16 and, if n = 1217, it is smaller than log \frac{6,5}{\sqrt{27}}, we are done. □

Step 3. If ⌈\frac{n}{4}⌉ ≥ 16 (equivalently, n ≥ 61), we have

4 ≤ π (3 ⌈\frac{n}{4}⌉) − π (2 ⌈\frac{n}{4}⌉) ≤ π (3 ⌈\frac{n}{4}⌉) − π (⌈\frac{n}{2}⌉) .

Step 4. We have 3 ≥ |\{p prime | p ≥ ⌈\frac{n}{2}⌉ and p | \frac{|G|}{χ(1)} = ∏_{h∈H(λ)} h\}| .

Proof of Step 4. We count the number of hook-lengths that are at least ⌈\frac{n}{2}⌉.

Denote the hook-length of the i-th row and j-th column by h_{i,j}(λ). Thus h_{i,j}(λ) = h_{j,i}(λ) = λ_{i} + λ_{j} − i − j + 1. Let i_{0} := max{k | λ_{k} ≥ k}. Given this notation, a hook-length which is at least ⌈\frac{n}{2}⌉ occurs only in the first i_{0} rows and first i_{0} columns.

The number of cells in the diagram is n = 2 · ∑_{j=1}^{i_{0}} λ_{j} − i_{0}^{2}. Consider this sum as a function in i, and write f(i) := 2 · ∑_{j=1}^{i} λ_{j} − i^{2}. Then it easy see that,

if i < i_{0}, then f(i) < f(i + 1).

If h_{1,2}(λ) = λ_{1} + λ_{2} − 2 < ⌈\frac{n}{2}⌉, then only the hook-length h_{1,1}(λ) could be at least ⌈\frac{n}{2}⌉.

Assume that h_{1,2}(λ) = λ_{1} + λ_{2} − 2 ≥ ⌈\frac{n}{2}⌉. This means that i_{0} is at least 2 and, if i_{0} > 2, the above monotonicity of f implies

n = 2 · ∑_{j=1}^{i_{0}} λ_{j} − i_{0}^{2} > 2(λ_{1} + λ_{2}) − 4 ≥ 2 ⌈\frac{n}{2}⌉ .

This is a contradiction. Thus i_{0} ≤ 2, and consequently there are at most 3 different hook-lengths that are at least ⌈\frac{n}{2}⌉. Clearly, there cannot be more prime divisors with the desired properties. □

Step 5. We know by Steps 3 and 4 that, if n ≥ 61, there exists a prime p ∈ (⌈\frac{n}{2}⌉ , 3 ⌈\frac{n}{4}⌉] such that p ∤ \frac{|S_{n}|}{χ(1)}, and so for this prime p we have

𝔈_{Co}^{C}(χ) ≥ p(n − p) ≥ p (n − 3 ⌈\frac{n}{4}⌉) ≥ p (⌊\frac{n}{4}⌋ − 2) .

According to [15, Cor. 3.1], we have p(k) > ^{\frac{e^{2\sqrt{k}}}{14}} . Therefore

𝔈_{Co}^{C}(χ) > \frac{e^{2\sqrt{(⌊\frac{n}{4}⌋−2)}}}{14} ≥ \frac{e^{\sqrt{n−11}}}{14} , for n ≥ 61.

Step 6. Since 𝔈_{Co}^{C}(χ) ≥ \frac{e^{\sqrt{n−11}}}{14} and 𝔈_{Ch}^{C}(χ) = ⌈\frac{n−1}{2}⌉ ≤ \frac{n}{2}, it is enough to prove that \frac{n}{2} ≤ \frac{e^{\sqrt{n−11}}}{14} . This inequality holds for n ≥ 44.

Thus, for n ≥ 61, we have

𝔈_{Co}^{C}(χ) ≥ 𝔈_{Ch}^{C}(χ).

Since in 𝔈_{Ch}^{G}(χ) we summed up the sizes of the 𝔈_{Ch}^{C}(χ) smallest different non-central conjugacy classes and in 𝔈_{Co}^{G} we summed up the sizes of 𝔈_{Co}^{C}(χ) different non-central conjugacy classes, the inequality 𝔈_{Ch}^{G}(χ) ≤ 𝔈_{Co}^{G}(χ) holds as well for n ≥ 61. □

Theorem 4.8. There exist only finitely many symmetric groups having a non-linear, irreducible character χ, with 𝔈_{Ch}^{G}(χ) > 𝔈_{Ga}^{G}(χ).

Proof. If n ≤ 4, then 𝔈_{Ch}^{G}(χ) ≤ 𝔈_{Ga}^{G}(χ). Let χ be an irreducible character of S_{n} corresponding to a symmetric partition. Then, by Lemma 3.4, we have 𝔈_{Ch}^{C}(χ) = ⌈\frac{n−1}{2}⌉. Hence 𝔈_{Ch}^{G}(χ) is less than the sum of the sizes of ⌈\frac{n−1}{2}⌉ + 1 ≤ \frac{n+2}{2} different conjugacy classes. Let k be the smallest number having at least \frac{n+2}{2} partitions.

According to [15, Cor. 3.1], we have p(k) > ^{\frac{e^{2\sqrt{k}}}{14}} . Thus,

k ≤ \frac{log^{2}(7n + 14)}{4} + 1.

Let P (k) denote the set of partitions of k. For every λ ∈ P (k), consider the conjugacy class (λ, 1^{n−k}). Let Cl(S_{n}) = {C_{i} | i ∈ I}. Then we have

𝔈_{Ch}^{G}(χ) = min_{\{J⊆I | |J|=⌈ \frac{n−1}{2} ⌉\}} \{∑_{j∈J} |C_{j}|\} − 1 < min_{\{J⊆I | |J|= \frac{n+2}{2} \}} ∑_{j∈J} |C_{j}|,

𝔈_{Ch}^{G}(χ) ≤ min_{{J⊆I | |J|=p(k)}} ∑_{j∈J} |C_{j}| ≤ ∑_{λ∈P (k)} | Cl_{S_{n}}(λ, 1^{n−k})|.

Using the inequality | Cl_{S_{n}}(λ, 1^{n−k})| ≤ \frac{n!}{k!(n−k)!}| Cl_{S_{k}}(λ)|, we get

𝔈_{Ch}^{G}(χ) < \frac{n!}{k!(n − k)!} ∑_{λ∈P (k)} | Cl_{S_{k}}(λ)| = \frac{n!}{(n − k)!} < n^{k}.

Thus

𝔈_{Ch}^{G}(χ) ≤ \frac{n!}{(n − k)!} − 1 < n^{\frac{log^{2}(7n+14)}{4} +1} − 1.

Furthermore, in Corollary 3.9 we have seen that 2^{n/2}−1 ≤ 𝔈_{Ga}^{G}(χ). Since n^{\frac{log^{2}(7n+14)}{4} +1}− 1 ≤ 2^{n/2} − 1 holds for n > 230, the inequality 𝔈_{Ga}^{G}(χ) ≥ 𝔈_{Ch}^{G}(χ) holds for χ ∈ Irr(S_{n}) and n > 230. If χ does not correspond to a symmetric partition, then by Lemma 3.2 we have 𝔈_{Ch}^{C}(χ) = 1. Thus 𝔈_{Ch}^{G}(χ) ≤ | Cl(2, 1^{n−2})| = \frac{n(n−1)}{2}. By Proposition 3.9, we conclude 𝔈_{Ga}^{G}(χ) ≥ n(n − 2). Therefore we are done. □

By the above results concerning the vanishing group elements, we have the following fact.

Remark 4.9. There are infinitely many symmetric groups which have a non-linear, irreducible character such that 𝔈_{Ch}^{G}(χ) < 𝔈_{Ga}^{G}(χ). For example, let n > 4 and χ be the irreducible character of S_{2n+1} corresponding to (2n, 1). Then 𝔈_{Ga}^{G}(χ) = (2n)^{2} − 1 and 𝔈_{Ch}^{G}(χ) ≤ | Cl(2, 1^{2n−1})| = (2n + 1)n.

Theorem 4.10. For each ordered pair from {𝔈_{Ch}^{G}, 𝔈_{Ga}^{G}, 𝔈_{Co}^{G}} there are infinitely many groups, having an irreducible character, for which the second estimate is better then the first, except for the pairs (𝔈_{Ga}^{G}, 𝔈_{Ch}^{G}), (𝔈_{Co}^{G}, 𝔈_{Ch}^{G}).

[figure: 𝔈_{Ch}^{G}( ); 4.9 S_{2n+1} (2n,1) n>4; 4.3; S_{p+2} (p,1^{2}); p>3 prime; 𝔈_{Ga}^{G}( ); 4.8; 4.5; S_{p+2} (p,1^{2}); p>3 prime; 𝔈_{Co}^{G}( ); 4.7; 4.6; S_{n} (n−1,1) n−1>4; not prime]

5. Acknowledgements

I thank my advisor Erzsébet Horváth for proposing this problem to work on, and for all the guidance and support throughout. I also thank Sándor Kiss for the help given in connection with number theory issues. I express my gratitude to Christine Bessenrodt, Mihály Szalay and Attila Maróti for their advice.

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Department of Algebra, Budapest University of Technology and Economics, H-1111 Budapest, Műegyetem rkp. 3–9., Hungary

  E-mail address: petenyi.franciska@gmail.com