Application of character estimates to the number of 𝑇₂-systems of the alternating group

Theorem 1.1.

Let n be sufficiently large. Then we have

Corollary 1.2.

Let n be sufficiently large. Then we have

Remark 1.3.

The lower bound in our theorem is the best result you can achieve with our method. This is because counting T2-systems of An is reduced to counting partitions of n. Upper bounds for the number of T2-systems of An are not known to the author.

Lemma 2.1.

Let G be a finite group with d⁢(G)⩽2. Then, for each T2-system S of G, the set of Aut⁡(G)-conjugates of {[g1,g2]±1} is invariant for all (g1,g2)∈S.

Lemma 2.2.

Let J:-{[π,σ]:(π,σ)∈N2⁢(An)}. For all sufficiently large n, we have

Proof.

Note that (cf. [9, Theorem 8.2A])

Thus we have

Taking into account that d⁢(An)⩽2 (cf. [8]), Lemma 2.1 yields

as desired. ∎

Definition 2.3.

Let C be a conjugacy class of Sn. Denote by f⁢(C) the number of fixed points of elements of C. For a function a:ℕ→ℝ, let Π⁢(a⁢(n)) be the set of all primes p in the range 12⁢n<p⩽a⁢(n).

Theorem 2.4.

Let p∈Π⁢(35⁢n), and let π∈An be a fixed permutation with cycle type (p,n-p-2,1,1) if n is even and with cycle type (p,n-p-2,2) if n is odd. In addition, suppose that C⊂An is a conjugacy class of Sn such that

Then, for all sufficiently large n, we have

Definition 2.5.

A partition of n is a sequence λ=(λ1,λ2,…,λl) of positive integers such that λ1⩾λ2⩾⋯⩾λl and λ1+λ2+⋯+λl=n. We write λ⊢n to indicate that λ is a partition of n. Denote by P⁢(n) the number of partitions of n.

Suppose λ=(λ1,λ2,…,λl)⊢n. The Ferrers diagram of λ is an array of n boxes having l left-justified rows with row i containing λi boxes for 1⩽i⩽l.

Lemma 2.6.

We have

Proof of Theorem 1.1.

First, we note that (see [1, Exercise 44])

Thus we have

Second, choose δ>0 sufficiently small. By Lemma 2.2, Theorem 2.4 and the above inequality, we obtain

Applying Lemma 2.6 yields our assertion. ∎

Lemma 3.1.

Let C1 and C2 be conjugacy classes of Sn, and let τ∈Sn. Then

Proof.

See [6, Proposition 9.33] or [20, Theorem 6.3.1]. ∎

Definition 3.2.

Let λ⊢n be a partition. A rim hookh is an edgewise connected part of the Ferrers diagram of λ, obtained by starting from a box at the right end of a row and at each step moving upwards or rightwards only, which can be removed to leave a proper Ferrers diagram denoted by λ\h. An r-rim hook is a rim hook containing r boxes.

The leg length of a rim hook h is

Let π∈Sn be a permutation with cycle type (1α1,…,qαq,…,nαn) and αq⩾1. Denote by π\q∈Sn-q a permutation with cycle type

Lemma 3.3 (Murnaghan–Nakayama rule).

Let λ⊢n be a partition. Suppose that π∈Sn is a permutation which contains a q-cycle. Then we have

Proof.

See [23, § 9] or [28, Theorem 4.10.2]. ∎

Definition 3.4.

Let λ⊢n be a partition. The hook of (i,j)∈λ is

Here we write (i,j)∈λ to indicate that (i,j) is a box in the Ferrers diagram of λ.

Lemma 3.5 (hook formula).

Let λ⊢n be a partition. Then

Proof.

See [13, Theorem 1] or [28, Theorem 3.10.2]. ∎

Lemma 3.6.

Let χ∈Irr⁡(Sn) be an irreducible character, let n be sufficiently large, and let σ∈Sn be an element with k fixed points. Then we have

Proposition 4.1.

Let p∈Π⁢(58⁢n), and let π∈An be a fixed permutation with cycle type (p,n-p-2,1,1) if n is even and with cycle type (p,n-p-2,2) if n is odd. In addition, suppose that C⊂An is a conjugacy class of Sn such that

Then, for all sufficiently large n, we have

Proof.

Due to the prime number theorem, there exists a prime p∈Π⁢(58⁢n) for all sufficiently large n. Denote by Kπ the conjugacy class of π in Sn. Because of the cycle type of π, Kπ remains a single conjugacy class inside An (cf. [31, pp. 16f.]). Taking into account that χ⁢(τ-1)=χ⁢(τ)¯=χ⁢(τ), it follows with Lemma 3.1 that

The contribution of the linear characters to (4.1) is |C|. This is the expected main term. We shall show that the remaining characters give terms which can be absorbed into the error term. More precisely, we will establish in Lemma 4.9 that

This completes our proof. ∎

Lemma 4.2.

Let p and π be chosen as in Proposition 4.1. Let λ⊢n be a partition with χλ⁢(π)≠0. Then the Ferrers diagram of λ has at most two boxes which do not belong to H1,1⁢(λ)∪H2,2⁢(λ). Moreover, let h be a p-rim hook of λ with χλ\h⁢(π\p)≠0 or an (n-p-2)-rim hook of λ with χλ\h⁢(π\(n-p-2))≠0. Then the Ferrers diagram of λ\h has at most two boxes which do not belong to H1,1⁢(λ\h). In particular, we have |H1,1⁢(λ)|⩾p.

Proof.

The claim follows from the special cycle type of π and the Murnaghan–Nakayama rule (Lemma 3.3). ∎

Lemma 4.3.

Let p and π be chosen as in Proposition 4.1, and let λ⊢n be a partition. Then we have |χλ⁢(π)|⩽2.

Proof.

If χλ⁢(π)=0, there is nothing to show. So let us assume that χλ⁢(π)≠0. Let

We have R≠∅ as a result of the Murnaghan–Nakayama rule. Due to

(see Lemma 4.2), a rim hook h∈R contains boxes of the hook H1,1⁢(λ). Hence

Now let h1∈R. Because of Lemma 4.2, there is exactly one (n-p-2)-rim hook h2 of λ\h1. Obviously, there exists exactly one 2-rim hook of (λ\h1)\h2 and exactly one 1-rim hook of (λ\h1)\h2.

The Murnaghan–Nakayama rule yields that χλ⁢(π) is equal to

Therefore, it follows that |χλ⁢(π)|⩽2. ∎

Definition 4.4.

Let λ⊢n. Define

Lemma 4.5.

Let p and π be chosen as in Proposition 4.1. Let λ⊢n be a partition such that the Ferrers diagram of λ contains the box (2,2) and χλ⁢(π)≠0. Then we have

Furthermore, if |H1,1⁢(λ)|⩾p+3, then there exists an i∈{0,1,2} such that

Proof.

Due to m1+m1′=|H1,1⁢(λ)|-1 the first two formulas are equivalent. In view of Lemma 4.2, we have the three cases |H3,3⁢(λ)|=0, |H3,3⁢(λ)|=1 and |H3,3⁢(λ)|=2. In each case, the hook formula (Lemma 3.5) yields

Hence our first result follows. We now turn to the second statement. It follows from the Murnaghan–Nakayama rule that there is an (n-p-2)-rim hook h of λ such that χλ\h⁢(π\(n-p-2))≠0. By assumption, we have |H1,1⁢(λ)|⩾p+3. Therefore, h contains either boxes from the first column or the first row of the Ferrers diagram of λ – but not both at the same time. Without loss of generality, we assume that the first occurs. With regard to Lemma 4.2, |H2,2⁢(λ\h)|∈{0,1,2}. If |H2,2⁢(λ\h)|=1, we obtain n-p-2=n-|H1,1⁢(λ)|-1+m1-m2 and hence m1=m2+|H1,1⁢(λ)|-p-1. The other cases are analogous. This proves our claim. ∎

Lemma 4.6.

Let p and π be chosen as in Proposition 4.1. Let λ⊢n, where λ∉{(n),(1n)}, be a partition such that χλ⁢(π)≠0 and the Ferrers diagram of λ does not contain the box (2,2).

1. If λ∈{(n-1,1),(2,1n-2)}, we obtain χλ⁢(1)=n-1.

2. If λ∈{(p+1,1n-p-1),(n-p,1p)}, we have χλ⁢(1)=(n-1p).

3. If λ∈{(n-p-1,1p+1),(p+2,1n-p-2)}, it follows χλ⁢(1)=(n-1p+1).

The cases specified above are the only ones that can occur.

Proof.

The computations for χλ⁢(1) follow immediately from the hook formula. Furthermore, the Murnaghan–Nakayama rule yields that there exists a p-rim hook h1 of λ such that there is an (n-p-2)-rim hook of λ\h1. This proves the last statement. ∎

Lemma 4.7.

Let p and π be chosen as in Proposition 4.1. Let λ⊢n be partition such that λ∉{(n),(1n),(n-1,1),(2,1n-2)} and χλ⁢(π)≠0. Then, for all sufficiently large n, we obtain

Proof.

If the Ferrers diagram of λ does not contain the box (2,2), then it follows from Lemma 4.6 that

Now let us assume that the Ferrers diagram of λ contains the box (2,2).

Case 1: |H1,1⁢(λ)|⩽78⁢n. By applying Lemmas 4.5 and 4.2, we get

Case 2: |H1,1⁢(λ)|⩾78⁢n. Without loss of generality, there is an i∈{0,1,2} such that m1=m2+|H1,1⁢(λ)|-p-i (see Lemma 4.5). Because of

we realize that

If 18⁢n⩽m1⩽12⁢(|H1,1⁢(λ)|-1), with Lemma 4.5, we obtain

The case 12⁢(|H1,1⁢(λ)|-1)⩽m1⩽12⁢n is analogous. Hence our result follows. ∎

Lemma 4.8.

Let C be a conjugacy class of Sn. We have χ(n-1,1)⁢(C)=f⁢(C)-1 and |χ(n-1,1)⁢(C)|=|χ(2,1n-2)⁢(C)|.

Proof.

See [28, Example 2.3.8] and [3, Lemma 3.6.10]. ∎

Lemma 4.9.

Let π, C and δ be chosen as in Proposition 4.1, and let n be sufficiently large. Then

Proof.

Let

By applying Lemmas 3.6 and 4.3, we get

We consider separately the two sums. Lemmas 4.2, 4.6 and 4.7 yield

which is far smaller than necessary. Due to Lemmas 4.6 and 4.8, we have

This completes the proof. ∎

Proposition 4.10.

Let p∈Π⁢(58⁢n), and let π∈Sn-1 be a fixed permutation with cycle type (p,n-p-2,1). In addition, suppose that C⊂An-1 is a conjugacy class of Sn-1 such that

Then, for all sufficiently large n, we have

Proposition 4.11.

Let p∈Π⁢(58⁢n), and let π∈Sn-2 be a fixed permutation with cycle type (p,n-p-2). In addition, suppose that C⊂An-2 is a conjugacy class of Sn-2 such that

Then, for all sufficiently large n, we have

Proof.

Apart from one estimate, the arguments are analogous to those in the proof of Proposition 4.1. We will discuss the missing detail below. ∎

Proposition 4.12.

Let π∈Sm be a fixed permutation with cycle type

In addition, suppose that C⊂Am is a conjugacy class of Sm such that

Then, for all sufficiently large m, we have

Proof.

We sketch the argument for the case that our permutation π∈Sm has cycle type (m). The other cases are similar.

Because of the cycle type of π, the Murnaghan–Nakayama rule and the hook formula yield for λ⊢m with χλ⁢(π)≠0 that

1. the Ferrers diagram of λ does not contain the box (2,2),

2. |χλ⁢(π)|=1,

3. χλ⁢(1)=(m-1m1).

Analogous to (4.1), we have

We now use Lemma 3.6 and (i)–(iii) to estimate the error term.