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# Conjugacy growth series for finitary wreath products

- Madeline Locus
^{1}Email authorView ORCID ID profile

**Received:**16 November 2016**Accepted:**15 December 2016**Published:**14 March 2017

The Erratum to this article has been published in Research in Number Theory 2017 3:15

## Abstract

We examine the conjugacy growth series of all wreath products of the finitary permutation groups \(\text {Sym}(X)\) and \(\text {Alt}(X)\) for an infinite set *X*. We determine their asymptotics, and we characterize the limiting behavior between the \(\text {Alt}(X)\) and \(\text {Sym}(X)\) wreath products. In particular, their ratios form a limit if and only if the dimension of the symmetric wreath product is twice the dimension of the alternating wreath product.

## Keywords

- Partitions
- Group theory
- Representation theory

## Mathematics Subject Classification

- 05A17
- 11P81
- 20B30
- 20B35
- 20C32

## 1 Introduction and statement of results

We begin by defining the infinite finitary symmetric and alternating groups and their corresponding wreath products, and then we state our results regarding growth series identities.

*X*, the

*finitary symmetric group*\(\text {Sym}(X)\) is the group of permutations of

*X*with finite support. The

*permutational wreath product*of a group

*H*with \(\text {Sym}(X)\) is the group \(H\wr _X\text {Sym}(X):=H^{(X)}\rtimes \text {Sym}(X)\) defined as follows:

- (i)
The group \(H^{(X)}\) is the group of functions from

*X*to*H*with finite support. - (ii)The action of permutations \(f\in \text {Sym}(X)\) on functions \(\psi \in H^{(X)}\) is defined by$$\begin{aligned} \psi \mapsto f(\psi ):=\psi \circ f^{-1}. \end{aligned}$$
- (iii)Multiplication in the semi-direct product is defined for \(\varphi ,\psi \in H^{(X)}\) and \(f,g\in \text {Sym}(X)\) by$$\begin{aligned} (\varphi ,f)(\psi ,g)=(\varphi f(\psi ),fg). \end{aligned}$$

*finitary alternating group*\(\text {Alt}(X)\) is the subgroup of \(\text {Sym}(X)\) of permutations with even signature, and the permutational wreath product \(H\wr _X\text {Alt}(X)\) is defined as above. In this paper, we only consider permutational wreath products with finite group

*H*. We now define some general terminology. For any group

*G*generated by a set

*S*, the

*word length*\(\ell _{G,S}(g)\) of any element \(g\in G\) is the smallest nonnegative integer

*n*such that there exist \(s_1,\dots ,s_n\in S\cup S^{-1}\) with \(g=s_1\cdots s_n\). The

*conjugacy length*\(\kappa _{G,S}(g)\) is the smallest word length appearing in the conjugacy class of

*g*. If

*n*is any natural number, we denote by \(\gamma _{G,S}(n)\in \mathbb {N}\cup \{0\}\cup \{\infty \}\) the number of conjugacy classes in

*G*with smallest word length

*n*. If \(\gamma _{G,S}(n)\) is finite for all

*n*, then we may define the conjugacy growth series of a group

*G*with generating set

*S*to be the following

*q*-series:

*G*. Bacher and de la Harpe [1] prove conjugacy growth series identities for sufficiently large

^{1}generating sets

*S*of \(\text {Sym}(X)\), \(S'\) of \(\text {Alt}(X)\), and \(S^{\left( W_S\right) }\) of \(W_S=H_S\wr _X\text {Sym}(X)\) relating the finitary permutation groups and their wreath products to the partition function. Explicitly, we have the fascinating identities

^{2}and

### Remark

*q*-series. Here we obtain a universal recurrence for these numbers. This result requires the ordinary divisor function \(\sigma _k(n)=\sum _{d\mid n}d^k\). We also must define, for \(n\ge 2\), the polynomial

### Remark

*n*; the first few are listed below.

### Remark

These polynomials have been used in earlier work [2, 4] on divisors of modular forms and the Rogers–Ramanujan identities.

### Theorem 1

*X*an infinite set, and \(W_S=H_S\wr _XSym (X)\) a wreath product generated by a sufficiently large set \(S^{\left( W_S\right) }\). Then we have

*L*. The hook length of a partition \(\lambda =(\lambda _1,\dots ,\lambda _n)\vdash L\) is defined using the Ferrers diagram of \(\lambda \). For example, Fig. 1 below is a Ferrers diagram of the partition \(\lambda =(6,4,3,1,1)\vdash 15\), Fig. 2 represents a hook length of 4, and Fig. 3 shows all hook lengths associated to \(\lambda \).

*v*in the Ferrers diagram of a partition \(\lambda \), its

*hook length*\(h_v(\lambda )\) is defined as the number of boxes

*u*such that

- (i)
\(u=v\),

- (ii)
*u*is in the same column as*v*and below*v*, or - (iii)
*u*is in the same row as*v*and to the right of*v*.

*hook length multi-set*\(\mathcal {H}(\lambda )\) is the set of all hook lengths of \(\lambda \). Theorem 1 implies the following formula for \(\gamma _{W_S}(n)\) in terms of hook lengths.

### Corollary 2

### Remark

Kostant observed [6] that the coefficients of the Nekrasov-Okounkov hook length identity are polynomials in the variable \(z=1-M_S\), but he did not give an explicit formula for computing them.

^{3}the corresponding growth series identity for the wreath product \(W_A=H_A\wr _X\text {Alt}(X)\) with sufficiently large generating set \(S^{\left( W_A\right) }\), namely

In analogy with Theorem 1, one may ask if the coefficients \(\gamma _{W_A}(n)\) in the alternating case can be seen as a function of the number of conjugacy classes \(M_A\). We obtain a similar recurrence relation in this case.

### Theorem 3

*X*an infinite set, and \(W_A=H_A\wr _XAlt (X)\) a wreath product generated by a sufficiently large set \(S^{\left( W_A\right) }\). Then we have

### Remark

It may be possible to interpret the coefficients \(\gamma _{W_A}(n)\) in terms of hook lengths from formulas of Han [5] or others, as in the symmetric case. The author does not make this connection here.

*modified exponential rate of conjugacy growth*

^{4}of a group

*G*generated by a set

*S*, namely

*X*is an infinite set. It is easy to see from Eq. (1.3) that the conjugacy growth series of such a wreath product is the generating function of the generalized partition function \(p(n)_\mathbf e \) for the vector \(\mathbf e =(M_S)\). This implies the following corollary.

### Corollary 4

*X*is an infinite set. If \(S^{\left( W_S\right) }\) is a sufficiently large generating set of \(W_S\), then we have

We now give the modified exponential rate of conjugacy growth for wreath products in the symmetric case using this asymptotic formula.

### Corollary 5

We can also apply the theorem to wreath products in the alternating case using Eq. (1.4).

### Corollary 6

*X*is an infinite set. If \(S^{\left( W_A\right) }\) is a sufficiently large generating set of \(W_A\), then we have

We also give the modified exponential rate of conjugacy growth in the alternating case using the above asymptotic formula.

### Corollary 7

We are interested in finding relationships between wreath products of \(\text {Sym}(X)\) and wreath products of \(\text {Alt}(X)\). Let \(W_S=H_S\wr _X\text {Sym}(X)\) and \(W_S'=H_S'\wr _X\text {Sym}(X)\) be two wreath products of \(\text {Sym}(X)\), where \(H_S,H_S'\) are finite groups with \(M_S,M_S'\) conjugacy classes respectively. Let \(W_A=H_A\wr _X\text {Alt}(X)\) and \(W_A'=H_A'\wr _X\text {Alt}(X)\) be two wreath products of \(\text {Alt}(X)\), where \(H_A, H_A'\) are finite groups with \(M_A, M_A'\) conjugacy classes respectively.

### Question 1

The asymptotic behavior of the ratios follows from Corollaries 4 and 6.

### Corollary 8

We now observe for which pairs \((M_S,M_S'),(M_S,M_A),(M_A,M_S),\) and \((M_A,M_A')\) these ratios asymptotically approach zero, infinity, or some nonzero finite number. Corollary 9 follows from the asymptotic behavior of the exponential functions in Corollary 8.

### Corollary 9

- (1)
If \(M_S<M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim 0\). If \(M_S>M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim \infty \). If \(M_S=M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim 1\).

- (2)
If \(M_S<2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim 0\). If \(M_S>2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim \infty \). If \(M_S=2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim 2^{M_A}\).

- (3)
If \(2M_A<M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim 0\). If \(2M_A>M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim \infty \). If \(2M_A=M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim \frac{1}{2^{M_A}}\).

- (4)
If \(M_A<M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim 0\). If \(M_A>M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim \infty \). If \(M_A=M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim 1\).

Given any two wreath products of \(\text {Sym}(X)\) or \(\text {Alt}(X)\), Corollary 9 guarantees the asymptotic behavior of the ratios between the coefficients of their conjugacy growth series. In other words, for any two wreath products *W* and \(W'\), we know the expected relationship between the number of conjugacy classes of *H* in *W* and the number of conjugacy classes of \(H'\) in \(W'\) with minimal word length *n* for any *n*.

### Remark

Although we know the asymptotic behavior of the above ratios, this does not mean that the ratios of the coefficients are always exactly equal to the above values.

| \(\gamma _{W_S}(n)\) | \(\gamma _{W_A}(n)\) | \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\) |
---|---|---|---|

1 | 10 | 5 | 2 |

10 | 1605340 | 176963 | 9.071613840 |

100 | \(0.2333013623\times 10^{28}\) | \(0.7541087996\times 10^{26}\) | 30.93736108 |

200 | \(0.1067904403\times 10^{42}\) | \(0.3346942881\times 10^{40}\) | 31.90686071 |

300 | \(0.4721905614\times 10^{52}\) | \(0.1476229954\times 10^{51}\) | 31.98624714 |

400 | \(0.5248644122\times 10^{61}\) | \(0.1640339890\times 10^{60}\) | 31.99729613 |

500 | \(0.5369981415\times 10^{69}\) | \(0.1678152777\times 10^{68}\) | 31.99935959 |

## 2 Proofs

We give the proofs of Eq. (1.4) and Theorems 1 and 3 here. We also explain what it means for a generating set to be sufficiently large and give remarks on Corollaries 2 and 6.

*S*of transpositions of a set

*X*is called

*partition-complete*(

*PC*) [1] if

- (i)
The transposition graph \(\Gamma (S)\) is connected, and

- (ii)
For every partition \(\lambda =(\lambda _1,\dots ,\lambda _k)\vdash L\), \(\Gamma (S)\) contains a forest of

*k*trees with \(\lambda _1+1,\dots ,\lambda _k+1\) vertices respectively.

*partition-complete for wreath products*(

*PCwr*) [1], we must first establish more notation. Let

*X*be an infinite set,

*H*a group, and \(W=H\wr _X\text {Sym}(X)\). The group

*W*acts naturally on the set \(H\times X\); namely, for \((\varphi ,f)\in W\), the action is defined by

*ah*,

*u*) if \(x=u\), and to (

*h*,

*x*) otherwise. Then \((\varphi _u^a)_{a\in H\setminus \{1\},\,u\in X}\) generates the group \(H^{(X)}\). Now, let \(H_u:=\{\varphi _u^a\mid a\in H\setminus \{1\}\}\), and define the subsets

*S*is said to be

*PCwr*if

- (i)
The transposition graph \(\Gamma (S_X)\) is connected, and

- (ii)
For all \(L\ge 0\) and partitions \(\lambda =(\lambda _1,\dots ,\lambda _k)\vdash L\), \(\Gamma (S_X)\) contains a forest of

*k*trees \(T_1,\dots ,T_k\), with \(T_i\) having \(\lambda _i\) vertices, including one vertex \(x^{(i)}\) such that \(\varphi _{x^{(i)}}^a\in S_H\) for all \(a\in H{\setminus }\{1\}\).

### Remark

The conditions *PC* and *PCwr* essentially require the generating set *S* to contain “enough” transpositions to represent all possible partitions in its transposition graph.

### Proof of equation (1.4)

^{5}from the proofs of eqs. (1.2) and (1.3) in [1]. For each \(w=(\phi , \sigma )\in W_A = H_A \wr _{X} \text {Alt}(X)\), we can split \(\sigma \) into a product of an even number of cycles of even length, denoted \(\sigma _{e}\), and a product of cycles of odd length, denoted \(\sigma _{o}\), so that \(w = (\phi , \sigma _{e} \sigma _{o})\). Let \({(H_A)}_{*}\) denote the set of conjugacy classes of \(H_A\); we write \(1\in {(H_A)}_{*}\) for the class \(\{1\}\in H_A\). To each conjugacy class in \(W_A\) we associate an \({(H_A)}_{*}\)-indexed family of partitions. Using the same notation as in [1], we associate the conjugacy classes in \(H_A\) to the family of partitions

*X*that is the union of the supports of \(\phi \) and \(\sigma \). Let \(\sigma \) be the product of the disjoint cycles \(c_{1}, ..., c_{k}\), where \(c_{i} = \left( x_{1}^{(i)}, x_{2}^{(i)}, ..., x_{v_{i}}^{(i)}\right) \) with \(x_{j}^{(i)} \in X^{(w)}\) and \(v_{i} = \text { length}(c_{i})\). We include cycles of length 1 for each \(x \in X\) such that \(x \in \text {sup}(\phi )\) and \(x \notin \text {sup}(\sigma )\), so that

*c*in \(\{c_{1}, ..., c_{k} \}\) such that \(\text {length}(c)=\ell \) and \(\eta _{*}^{w}(c) = \eta \). Let \(\mu ^{w, \eta } \vdash n^{w, \eta }\) be the partition with \(m_{\ell }^{w, \eta }\) parts equal to \(\ell \), for all \(\ell \ge 1\). Note that

*type*of

*w*to be the family \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*} \setminus 1} \right) \). Then two elements in \(W_A\) are conjugate if and only if they have the same type. Thus, each \({(H_A)}_{*}\)-indexed family of partitions \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*} \setminus 1} \right) \) is the type of one conjugacy class in \(W_A\).

Consider an \({(H_A)}_{*}\)-indexed family of partitions \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*}\setminus 1} \right) \) and the corresponding conjugacy class in \(W_A\). Let \(u^{(1)}, v^{(1)}, u^{(\eta )}, v^{(\eta )}\) be the sums of the parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) respectively, and let \(k^{(1)}, t^{(1)}, k^{(\eta )}, t^{(\eta )}\) be the number of parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) respectively.

*generalized partition function*\(p(n)_\mathbf e \) is defined for the vector \(\mathbf e =(e_1,\dots ,e_k)\in \mathbb {Z}^k\) by its generating function

^{6}

**Theorem**(Cotron-Dicks-Fleming [3]) Let \(\mathbf e =(e_1,\dots ,e_k)\) be any nonzero vector with nonnegative integer entries, and let \(d:=\gcd \{m:e_m\ne 0\}\). Define the quantities

### A Remark on Corollary 6

*k*in the exponential function is negative. Therefore, the above sum is asymptotic to the \(k=0\) term, so we have

*q*-series. Their proof surprisingly only requires properties of logarithmic derivatives applied to a

*q*-series infinite product identity. The proof below is adapted from the proof in [4] and can be applied to any

*q*-series infinite product identity, including the famous identity of Nekrasov and Okounkov [7].

### Proof of Theorem 1

*q*-series identity

^{7}

*r*, and we ignore its implications for finite groups. Then the above proof also applies to the coefficients of the Nekrasov-Okounkov hook length formula [7]

### Proof of Theorem 3

*q*-series identity

### Remark

This recurrence relation gives the coefficients \(\gamma _{W_A}(n)\) in terms of the coefficients \(a_k(1)\), \(\dots \), \(a_k(n-1)\) of each summand. Since the linear combination of infinite products is raised to the \((M_A)\)th power in the conjugacy growth series, presumably there is no simple way to obtain a recurrence relation for \(\gamma _{W_A}(n)\) in terms of \(\gamma _{W_A}(1),\dots ,\gamma _{W_A}(n-1)\) as in the symmetric case.

### Acknowledgements

The author would like to thank Ken Ono for his invaluable advice and guidance on this project, and also Pierre de la Harpe and the referee for their helpful comments.

The condition that the generating sets are sufficiently large refers to the properties defined in Sect. 2.

Cotron, Dicks, and Fleming [3] modify Bacher’s and de la Harpe’s definition [1] by changing the denominator from *n* to \(\sqrt{n}\). With denominator *n*, most of the growth series that we study have exponential rate of conjugacy growth zero.

A careful description of the conjugacy classes of wreath products can also be found in Macdonald’s 1995 book, *Symmetric Functions and Hall Polynomials*.

## Notes

## Declarations

### Open Access

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## Authors’ Affiliations

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