Research ContributionArnold Mathematical Journal

Received: 10 August 2016 / Revised: 27 December 2016 / Accepted: 1 February 2017

Polynomial Splitting Measures and Cohomology of the Pure Braid Group
tghyde@umich.edu
Department of Mathematics, University of Michigan, Ann Arbor MI 48109-1043 USA
and lagarias@umich.edu
Department of Mathematics, University of Michigan, Ann Arbor MI 48109-1043 USA
###### Abstract
We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\FF_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \QQ)$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations).
###### Keywords
Symmetric groups, Braid group, Configuration space
###### Mathematics Subject Classification
Primary 11R09, Secondary 11R32, 12E20, 12E25

## 1. Introduction

The purpose of this paper is to study for each $n \ge 1$ a one-parameter family of complex-valued measures on the symmetric group $S_n$ arising from a problem in number theory, and to exhibit an explicit representation-theoretic connection between these measures and the characters of the natural $S_n$-action on the rational cohomology of the pure braid group $P_n$.

This family of measures, denoted $\nu_{n, z}^{\ast}$, was introduced by [18], where they were called $z$ -splitting measures , with parameter $z$. The measures interpolate from prime power values $z=q$ the probability of a monic, degree $n$, square-free polynomial in $\FF_q[x]$ having a given factorization type. Square-free factorization types are indexed by partitions $\lambda$ of $n$ specifying the degrees of the irreducible factors. Each partition $\lambda$ of $n$ corresponds to a conjugacy class $C_{\lambda}$ of the symmetric group $S_n$; distributing the probability of a factorization of type $\lambda$ equally across the elements of $C_{\lambda}$ defines a probability measure on $S_n$. A key property of the resulting probabilities is that their values for each fixed partition $\lambda$ are described by a rational function in the size of the field $\FF_q$ as $q$ varies. This property permits interpolation from $q$ to a parameter $z \in \PP^{1}(\CC)$ on the Riemann sphere, to obtain a family of complex-valued measures $\nu_{n, z}^{\ast}$ on $S_n$ given in Definition 2.3 below.

On the number theory side, these measures connect with problems on the splitting of ideals in $S_n$-number fields, which are degree $n$ number fields formed by adjoining a root of a degree $n$ polynomial over $\ZZ[x]$ whose splitting field has Galois group $S_n$. The paper ([18], Theorem 2.6) observed that for primes $p < n$ these measures vanish on certain conjugacy classes, corresponding to the phenomenon of essential discriminant divisors of polynomials having Galois group $S_n$, first noted by [8]. These measures converge to the uniform measure on the symmetric group as $z= p \to \infty$, and in this limit agree with a Conjecture 1.3 of [2] on the distribution of splitting types of the prime $p$ in $S_n$-extensions of discriminant $|D| \le B$ as the bound $B \to \infty$, conditioned on $(D, p)=1$.

The second author subsequently studied these measures interpolated at the special value $z = 1$, viewed as representing splitting probabilities for polynomials over the (hypothetical) "field with one element $\FF_1$" ([17]). These measures, called $1$ -splitting measures , turn out to be signed measures for all $n \ge 3$. They are supported on a small set of conjugacy classes, the Springer regular elements of $S_n$ which are those conjugacy classes $C_{\lambda}$ for which $\lambda$ has a rectangular Young diagram or a rectangle plus a single box. Treated as class functions on $S_n$, rather than as measures, they were found to have a representation-theoretic interpretation: after rescaling by $n!$, the 1-splitting measures are virtual characters of $S_n$ corresponding to explicitly determined representations. As $n$ varies, their values on conjugacy classes were observed to have arithmetic properties compatible with the multiplicative structure of $n$; letting $n = \prod_{p} p ^{e_p}$ be the prime factorization of $n$, the value of the measure on each conjugacy class factors as a product of values on classes of smaller symmetric groups $S_{p^{e_p}}$. That paper also showed the rescaled $z$-splitting measures at $z=-1$ have a related representation-theoretic interpretation.

In this paper we extend the representation-theoretic interpretation to the entire family of $z$-splitting measures and relate it to the cohomology of the pure braid group. Our starting point is the observation made in Lemma 2.5 of [17] that for a fixed conjugacy class the $z$-splitting measures are Laurent polynomials in $z$. They have degree at most $n-1$, so may be written \begin{eqnarray*} \nu_{n, z}^{\ast} (C_{\lambda}) = \sum_{k=0}^{n-1} \alpha_n^k(C_{\lambda})\left(\tfrac{1}{z}\right)^k, \end{eqnarray*} with rational coefficients $\alpha_n^k(C_{\lambda})$, where $\lambda$ is a partition of $n$. We call the $\alpha_n^k(C_{\lambda})$ splitting measure coefficients. A main observation of this paper is that each splitting measure coefficient $\alpha_n^k(C_{\lambda})$, viewed as a function of $\lambda$, is a rescaled character $\chi_n^k$ of a certain $S_n$-subrepresentation $A_n^k$ of the cohomology of the pure braid group $H^k(P_n, \QQ)$. The pure braid groups $P_n$ and their cohomology, along with the subrepresentations $A_n^k$, are defined and discussed in Sect. 4. In Sect. 4.3 we identify the $S_n$-representation $A_n^k$ with the cohomology of a complex manifold $Y_n$ carrying an $S_n$-action. We deduce as a consequence a topological interpretation of the $1$-splitting measure as a rescaled version of the $S_n$-equivariant Euler characteristic of $Y_n$. We also deduce that the rescaled $z$-splitting measure is a character of $S_n$ at $z = -\frac{1}{m}$ and is a virtual character of $S_n$ at $z = \frac{1}{m}$, for all integers $m\geq 1$.

The last result extends the representation-theoretic connection of [17] for $z= \pm 1$ to parameter values $z= \pm \frac{1}{m}$ for all $m \ge 1$.

### 1.1. Results

a

The $z$ -splitting measure on a conjugacy class $C_{\lambda}$ of $S_n$ is the rational function of $z$ \begin{eqnarray*} \nu_{n, z}^{\ast} (C_{\lambda}) := \frac{N_{\lambda}(z)}{ z^n - z^{n-1}}, \end{eqnarray*} where $N_{\lambda}(z) \in \QQ[z]$ denotes the cycle polynomial associated to a partition $\lambda$ describing the cycle lengths of $C_\lambda$. Given $\lambda= \left(1^{m_1(\lambda)} 2^{m_2(\lambda)} \cdots n^{m_n(\lambda)} \right)$, the associated cycle polynomial is

 \begin{eqnarray}\label{eq:11} N_{\lambda}(z) := \prod_{j\geq 1} {{M_j(z)}\choose{m_j(\lambda)}}, \end{eqnarray} (1.1)
where $M_j(z)$ denotes the $j$th necklace polynomial. The necklace polynomial $M_j(z)$ of order $j$ is given by \begin{eqnarray*} M_j(z) := \frac{1}{j} \sum_{d \mid j} \mu(d) z^{j/d}. \end{eqnarray*} where $\mu(d)$ is the M[U+00A8]obius function.

To avoid confusion we make a remark on values of measures. Given a class function $f$ on $S_n$ we write $f(C_{\lambda})$ to mean the sum of the values of $f$ on $C_\lambda$, and write $f(\lambda)$ to mean the value $f(g)$ taken at one element $g \in C_{\lambda}$; the latter notation is standard for characters. Thus $\nu_{n,z}^{\ast}(C_{\lambda}) = |C_\lambda| \nu_{n, z}^{\ast}(\lambda)$.

In Sect. 3 we express the coefficients of the family of cycle polynomials $N_\lambda(z)$ in terms of characters of the cohomology of the pure braid group $P_n$ viewed as an $S_n$-representation.

###### Theorem 1.1.
(Character interpretation of cycle polynomial coefficients) Let $\lambda$ be a partition of $n$ and $N_\lambda(z)$ be a cycle polynomial. Then \begin{eqnarray*} N_\lambda(z) = \frac{|C_{\lambda}|}{n!}\sum_{k=0}^n{(-1)^k h_n^k(\lambda) z^{n-k}}. \end{eqnarray*} where $h_n^k$ is the character of the $k$th cohomology of the pure braid group $H^k(P_n,\QQ)$, viewed as an $S_n$-representation.

Theorem 1.1 is a rescaled version of a result of Theorem 5.5 in [19]. Lehrer arrived at it from his study of the Poincaré polynomials associated to the elements of a Coxeter group acting on the complements of certain complex hyperplane arrangements. We arrived at it through a direct study of the cycle polynomial $N_{\lambda}(z)$ appearing in the definition of the $z$-splitting measure, relating it to representation stability using the twisted Grothendieck--Lefschetz formula of Prop. 4.1 in [6]. We include a proof of Theorem 1.1 (as Theorem 3.2); the method behind this proof also traces back to work of [20].

At the end of Sect. 3 we apply Theorem 1.1 together with the formula (1.1) for $N_{\lambda}(z)$ to obtain explicit expressions for various characters $h_n^k$ showing number-theoretic structure, and to determine restrictions on the support of various $h_n^k$.

In Sect. 4 we review Arnol'd's presentation of the cohomology ring of the pure braid group. In Sect. 4.2 we use it derive an exact sequence determining certain $S_n$-subrepresentations $A_n^k$ of $H^k(P_n,\QQ)$ which play the main role in our results. These subrepresentations lead to a direct sum decomposition $H^k(P_n, \QQ) \simeq A_n^{k-1}\oplus A_n^k$, for each $k \ge 0$. In Sect. 4.3 we interpret the $A_n^k$ as the cohomology of an $(n-1)$-dimensional complex manifold $Y_n$ that carries an $S_n$-action. The manifold $Y_n$ is the quotient of the pure configuration space $\pconf_n(\CC)$ of $n$ distinct (labeled) points in $\CC$ by a free action of $\CC^{\times}$.

The main result of this paper, given in Sect. 5, expresses the $z$-splitting measures $\nu_{n, z}^{\ast}$ in terms of the characters $\chi_n^k$ of the $S_n$-representations $A_n^k$.

###### Theorem 1.2.
(Character interpretation of splitting measure coefficients) For each $n \ge 1$ and $0 \le k \le n-1$ there is an $S_n$-subrepresentation $A_n^k$ of $H^k(P_n,\QQ)$ $($constructed explicitly in Proposition 4.2) with character $\chi_n^k$ such that for each partition $\lambda$ of $n,$ \begin{eqnarray*} \nu_{n,z}^\ast(C_\lambda) = \frac{|C_{\lambda}|}{n!} \sum_{k=0}^{n-1}\chi_n^k(\lambda)\left(-\tfrac{1}{z}\right)^k. \end{eqnarray*} Thus the splitting measure coefficient $\alpha_n^k(C_{\lambda})=|C_{\lambda}| \,\alpha_n^k(\lambda)$ is given by \begin{eqnarray*} \alpha_n^k(C_{\lambda}) =(-1)^k \frac{|C_{\lambda}|}{n!} \chi_n^{k}(\lambda). \end{eqnarray*}

In Sect. 5.2 we interpret this result in terms of cohomology of the manifold $Y_n$. On setting $t = -\frac{1}{z}$, we have that for each $g \in S_n$, \begin{eqnarray*} \nu_{n, z}^{\ast}(g) = \frac{1}{n!} \sum_{k=0}^{n-1} \tr(g, H^k(Y_n, \QQ)) t^k, \end{eqnarray*} which is a value of the equivariant Poincaré polynomial for $Y_n$ with respect to the $S_n$-action (Theorem 5.2). In particular we obtain the following topological interpretation of the $1$-splitting measure, as the special case $t=-1$.

###### Theorem 1.3.
(Topological interpretation of $1$-splitting measure) Let $Y_n$ denote the open complex manifold $\pconf_n(\CC)/\CC^{\times},$ which carries an $S_n$-action under permutation of the $n$ points. Then the rescaled $1$-splitting measure $\nu^*_{n,1}(\cdot)$ evaluated at elements $g \in S_n$ is the equivariant Euler characteristic of $Y_n,$ \begin{eqnarray*} \nu^*_{n,1}(g) = \frac{1}{n!}\sum_{k=0}^{n-1}{ (-1)^k \tr(g, H^k( Y_n, \QQ))}, \end{eqnarray*} with respect to its $S_n$-action.

In Sect. 5.3 we obtain another corollary of Theorem 1.2. For $z = -\frac{1}{m}$ with $m \ge 1$, the rescaled splitting measure $\frac{n!}{|C_\lambda|}\nu_{n,z}^{k} (C_\lambda)$ is the character of an $S_n$-representation, and when $z= \frac{1}{m}$ it is the character of a virtual $S_n$-representation (Theorem 5.3).

In Sect. 5.4 we deduce an interesting consequence concerning the $S_n$-action on the full cohomology ring $H^{\bullet}(P_n, \QQ)$. The structure of the cohomology ring of the pure braid group $H^{\bullet}(P_n, \QQ)$ as an $S_n$-module has an extensive literature. [26] noted that $H^{\bullet}(P_n, \QQ) \simeq H^{\bullet} (M(\sA_n), \QQ)$ as $S_n$-modules, where \begin{eqnarray*} M(\sA_n) = \CC^n \smallsetminus \cup_{H \in \sA_n} H \end{eqnarray*} is the complement of the (complexified) braid arrangement $\sA_n$, i.e. the arrangement of $n(n-1)/2$ hyperplanes $z_i = z_j$ in $\CC^n$ where $1\leq i < j \leq n$ are the coordinate functionals of $\CC^n$. The structure of the cohomology groups $H^k(M(\sA_n), \CC) = H^k(M(\sA_n),\QQ)\otimes \CC$ as $S_n$-representations was determined in 1986 by [21], Theorem 4.5 in terms of induced representations $\Ind_{Z(C_\lambda)}^{S_n}(\xi_{\lambda})$ for specific linear representations $\xi_{\lambda}$ on the centralizers $Z(C_{\lambda})$ of conjugacy classes $C_{\lambda}$ having $n-k$ cycles. In 1987 [19], p. 276 noted that his results on Poincaré polynomials implied the "curious consequence" that the action of $S_n$ on $\bigoplus_k H^{k}(M(\sA_n, \CC))$ is "almost" the regular representation in the sense that the dimension is $n!$ and the character $\theta(g)$ of this representation is $0$ unless $g$ is the identity element or a transposition, see also [19], Corollary (5.5)', Prop. (5.6). where $r$ is a reflection and $1$ is the trivial representation. In Sect. 5.4 we apply Theorem 1.2 together with values of the $(-1)$-splitting measure computed in [17] to make a precise connection between the $S_n$-representation structure on pure braid group cohomology and the regular representation $\QQ[S_n]$.

###### Theorem 1.4.
Let $\triv_n,$ $\Sign_n,$ and $\QQ[S_n]$ be the trivial$,$ sign$,$ and regular representations of $S_n$ respectively. Then there is an isomorphism of $S_n$-representations$,$ \begin{eqnarray*} \bigoplus_{k=0}^n{H^k(P_n,\QQ)\otimes \Sign_n^{\otimes k}} \cong \QQ[S_n]. \end{eqnarray*} Here $\Sign_n^{\otimes k}\cong \triv_n$ or $\Sign_n$ according to whether $k$ is even or odd.

When combined with [19], Prop. 5.6 (i) determination of the character $\theta$ as $2 \,\Ind_{\langle \tau \rangle}^{S_n}(1)$, where $\tau$ is a transposition, this result implies that each of the characters of the $S_n$-representations acting on the even-dimensional cohomology, resp. odd-dimensional cohomology are supported on the identity element plus transpositions. We comment on other related work in Sect. 1.2.

In Sect. 6 we describe further interpretations of the representations $A_n^k$ in terms of other combinatorial homology theories. For fixed $k$ and varying $n$, the sequence of $S_n$-representations $H^{k} (P_n, \QQ)$ was one of the basic examples exhibiting representation stability in the sense of [5], see [6], [7]. We show in Proposition 6.2 that the representations $A_n^k$ are isomorphic to others appearing in the literature known to exhibit representation stability. [15], Corollary 5.4 determine the precise rate of stabilization of these representations, yielding the following result.

###### Theorem 1.5.
(Representation stability for $A_n^k$) For each fixed $k \ge 1,$ the sequence of $S_n$-representations $A_n^k$ with characters $\chi_n^k$ are representation stable$,$ and stabilize sharply at $n=3k+1$.

To summarize these results:

1. (i) We start from a construction in number theory: a set of probability measures on $S_n$ that describe the distribution of degree $n$ squarefree monic polynomial factorizations $\pmod{p}$ defined for a parameter $z$ being a prime $p$. These measure values interpolate at each fixed $g \in S_n$ in the $z$-variable as polynomials in $1/z$ to define complex-valued measures on $S_n$.
2. (ii) We make a connection of the interpolated measures as functions of $z$ to topology and representation theory: For fixed $n$ the $k$th Laurent coefficients of the $z$-parametrization at $g \in S_n$ (rescaled by $n!$) coincide with the character of an $S_n$-subrepresentation $A_n^k$ of the cohomology of the pure braid group $P_n$, which is an $S_n$-representation on the cohomology of the complex manifold $Y_n= \pconf_n(\CC) / \CC^{\times}$. As $n$ varies with $k$ fixed these coefficients exhibit representation stability as $n \to \infty.$
3. (iii) We deduce that (rescaled) measure values at values $z= -\frac{1}{m}$ for $m \ge 1$ coincide with characters of certain $S_n$-representations; those at $z= \frac{1}{m}$ with $m \ge 1$ coincide with certain virtual $S_n$-representations. For each $n$ these representations combine stable and unstable cohomology of $P_n$.
4. (iv) As a by-product we find a precise connection between the (total) cohomology of the pure braid group as an $S_n$-representation and the regular representation of $S_n$.
The main observation of this paper is the relation of these interpolation measures to representation theory. We demonstrate this relation by calculation, and leave open the problem of finding a deeper conceptual explanation for its existence.

### 1.2. Related work.

The representations $A_n^k$ have appeared in the literature in numerous places. In particular, a 1995 result of [13], Corollary 3.10 permits an identification of $A_n^k$ as an $S_n$-module with the $k$th cohomology group of the moduli space $\sM_{0, n+1}$ of the Riemann sphere with $n+1$ marked points, viewed as an $S_n$-module, holding one point fixed. Getzler identifies this cohomology with the $S^1$-equivariant cohomology of $\pconf_n(\CC)$, which is the cohomology of $Y_n$ given in Theorem 5.2. Some more recent occurrences of $A_n^k$ are discussed in Sect. 6.

In connection with Theorem 1.4, in [12] further explained Lehrer's formula $\theta= 2 \,\Ind_{\langle \tau \rangle}^{S_n}(1)$ by showing that \begin{eqnarray*} H^{\bullet}( M(\sA_{n-1}), \CC) \simeq H^{\bullet}( M(d\sA_{n-1}), \CC) \otimes \left(\CC \oplus \tfrac{\CC[\varepsilon]}{ \varepsilon^2}\right), \end{eqnarray*} as $S_n$-modules, where $d\sA_{n-1}$ is obtained by a deconing construction, while the class $\varepsilon$ has degree $1$ and carries the trivial $S_n$-action. (His space $M(\sA_{n-1})$ lies in $\CC^{n-1}$ and is obtained by restricting the braid arrangement on $\CC^n$ to the hyperplane $x_1+ x_2 + \cdots + x_n=0$ in $\CC^{n}$, and the deconed configuration space $M(d\sA_{n-1}) \subset \CC^{n-2}$.) On comparison with our direct sum decomposition we have $H^k(d \sA_{n-1}, \CC) \simeq A_n^k$ as $S_n$-modules, showing that the deconed space $d\sA_{n-1}$ has an isomorphic cohomology ring as the complex manifold $Y_n$ with an appropriate $S_n$-module structure. Gaiffi and also [23] showed there is a "hidden" $S_{n+1}$-action on this cohomology ring. For more recent developments on the "hidden" action see [3].

### 1.3. Plan of the Paper.

In Sect. 2 we recall properties of the $z$-splitting measures from [18]. In Sect. 3 we use the twisted Grothendieck--Lefschetz formula to relate the coefficients of cycle polynomials to the characters of the $S_n$-representations $H^k(P_n,\QQ)$. In Sect. 4 we discuss the cohomology $H^k(P_n, \QQ)$ of the pure braid group $P_n$, and derive an exact sequence leading to the construction of the $S_n$-representations $A_n^k$. In Sect. 5 we express the splitting measure coefficients $\alpha_n^k(C_\lambda)$ in terms of the character $\chi_n^k$ of the representation $A_n^k$. In Sect. 6 we discuss representation stability and connect the $S_n$-representations $A_n^k$ with others in the literature.

### 1.4. Notation.

1. 1. $q=p^f$ denotes a prime power.
2. 2. The set of monic, degree $n$, square-free polynomials in $\FF_q[x]$ is denoted $\conf_n(\FF_q)$.
3. 3. We write partitions either as $\lambda =[\lambda_1, \lambda_2, \ldots, \lambda_{\ell}]$, with parts $\lambda_1 \ge \lambda_2 \ge \cdots$ eventually $0$, or as $\lambda = (1^{m_1} 2^{m_2}\cdots)$ where $m_j = m_j(\lambda)$ is the number of parts of $\lambda$ of size $j$. The length of $\lambda$ is $\ell(\lambda) = \max\{ r : \lambda_r \ge 1\}$, the size of $\lambda$ is $|\lambda| = \sum_{i} \lambda_i = \sum_j{jm_j}$, and $\lambda_i$ is the $i$th largest part of $\lambda$. (Compare [22].)
4. 4. Each partition $\lambda$ of $n$ corresponds to a conjugacy class $C_{\lambda}$ of $S_n$ given by the common cycle structure of the elements in $C_\lambda$. We let $Z_\lambda$ denote the centralizer of $C_{\lambda}$ in $S_n$. The size of the centralizer and conjugacy class are \begin{eqnarray*} z_\lambda := |Z_{\lambda}| = \prod_{j \geq 1}{j^{m_j(\lambda)}m_j(\lambda)!} \hspace{2em} c_\lambda:= |C_{\lambda}|= \frac{n!}{z_\lambda} \end{eqnarray*} respectively. Note that $c_\lambda z_\lambda = n!$.
5. 5. Following [32], we let $\partakak(n)$ denote the set of partitions of $n$ and $\partakak = \bigcup_n{\partakak(n)}$ the set of all partitions. However in Sect. 6, we let $\Pi_n$ denote the set of partitions of $n$, partially ordered by refinement.

## 2. Splitting Measures

We review the splitting measures introduced in [18], summarize their properties, and introduce the normalized splitting measures.

### 2.1. Necklace Polynomials and Cycle Polynomials.

###### Definition 2.1.
For $j\geq 1$, the $j$th necklace polynomial $M_j(z)\in \frac{1}{j}\ZZ[z]$ is \begin{eqnarray*} M_j(z) := \frac{1}{j}\sum_{d\mid j}{\mu(d) z^{j/d}}, \end{eqnarray*} where $\mu(d)$ is the M[U+00A8]obius function.
[25] noted in 1872 that for all integers $m \ge 1$, $M_j(m)$ is the number of distinct necklaces having $j$ beads drawn from a set of $m$ colors, up to cyclic permutation. This fact motivated [24] to name them necklace polynomials. Relevant to the present paper, $M_j(q)$ is the number of monic, degree $j$, irreducible polynomials in $\FF_q[X]$ [28], Prop. 2.1. The factorization type of a polynomial $f\in \conf_n(\FF_q)$ is the partition formed by the degrees of its irreducible factors, which we write $[f]$.
###### Definition 2.2.
Given a partition $\lambda$ of $n$, the cycle polynomial $N_\lambda(z) \in \frac{1}{z_\lambda}\ZZ[z]$ is \begin{eqnarray*} N_\lambda(z) := \prod_{j\geq 1}{\binom{M_j(z)}{m_j(\lambda)}}, \end{eqnarray*} where $\binom{\alpha}{m}$ is the usual extension of a binomial coefficient, \begin{eqnarray*} \binom{\alpha}{m} := \frac{1}{m!}\prod_{k=0}^{m-1}{(\alpha - k)}. \end{eqnarray*}

The cycle polynomial $N_\lambda(z)$ has degree $n = |\lambda|$ and is integer valued for $z\in\ZZ$. The number of $f\in \conf_n(\FF_q)$ with $[f] = \lambda$ is $N_\lambda(q)$ (see [18], Sect. 4).

### 2.2. $z$-Splitting Measures.

If $\lambda$ a partition of $n$, then the probability of a uniformly chosen $f\in \conf_n(\FF_q)$ having factorization type $\lambda$ is \begin{eqnarray*} \mathrm{Prob}\{f\in \conf_n(\FF_q) : [f] = \lambda\} = \frac{N_\lambda(q)}{|\conf_n(\FF_q)|}. \end{eqnarray*} When $n = 1$, $|\conf_n(\FF_q)| = q$ and for $n\geq 2$ we have $|\conf_n(\FF_q)| = q^n - q^{n-1}$. (See [28], Prop. 2.3 for a proof via generating functions. A proof due to Zieve appears in [35], Lem. 4.1.) Hence, the probability is a rational function in $q$. Replacing $q$ by a complex-valued parameter $z$ yields the $z$-splitting measure.

###### Definition 2.3.
For $n \ge 2$ the $z$ -splitting measure $\nu_{n,z}^*(C_{\lambda}) \in \QQ(z)$ is given by \begin{eqnarray*} \nu_{n,z}^{\ast}(C_{\lambda}) := \frac{N_\lambda(z)}{z^n - z^{n-1}}. \end{eqnarray*}
###### Proposition 2.4.
For each partition $\lambda$ of $n \ge 1,$ the rational function $\nu_{n, z}^{\ast}(C_{\lambda})$ is a polynomial in $\frac{1}{z}$ of degree at most $n-1$. Thus it may be written as \begin{eqnarray*} \nu_{n, z}^{\ast}(C_{\lambda}) = \sum_{k=0}^{n-1} \alpha_n^k(C_{\lambda})\left(\tfrac{1}{z}\right)^k. \end{eqnarray*} The function $\nu_{1, z}^{\ast}(C_{1}) = 1$ is independent of $z$.

###### Proof.
The case $n=1$ is clear. For $n\geq 2$ we have $N_\lambda(1) = 0$ by [17], Lemma 2.5, whence $\frac{N_\lambda(z)}{z-1}$ is a polynomial of degree at most $n-1$ in $z$. Therefore, \begin{eqnarray*} \nu_{n, z}^{\ast}(C_\lambda) = \frac{N_\lambda(z)}{z^n - z^{n-1}} = \frac{1}{z^{n-1}}\left(\frac{N_\lambda(z)}{z-1}\right) \end{eqnarray*} is a polynomial in $\frac{1}{z}$ of degree at most $n - 1$.

For $n \ge 2$ the Laurent polynomial $\nu_{n, z}^{\ast}(C_{\lambda})$ is of degree at most $n-2$ since $z \mid N_{\lambda}(z)$ ([18], Lemma 4.3); that is, $\alpha_n^{n-1}(C_{\lambda}) =0$. Tables 1 and 2 give $\nu^*_{n,z}(C_{\lambda})$, exhibiting the splitting measure coefficients $\alpha_n^k(C_{\lambda})$ for $n=4$ and $n=5$.

Table 1 Values of the $z$-splitting measures $\nu_{4,z}^{\ast}(C_\lambda)$ on partitions $\lambda$ of $n=4$..
 $\lambda$ $|C_\lambda|$ $z_\lambda$ $\nu^*_{4,z}(C_{\lambda})$ $[1,1,1,1]$ $1$ 24 $\frac{1}{24}\left(1 -\frac{5}{z} + \frac{6}{z^2}\right)$ $[2,1,1]$ $6$ 4 $\frac{1}{4}\left(1 - \frac{1}{z}\right)$ $[2,2]$ $3$ 8 $\frac{1}{8}\left(1 - \frac{1}{z} - \frac{2}{z^2}\right)$ $[3,1]$ $8$ 3 $\frac{1}{3}\left(1 + \frac{1}{z}\right)$ $[4]$ $6$ 4 $\frac{1}{4}\left(1 + \frac{1}{z}\right)$
Table 2 Values of the $z$-splitting measures $\nu_{5,z}^{\ast}(C_\lambda)$ on partitions $\lambda$ of $n=5$..
 $\lambda$ $|C_\lambda|$ $z_\lambda$ $\nu^*_{5,z}(C_\lambda)$ $[1,1,1,1,1]$ 1 120 $\frac{1}{120}\left(1 - \frac{9}{z} + \frac{26}{z^2} - \frac{24}{z^3}\right)$ $[2,1,1,1]$ 10 12 $\frac{1}{12}\left(1 - \frac{3}{z} + \frac{2}{z^2}\right)$ $[2,2,1]$ 15 8 $\frac{1}{8}\left(1 - \frac{1}{z} - \frac{2}{z^2}\right)$ $[3,1,1]$ 20 6 $\frac{1}{6}\left(1 - \frac{1}{z^2}\right)$ $[3,2]$ 20 6 $\frac{1}{6}\left(1 - \frac{1}{z^2}\right)$ $[4,1]$ 30 4 $\frac{1}{4}\left(1 + \frac{1}{z}\right)$ $[5]$ 24 5 $\frac{1}{5}\left(1 + \frac{1}{z} + \frac{1}{z^2} + \frac{1}{z^3}\right)$

## 3. Interpretation of Cycle Polynomial Coefficients

In Sect. 2.1 we defined the cycle polynomials $N_\lambda(z) \in \frac{1}{z_\lambda}\ZZ[z]$ for each partition $\lambda$ of $n$. In this section we express the coefficients of $N_\lambda(z)$ as a function of $\lambda$ in terms of characters $h_n^k$ of the cohomology of the pure braid group $P_n$ viewed as an $S_n$-representation. We establish this connection using the twisted Grothendieck--Lefschetz formula of [6]. Using explicit formulas for the cycle polynomials we obtain constraints on the support of $h_n^k$, and we compute $h_n^k(\lambda)$ for varying $n$ in several examples.

### 3.1. Cohomology of the Pure Braid Group.

Given a set $X$ of $n$ distinct points in $3$-dimensional affine space, the braid group $B_n$ consists of homotopy classes of simple, non-intersecting paths beginning and terminating in $X$, with concatenation as the group operation. Each element of $B_n$ determines a permutation of $X$, giving a short exact sequence of groups \begin{eqnarray*} 0 \rightarrow P_n \rightarrow B_n \xrightarrow{\pi} S_n \rightarrow 0. \end{eqnarray*} Then $P_n := \ker \pi$ is called the pure braid group . $P_n$ consists of homotopy classes of simple, non-intersecting loops based in $X$. The action of $S_n$ on $X$ induces an action on $P_n$ by permuting the loops. Thus, for each $k$, the $k$th group cohomology $H^k(P_n,\QQ)$ carries an $S_n$-representation whose character we denote by $h_n^k$.

### 3.2. Twisted Grothendieck--Lefschetz Formula.

A character polynomial is a polynomial $P(x) \in \QQ[x_j : j\geq 1]$. Character polynomials induce functions $P: \partakak \rightarrow \QQ$ by \begin{eqnarray*} P(\lambda) := P\left(m_1(\lambda), m_2(\lambda), \ldots\right), \end{eqnarray*} noting that $m_i(\lambda) = 0$ for all but finitely many $i$. For $f \in \conf_n(\FF_q)$ we let $P(f) := P([f])$. Given two $\QQ$-valued functions $F$ and $G$ defined on $S_n$ let \begin{eqnarray*} \langle F, G \rangle := \frac{1}{n!}\sum_{g \in S_n}{F(g)G(g)}. \end{eqnarray*} The following theorem is due to [6], Prop. 4.1.

###### Theorem 3.1.
(Twisted Grothendieck--Lefschetz formula for $\pconf_n$) Given a prime power $q,$ an integer $n\geq 1,$ and a character polynomial $P,$ we have

 \begin{eqnarray}\label{twistEq} \sum_{f\in \conf_n(\FF_q)}{P(f)} = \sum_{k=0}^n{(-1)^k \langle P, h_n^k \rangle\, q^{n-k}}, \end{eqnarray} (3.1)
where $h_n^k$ is the character of the cohomology of the pure braid group $H^k(P_n, \QQ)$.

The classic Lefschetz trace formula counts the fixed points of an endomorphism $f$ on a compact manifold $M$ by the trace of the induced map on the singular cohomology of $M$. One may interpret the $\overline{\FF}_q$ points on an algebraic variety $V$ defined over $\FF_q$ as the fixed points of the geometric Frobenius endomorphism of $V$. Using the machinery of $\ell$-adic étale cohomology, [14] generalized Lefschetz's formula to count the number of points in $V(\FF_q)$ by the trace of Frobenius on the étale cohomology of $V$. For nice varieties $V$ defined over $\ZZ$, there are comparison theorems relating the étale cohomology of $V(\overline{\FF}_q)$ to the singular cohomology of $V(\CC)$. This connects the topology of a complex manifold to point counts of a variety over a finite field. For hyperplane complements the connection was made in 1992 by [20], and for equivariant actions of a finite group on varieties the equivariant Poincaré polynomials were determined by [16]. [6] build upon Grothendieck's extension of the Lefschetz formula to relate point counts on natural subsets of $\conf_n(\FF_q)$ to the singular cohomology of the covering space $\pconf_n(\CC) \rightarrow \conf_n(\CC)$. $\pconf_n(\CC)$ is the space of $n$ distinct, labelled points in $\CC$. The space $\pconf_n(\CC)$ has fundamental group $P_n$, the pure braid group, and is a $K(\pi,1)$ for this group. Hence, the singular cohomology of $\pconf_n(\CC)$ is the same as the group cohomology of $P_n$. This fact yields the connection between $\conf_n(\FF_q)$ on the left hand side of (3.1) and the character of the pure braid group cohomology.

### 3.3. Cycle Polynomials and Pure Braid Group Cohomology.

We express the coefficients of the cycle polynomials $N_\lambda(z)$ in terms of the characters $h_n^k$ as an application of Theorem 3.1. Theorem 3.2 is equivalent to Lehrer's ([19], Theorem 5.5) by comparing numerators and making a slight change of variables.

###### Theorem 3.2.
Let $\lambda$ be a partition of $n,$ then \begin{eqnarray*} N_\lambda(z) = \frac{1}{z_\lambda}\sum_{k=0}^n{(-1)^k h_n^k(\lambda) z^{n-k}}, \end{eqnarray*} where $h_n^k$ is the character of the $S_n$-representation $H^k(P_n,\QQ)$.

###### Proof.
Define the character polynomial $1_\lambda(x) \in \QQ[x_j : j \geq 1]$ by \begin{eqnarray*} 1_\lambda(x) = \prod_{j\geq 1}{\binom{x_j}{m_j(\lambda)}}. \end{eqnarray*} Observe that for a partition $\mu \in \partakak(n)$ we have \begin{eqnarray*} 1_\lambda(\mu) = \begin{cases} 1 & \text{if }\mu = \lambda,\\ 0 & \text{otherwise.} \end{cases} \end{eqnarray*} Therefore, \begin{eqnarray*} N_\lambda(q) = \sum_{f\in \conf_n(\FF_q)}{1_\lambda(f)}. \end{eqnarray*} On the other hand, by Theorem 3.1 we have \begin{eqnarray*} \sum_{f\in \conf_n(\FF_q)}{1_\lambda(f)} = \sum_{k=0}^n{(-1)^k \langle 1_\lambda, h_n^k \rangle q^{n-k} }. \end{eqnarray*} If $g \in S_n$, let $[g] \in \partakak(n)$ be the partition given by the cycle lengths of $g$. Thus, \begin{eqnarray*} \langle 1_\lambda, h_n^k \rangle = \frac{1}{n!}\sum_{g \in S_n}{1_\lambda(g)h_n^k(g)} = \frac{1}{n!}\sum_{\substack{g \in S_n \\ \, [g]=\lambda}}{h_n^k(g)} = \frac{c_\lambda}{n!}h_n^k(\lambda) = \frac{1}{z_\lambda}h_n^k(\lambda). \end{eqnarray*} Therefore the identity \begin{eqnarray*} N_\lambda(q) = \frac{1}{z_\lambda}\sum_{k=0}^n{(-1)^k h_n^k(\lambda) q^{n-k}} \end{eqnarray*} holds for all prime powers $q$, giving the identity as polynomials in $\QQ[z]$.

###### Remark.
A recent result of [4], Theorem 1 also yields the identity in Theorem 3.2 by specializing at $t=0$.

One can explicitly compute $h_n^k(\lambda)$ using Theorem 3.2 by expanding the formula (1.1) for $N_\lambda(z)$ and comparing coefficients. [19] derives several corollaries this way. Here we give further examples intended to explore possible connections with number theory. We obtain restrictions on the support of $h_n^k$ in Proposition 3.3. Then we compute values of $h_n^k(\lambda)$ in Sects. 3.5 and 3.6. For any fixed $k$, the $h_n^k$ are given by character polynomials, while $h_n^{n-k}$ for $k < 2n/3$ exhibit interesting arithmetic structure.

### 3.4. Support Restrictions on Characters $h_n^k$ .

The character $h_n^k$ is supported on partitions with at least one small part, while $h_n^{n-k}$ is supported on partitions having at most $k$ different parts. The latter are multi-rectangular Young diagrams having at most $k$ steps, using the terminology of [10], Sect. 1.7 and [30].

###### Proposition 3.3.
Let $0 \le k \le n$ and $h_n^k$ be the character of the $S_n$-representation $H^k(P_n, \QQ),$ then
1. 1. $h_n^k$ is supported on partitions having at least one part of size at most $2k$. The value $h_n^k(\lambda)$ is determined by $m_j(\lambda)$ for $1 \le j \leq 2k$.
2. 2. $h_n^{n-k}$ is supported on multi-rectangular partitions $\lambda$ having at most $k$ distinct values of $j$ with $m_j(\lambda)>0$.

###### Proof.
1. (1) Theorem 3.2 implies $h_n^k(\lambda)$ is nonzero iff the coefficient of $z^{n-k}$ in $N_{\lambda}(z)$ is nonzero. The degree of $M_j(z) - \frac{1}{j}z^j$ is at most $\lfloor j/2 \rfloor$. Hence if $j > 2k$, then the coefficient of $z^{n-k}$ in $\left( {\substack{M_j(z) \\ m_j(\lambda)}} \right)$ is zero. Thus the only $j$ contributing to the coefficient of $z^{n-k}$ in $N_{\lambda}(z)$ in (1.1) are those with $1 \le j\leq 2k$.
2. (2) Theorem 3.2 implies $h_n^{n-k}(\lambda)$ is nonzero iff the coefficient of $z^k$ in $N_\lambda(z)$ is nonzero. If $m_j(\lambda) > 0$, then $z$ divides $\left( {\substack{M_j(z) \\ m_j(\lambda)}} \right)$. Hence if $m_j(\lambda) > 0$ for more than $k$ values of $j$, then $h_n^{n-k}(\lambda) = 0$.

###### Remark.
Property (1) is a manifestation of representation stability of $h_n^k$, which says that for fixed $k$ and all sufficiently large $n$, the values of $h_n^k(\lambda)$ are described by a character polynomial in $\lambda$. A character polynomial for a partition $\lambda=(1^{m_1}2^{m_2} \cdots n^{m_n})$ is a polynomial in the variables $m_j$, see Example 3.7. [11] raised the problem of explicitly determining such character polynomials. Proposition 3.3 bounds which variables $m_j$ may occur in the character polynomial for $h_n^k$. A known sharp representation stability property of $h_n^k$ is that it equals such a character polynomial for all $n \ge 3k+1$, as shown in [15], Theorem 1.1, taking dimension $d=2$.

### 3.5. Character Values $h_n^k(\lambda)$ for Fixed $\lambda$ and Varying $k$.

We give special cases of explicit determinations for $h_n^k(\lambda)$ for various fixed $\lambda$ and varying $k$ by directly expanding the cycle polynomial $N_\lambda(z)$.

###### Example 3.4.
(Dimensions of cohomology) The dimension of $H^k(P_n,\QQ)$ is the value of $h_n^k$ at the identity element, corresponding to the partition $(1^n)$. Since $M_1(z) = z$ and the centralizer of the identity in $S_n$ has order $z_{(1^n)} = n!$, we have \begin{eqnarray*} N_{(1^n)}(z) = \binom{z}{n} = \frac{1}{n!}\prod_{i=0}^{n-1}(z - i) = \frac{1}{n!}\sum_{k=0}^n{(-1)^k{n \brack n-k}z^{n-k}}, \end{eqnarray*} where ${n \brack n-k}$ is an unsigned Stirling number of the first kind . Theorem 3.2 says \begin{eqnarray*} N_{(1^n)}(z) = \frac{1}{n!}\sum_{k=0}^n{(-1)^k h_n^k\left((1^n)\right) z^{n-k}}. \end{eqnarray*} Comparing coefficients recovers the well-known formula due to [1] for the dimension of the pure braid group cohomology: \begin{eqnarray*} {\mathrm{dim}}\,H^k(P_n,\QQ) = h_n^k\left((1^n)\right) = {n \brack n-k}. \end{eqnarray*} These values are given in Table 3.
Table 3 Betti numbers of pure braid group cohomology $H^k(P_n, \QQ)$ .
 $n$ $\backslash$ $k$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ 1 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 3 1 3 2 0 0 0 0 0 0 4 1 6 11 6 0 0 0 0 0 5 1 10 35 50 24 0 0 0 0 6 1 15 85 225 274 120 0 0 0 7 1 21 175 735 1624 1764 720 0 0 8 1 28 322 1960 6769 13,132 13,068 5040 0 9 1 36 546 4536 22,449 67,284 118,124 109,584 40,320
###### Example 3.5.
The partition $\lambda = [n]$ corresponds to an $n$-cycle in $S_n$. The centralizer of an $n$-cycle has order $z_{[n]} = n$ and

 \begin{eqnarray}\label{firstspsexp} N_{[n]}(z) = \binom{M_n(z)}{1} = M_n(z) = \frac{1}{n}\sum_{d\mid n}{\mu(d) z^{n/d}}. \end{eqnarray} (3.2)
Theorem 3.2 gives us

 \begin{eqnarray}\label{secondspsexp} N_{[n]}(z) = \frac{1}{n}\sum_{k=0}^n{(-1)^kh_n^k\left([n]\right)z^{n-k}}. \end{eqnarray} (3.3)
Comparing coefficients, we find that \begin{eqnarray*} h_n^{n-k}\left([n]\right) = \left\{ \begin{array}{ll} (-1)^{n-k} \mu(\frac{n}{k}) &\quad \hbox{if } k \mid n, \\ 0 &\quad \hbox{if }k\nmid n. \end{array} \right. \end{eqnarray*}

### 3.6. Character Values $h_n^k(\lambda)$ for Fixed $k$ and Varying $\lambda$.

We now compute $h_n^k(\lambda)$ for fixed $k$ and varying $\lambda$.

###### Example 3.6.
(Computing $h_n^0$ and $h_n^n$) The cases $k=0$ and $n$ are both constant: $h_n^0 = 1$ and $h_n^n = 0$. The leading coefficient of $N_\lambda(z)$ is $1/z_\lambda$, hence Theorem 3.2 tells us $h_n^0(\lambda) = 1$ for all $\lambda$. For $j\geq 1$, we have $z \mid M_j(z)$, from which it follows that $z \mid N_\lambda(z)$ for all partitions $\lambda$ of $n \geq 1$. In other words, for all $m_j \geq 1$ \begin{eqnarray*} \frac{1}{z_\lambda}(-1)^n h_n^n(\lambda) = N_\lambda(0) = 0. \end{eqnarray*} Thus $h_n^n(\lambda) = 0$ for all $\lambda$, and $H^n(P_n, \QQ) =0.$
###### Example 3.7.
(Computing $h_n^1$ and $h_n^2$) Taking $\lambda= ( 1^{m_1} 2^{m_2} \cdots)$, a careful analysis of the $z^{n-1}$ and $z^{n-2}$ coefficients in $N_\lambda(z)$ and Theorem 3.2 yields the following formulas \begin{eqnarray*} h_n^1(\lambda) &=& \binom{m_1}{2} + \binom{m_2}{1}\\ h_n^2(\lambda) &=& 2\binom{m_1}{3} + 3\binom{m_1}{4} +\binom{m_1}{2}\binom{m_2}{1}-\binom{m_2}{2} - \binom{m_3}{1} - \binom{m_4}{1}, \end{eqnarray*} where $m_j = m_j(\lambda)$. These formulas represent $h_n^1$ and $h_n^2$ as character polynomials, and they appear in [6], Lemma 4.8. Note that $h_n^1(\lambda) = h_n^2(\lambda) = 0$ for partitions $\lambda$ having all parts larger than 2 and 4 respectively, illustrating Proposition 3.3(1).
###### Example 3.8.
(Computing $h_n^{n-1}$) The $z$ coefficient of $N_\lambda(z)$ determines the value of $h_n^{n-1}(\lambda)$. Since each $j$ with $m_j(\lambda)>0$ contributes a factor of $z$ to $N_\lambda(z)$, $h_n^{n-1}$ is supported on partitions of the form $\lambda = (j^{m})$. Note that the $z$ coefficient of the necklace polynomial $M_j(z)$ is $\mu(j)/j$. Let $\lambda = (j^{m})$, then the $z$ coefficient of \begin{eqnarray*} N_\lambda(z) = \binom{M_j(z)}{m} = \frac{M_j(z)(M_j(z) - 1)\cdots(M_j(z) - m + 1)}{m!} \end{eqnarray*} is $(-1)^{m -1}\frac{\mu(j)}{j m}$. Since $z_\lambda = j^{m}m!$, we conclude \begin{eqnarray*} h_n^{n-1}(\lambda) = \left\{ \begin{array}{ll} (-1)^{m - n}\mu(j) j^{m - 1} (m - 1)! & \quad\hbox{if } \lambda= (j^{m}), \\ 0 & \quad\hbox{otherwise}. \end{array} \right. \end{eqnarray*} By [19], Corollary $(5.5)^{\prime\prime}$ $h_n^{n-1} = \Sign_n \otimes \Ind_{c_{n}}^{S_n}(\zeta_n),$ where $c_n$ is a cyclic group of order $n$ and $\zeta_n$ is a faithful character on it, noted earlier by [31].
###### Example 3.9.
(Computing $h_n^{n-2}$) The $z^2$ coefficient of $N_\lambda(z)$ determines $h_n^{n-2}(\lambda)$. Proposition 3.3(2) tells us that $h_n^{n-2}(\lambda) = 0$ when $m_j(\lambda) > 0$ for at least three $j$. We treat the two remaining cases $\lambda = (i^{m_i} j^{m_j})$ and $\lambda = ( j^{m})$ in turn. If $\lambda = (i^{m_i}j^{m_j})$, then the $z$ coefficient of $\left( {\substack{M_i(z) \\ m_i}} \right)$ is $(-1)^{m_i -1}\frac{\mu(i)}{i m_i}$, and similarly for $\left( {\substack{M_j(z) \\ m_j}} \right)$. We have $z_\lambda = (i^{m_i} m_i!)(j^{m_j}m_j!)$. Thus, by Theorem 3.2 \begin{eqnarray*} h_n^{n-2}\left( (i^{m_1} j^{m_j}) \right)&=& (-1)^{m_i + m_j - n} z_\lambda \frac{\mu(i)\mu(j)}{(i m_i )(jm_j)}\\ &=& (-1)^{m_i + m_j - n} \left(\mu(i)i^{m_i -1}(m_i -1)!\right)\left(\mu(j) j^{m_j-1} (m_j-1)!\right). \end{eqnarray*}

If $\lambda = (j^{m})$, then the $z^2$ coefficient of $N_\lambda(z)$ receives a contribution of $(-1)^{m-1}\frac{\mu(j/2)}{j m}$ from the quadratic term of $M_j(z)$ if $j$ is even. The $z$ coefficient of $\left({\substack{M_j(z) \\ m_j}}\right)/M_j(z)$ is \begin{eqnarray*} \frac{\mu(j)}{j m!}\left( \sum_{i=1}^{m-1}{\frac{(-1)^{m-2}(m-1)!}{i}} \right) = (-1)^{m} \frac{\mu(j)}{j m} H_{m -1}, \end{eqnarray*} where $H_{m-1}=\sum_{i=1}^{m-1} \frac{1}{i}$ denotes the $(m-1)$ th harmonic number . The $z$ coefficient of $M_j(z)$ is $\frac{\mu(j)}{j}$. Using the convention that the M[U+00A8]obius function $\mu(\alpha)$ vanishes at non-integral $\alpha$, we arrive at the following expression for $h_n^{n-2}(\lambda)$: \begin{eqnarray*} h_n^{n-2}\left( (j^{m}) \right) &=& z_\lambda (-1)^{m - n}\frac{\left(\mu(j)^2 H_{m-1} - \mu(\frac{j}{2})\right)}{jm}\\ &=& (-1)^{m - n}\left(\mu(j)^2 H_{m-1} - \mu(\tfrac{j}{2})\right)j^{m - 1} (m -1)!. \end{eqnarray*}

## 4. Submodules $A_n^k$ of Pure Braid Group Cohomology

Starting from Arnol'd's presentation for the $S_n$-algebra $H^\bullet(P_n,\QQ)$ we obtain a decomposition $H^k (P_n, \QQ) = A_n^{k-1} \oplus A_n^k$ of $S_n$-modules. The characters of the sequence $A_n^k$ of $S_n$-modules determine the splitting measure coefficients $\alpha_n^k(C_\lambda)$. In Sect. 4.3 we interpret $A_n^\bullet$ as the cohomology of $\pconf_n(\CC)/\CC^\times$, where $\CC^\times$ acts freely on $\pconf_n(\CC)$ by scaling coordinates.

### 4.1. Presentation of Pure Braid Group Cohomology Ring.

[1] gave the following presentation of the cohomology ring $H^{\bullet}(P_n,\QQ)$ of the pure braid group $P_n$ as an $S_n$-algebra.
###### Theorem 4.1.
(Arnol'd) There is an isomorphism of graded $S_n$-algebras \begin{eqnarray*} H^{\bullet}(P_n,\QQ) \cong \Lambda^{\bullet}[\omega_{i,j}]/\langle R_{i,j,k} \rangle, \end{eqnarray*} where $1 \leq i,j,k \leq n$ are distinct$,$ $\omega_{i,j} = \omega_{j,i}$ have degree $1,$ and \begin{eqnarray*} R_{i,j,k} = \omega_{i,j}\wedge\omega_{j,k} + \omega_{j,k}\wedge\omega_{k,i} + \omega_{k,i}\wedge\omega_{i,j}. \end{eqnarray*} An element $g \in S_n$ acts on $\omega_{i,j}$ by $g\cdot \omega_{i,j} = \omega_{g(i),g(j)}$.

In what follows, we identify $H^{\bullet}(P_n,\QQ)$ with this presentation as a quotient of an exterior algebra. The ring $\Lambda^{\bullet}[\omega_{i,j}]/\langle R_{i,j,k} \rangle$ is an example of an Orlik--Solomon algebra , which arise as cohomology rings of complements of hyperplane arrangements (see [26], [9], [36]).

### 4.2. $S_n$-Modules $A_n^k$ Inside Braid Group Cohomology.

Let $\tau = \sum_{1\leq i< j \leq n}{\omega_{i,j}} \in H^1(P_n, \QQ)$. The element $\tau$ generates a trivial $S_n$-subrepresentation of $H^1(P_n,\QQ)$. We define maps $d^k: H^k(P_n,\QQ) \rightarrow H^{k+1}(P_n,\QQ)$ for each $k$ by $\nu \mapsto \nu \wedge \tau$. This map is linear and $S_n$-equivariant, since \begin{eqnarray*} g \cdot d^k(\nu) = g\cdot (\nu \wedge \tau) = (g\cdot \nu) \wedge (g \cdot \tau) = (g\cdot \nu) \wedge \tau = d^k(g\cdot \nu). \end{eqnarray*} From $d^{k+1}\circ d^k = 0$ we conclude that \begin{eqnarray*} 0 \rightarrow H^0(P_n, \QQ) \xrightarrow{d^0} H^1(P_n, \QQ) \xrightarrow{d^1} \cdots \xrightarrow{d^{n-1}} H^n(P_n, \QQ) \xrightarrow{d^n} 0 \end{eqnarray*} is a chain complex of $S_n$-representations. It follows from the general theory of Orlik-Solomon algebras that the above sequence is exact ([9], Thm. 5.2). We include a proof in this case for completeness.

###### Proposition 4.2.
In the above notation$,$

 \begin{eqnarray} \label{Exact} 0 \rightarrow H^0(P_n, \QQ) \xrightarrow{d^0} H^1(P_n, \QQ) \xrightarrow{d^1} \cdots \xrightarrow{d^{n-1}} H^n(P_n, \QQ) \xrightarrow{d^n} 0 \end{eqnarray} (4.1)
is an exact sequence of $S_n$-representations. Set $A_n^k := \im(d^k) \subset H^{k+1}(P_n, \QQ)$. Hence we have an isomorphism of $S_n$-representations for each $k,$ \begin{eqnarray*} H^k(P_n, \QQ) \cong A_n^{k-1} \oplus A_n^{k}. \end{eqnarray*}

###### Proof.
[1], Cor. 3 describes an additive basis $\cB_k$ for $H^k(P_n, \QQ)$ comprised of all simple wedge products \begin{eqnarray*} \omega_{i_1, j_1} \wedge \cdots \wedge \omega_{i_k, j_k}\text{ such that } i_s < j_s\text{ for each } s, \text{ and }j_1 < j_2 < \cdots < j_k. \end{eqnarray*} Let \begin{eqnarray*} U_k = \{ \omega_{i_1,j_1} \wedge \cdots \wedge \omega_{i_k,j_k} \in \cB_k : (i_s, j_s) \neq (n-1, n)\}, \end{eqnarray*} for $k >0$ and $U_0 = \{1\}$. Then set \begin{eqnarray*} \cC_k = U_k \cup \{\omega \wedge \tau : \omega \in U_{k-1}\}. \end{eqnarray*} Claim. $\cC_k$ is a basis of $H^k(P_n,\QQ)$.

For example, we have \begin{eqnarray*} \cC_1 = \{\omega_{i,j} : (i,j) \neq (n-1, n)\} \cup \{\tau\}, \end{eqnarray*} which is clearly a basis for $H^1(P_n,\QQ)$.

To prove the claim, since $|\cB_k| = |\cC_k|$, it suffices to show $\cC_k$ spans $H^k(P_n, \QQ)$. Note that \begin{eqnarray*} \cB_k = U_k \cup \{\omega \wedge \omega_{n-1,n} : \omega \in U_{k-1}\}, \end{eqnarray*} further reducing the problem to expressing $\omega \wedge \omega_{n-1,n}$ as a linear combination of $\cC_k$ for each $\omega \in U_{k-1}$. Given $\omega = \omega_{i_1,j_1} \wedge \cdots \wedge \omega_{i_{k-1},j_{k-1}} \in U_{k-1}$, we use the relation \begin{eqnarray*} \omega_{i_s,j}\wedge \omega_{i,j} = \omega_{i_s,i}\wedge\omega_{i,j} - \omega_{i_s,i}\wedge\omega_{i_s,j} \end{eqnarray*} to express $\omega \wedge \omega_{i,j}$ in terms of elements of $U_k$ as follows: \begin{eqnarray*} \omega \wedge \omega_{i,j} = \begin{cases} \pm \omega_{i_1,j_1}\wedge \cdots \wedge \omega_{i_s,j_s}\wedge \omega_{i,j} \wedge \omega_{i_{s+1},j_{s+1}} \wedge \cdots \wedge \omega_{i_{k-1},j_{k-1}}\\ \quad\quad\quad \quad\quad\quad\quad\quad \quad \hbox{for} \quad j_s < j < j_{s+1},\\ \pm \omega_{i_1,j_1} \wedge \cdots \wedge (\omega_{i_s,i}\wedge \omega_{i,j} - \omega_{i_s,i}\wedge \omega_{i_s,j}) \wedge \cdots \wedge \omega_{i_{k-1},j_{k-1}}\\ \quad\quad\quad \quad\quad\quad\quad\quad \quad \hbox{for} \quad j_s = j, i_s \neq i,\\ 0 \quad\quad \quad\quad\quad \quad\quad\quad\,\, \hbox{for} \quad (i_s, j_s) = (i,j). \end{cases} \end{eqnarray*} The first and third cases are easily seen to belong in the span of $U_k$. Since $i_s, i < j$ and $j$ does not occur twice as a largest subscript in $\omega$, we see inductively that the second case also belongs in the span of $U_k$. Therefore, $\omega \wedge \tau = \omega \wedge \omega_{n-1,n} + \nu$, where $\nu$ is in the span of $U_k$. Hence $\omega\wedge \omega_{n-1,n} = \omega \wedge \tau -\nu$ is in the span of $\cC_k$ and we conclude that $\cC_k$ is a basis, proving the claim.

We now show the sequence (4.1) is exact. Suppose $\nu \in \ker(d^k)$. Express $\nu$ in the basis $\cC_k$ as \begin{eqnarray*} \nu = \sum_{\omega \in U_k}{a_\omega\, \omega} + \sum_{\omega\in U_{k-1}}{b_\omega\, \omega}\wedge \tau. \end{eqnarray*} Then \begin{eqnarray*} 0 = d^k(\nu) = \nu \wedge \tau = \sum_{\omega \in U_k}{a_\omega \, \omega \wedge \tau}. \end{eqnarray*} Since $\omega \wedge \tau$ is an element of the basis $\cC_{k+1}$ for each $\omega \in U_k$, we have $a_\omega= 0$. Hence, $\nu = \mu \wedge \tau = d^{k-1}(\mu)$ where \begin{eqnarray*} \mu = \sum_{\omega \in U_{k-1}}{b_\omega \, \omega}, \end{eqnarray*} so $\ker(d^k)= \im(d^{k-1}).$

Recall from Sect. 3.5 that the dimension of $H^k(P_n, \QQ)$ is given by an unsigned Stirling number of the first kind \begin{eqnarray*} \dim\left(H^k(P_n,\QQ)\right)= {n \brack n - k}, \end{eqnarray*} where the unsigned Stirling numbers are determined by the identity $\prod_{k=0}^{n-1}{(x + k)} = \sum_{k=0}^{n-1}{{n \brack k} x^k}.$ The exact sequence in Proposition 4.2 shows the dimension of $A_n^k$ is \begin{eqnarray*} \dim(A_n^k) = \sum_{j=0}^k {(-1)^j {n \brack n - k + j}}. \end{eqnarray*} Table 4 gives values of $\dim(A_n^k)$ for small $n$ and $k$; here $\dim (A_n^{n-1})=0$ for $n \ge 2$.

Table 4 $\dim(A_n^k)$ .
 $n\backslash k$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ 1 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 3 1 2 0 0 0 0 0 0 4 1 5 6 0 0 0 0 0 5 1 9 26 24 0 0 0 0 6 1 14 71 154 120 0 0 0 7 1 20 155 580 1044 720 0 0 8 1 27 295 1665 5104 8028 5040 0 9 1 35 511 4025 18,424 48,860 69,264 40,320

### 4.3. $A_n^k$ as Cohomology of a Complex Manifold with an $S_n$-Action.

Recall from Sect. 3.2 that the pure configuration space $\pconf_n(\CC)$ is defined by \begin{eqnarray*} \pconf_n(\CC) = \{(z_1, z_2, \ldots, z_n) \in \CC^n : z_i \neq z_j \text{ when } i\neq j\}. \end{eqnarray*} It is an open complex manifold, and the symmetric group $S_n$ acts on $\pconf_n(\CC)$ by permuting coordinates. There is also a free action of $\CC^\times$ on $\pconf_n(\CC)$ defined by \begin{eqnarray*} c\cdot(z_1,z_2,\ldots, z_n) = (c z_1, c z_2, \ldots, c z_n). \end{eqnarray*} This action commutes with the $S_n$-action, hence induces an action of $S_n$ on the quotient complex manifold $\pconf_n(\CC)/\CC^\times$. Therefore $H^\bullet(\pconf_n(\CC)/\CC^\times,\QQ)$ is an $S_n$-algebra. We now relate the graded components $H^k(\pconf_n(\CC)/\CC^\times, \QQ)$ to the $S_n$-submodules $A_n^k$ of $H^k(\pconf_n(\CC),\QQ) = H^k(P_n,\QQ)$ constructed in Proposition 4.2.

###### Theorem 4.3.
Let $\pconf_n(\CC)/\CC^\times$ be the quotient of pure configuration space by the free $\CC^\times$ action. The symmetric group $S_n$ acts on $\pconf_n(\CC)/\CC^\times$ by permuting coordinates. Let $A_n^\bullet$ be the sequence of $S_n$-modules constructed in Proposition 4.2. Then for each $k\geq 0$ we have an isomorphism of $S_n$-modules \begin{eqnarray*} H^k(\pconf_n(\CC)/\CC^\times,\QQ)\cong A_n^k. \end{eqnarray*}

###### Proof.
We regard $X_n := \pconf_n(\CC)$ as the total space of a $\CC^{\times}$-bundle over the base space $Y_n := \pconf_n(\CC)/\CC^\times$. As noted in Sect. 3.2 the cohomology of $X_n$ is that of the pure braid group, with its $S_n$-action. Viewing $\CC^{\times}$ as $\RR^+ \times S^1$, we see that $X_n$ is an $\RR^+$-bundle over the base space $Z_n := \pconf_n(\CC)/\RR^+$, such that $Z_n$ is an $S^1$-bundle over $Y_n$. The space $Z_n$ is a real-analytic manifold which inherits the $S_n$-action. For any $(z_1,z_2, \ldots, z_n) \in \pconf_n(\CC)$, let $[[z_1,z_2, \ldots, z_n]]$ denote its image in $Z_n$ Since $z_1 \neq z_2$, we may rescale this vector by $c = \frac{1}{|z_1- z_2|} \in \CC^{\times}$ to get $(\tilde{z}_1, \tilde{z}_2, \ldots, \tilde{z}_n) = \frac{1}{|z_1 - z_2|} (z_1, \ldots, z_n)$, which comprise exactly the set of all $(\tilde{z}_1, \tilde{z}_2, \ldots, \tilde{z}_n) \in X_n$ satisfying the linear constraint $\tilde{z}_1 - \tilde{z}_2 \in U(1) = \{ z \in \CC: \, |z|=1\}$. We obtain a global section $Z_n \to X_n$ by mapping $[[z_1,z_2, \ldots, z_n]] \mapsto \frac{1}{|z_1 - z_2|}(z_1, \ldots, z_n)$, so may regard $Z_n \subset X_n$, noting that it is invariant under the $S_n$-action. Under this embedding we see that $Z_n$ is a strong deformation retract of $X_n$, so has the same homotopy type as $X_n$. The retraction map is: \begin{eqnarray*} h_t(z_1, z_2,\ldots, z_n) := \left((1-t) |z_1 -z_2| + t\right) \frac{1}{|z_1-z_2|} (z_1, z_2, \ldots, z_n) \quad \hbox{for }0 \le t\le 1. \end{eqnarray*} Consequently $H^k( X_n, \QQ) \cong H^k(Z_n, \QQ)$, for each $k \ge 0$ as $S_n$-modules.

For any $(z_1,z_2, \ldots, z_n) \in X_n$, let $[z_1,z_2, \ldots, z_n]$ denote its image in $Y_n$. Since $z_1 \neq z_2$, we may rescale this vector by $\frac{1}{z_1- z_2} \in \CC^{\times}$ to get $(\tilde{z}_1, \tilde{z}_2, \ldots, \tilde{z}_n) = \frac{1}{z_1 - z_2}(z_1, \ldots, z_n)$, which comprise exactly the set of all $(\tilde{z}_1, \tilde{z}_2, \ldots, \tilde{z}_n) \in X_n$ satisfying the linear constraint $\tilde{z}_1 - \tilde{z}_2 = 1$. These define a global coordinate system for $Y_n$, identifying it as an open complex manifold, and the map $Y_n \to X_n$ sending $[z_1, z_2 \ldots, z_n] \mapsto (\tilde{z}_1, \tilde{z}_2, \ldots, \tilde{z}_n)$ is a nowhere vanishing global section of this bundle, so we may view $Y_n \subset Z_n \subset X_n$. This map is a nowhere vanishing section of $Y_n$ inside the $S^1$-bundle $Z_n$ as well.

The Gysin long exact sequence for $Z_n$ as an $S^1$-bundle over $Y_n$ is \begin{eqnarray*} \xrightarrow{e_\wedge} H^{k}(Y_n, \QQ) {\rightarrow} H^k(Z_n, \QQ) {\rightarrow} H^{k-1}(Y_n, \QQ) \xrightarrow{e_\wedge} H^{k+1}(Y_n, \QQ) {\rightarrow} H^{k+1}(Z_n, \QQ) {\rightarrow} \end{eqnarray*} The Euler class $e \in H^2(Y_n, \QQ)$ of this is zero since the bundle has a nowhere vanishing global section in $Z_n$. Thus $e_\wedge$ is the zero map and the Gysin sequence splits into short exact sequences \begin{eqnarray*} 0 \longrightarrow H^{k}(Y_n, \QQ) \longrightarrow H^k(Z_n, \QQ) \longrightarrow H^{k-1}(Y_n, \QQ) \longrightarrow 0. \end{eqnarray*} The maps are $S_n$-equivariant, since the Gysin sequence is functorial. It follows from Maschke's theorem that

 \begin{eqnarray}\label{sumdecomp} H^k(X_n,\QQ) \cong H^k(Z_n, \QQ) \cong H^{k-1}(Y_n,\QQ) \oplus H^k(Y_n,\QQ) \end{eqnarray} (4.2)
as $S_n$-modules. Since $H^{-1}(Y_n,\QQ) = A_n^{-1} = 0$ by convention, we have $H^0(Y_n,\QQ) \cong A_n^0 \cong H^0(Z_n,\QQ) \cong H^0(X_n, \QQ)$. It then follows inductively from (4.2) and \begin{eqnarray*} H^k(X_n,\QQ) \cong A_n^{k-1} \oplus A_n^k, \end{eqnarray*} that $H^k(Y_n,\QQ) \cong A_n^k$ as $S_n$-modules for all $k\geq 0$.

###### Remark.
The configuration space $\pconf(\CC)$ is a hyperplane complement as treated in the book of [27]. It equals \begin{eqnarray*} M(\sA_n): = \CC^n \smallsetminus \bigcup_{H_{i,j} \in \sA_n} H_{i,j}, \end{eqnarray*} where $\sA_n :=\{ H_{i,j}: \, 1 \le i < j \le n\}$ denotes the braid arrangement of hyperplanes $H_{i,j} : z_i = z_j$ for $1 \le i< j\le n$.

## 5. Polynomial Splitting Measures and Characters

We now express the splitting measure coefficients $\alpha_n^k(C_\lambda)$ in terms of the character values $\chi_n^k(\lambda)$ where $\chi_n^k$ is the character of the $S_n$-representation $A_n^k$ constructed in Proposition 4.2. As a corollary we deduce that the rescaled $z$-splitting measures are characters when $z = -\frac{1}{m}$ and virtual characters when $z = \frac{1}{m}$, generalizing results from [17].

### 5.1. Expressing Splitting Measure Coefficients by Characters.

Recall, \begin{eqnarray*} \nu_{n, z}^{\ast}(C_{\lambda}) = \frac{N_\lambda(z)}{z^n - z^{n-1}}= \sum_{k=0}^{n-1} \alpha_n^k(C_{\lambda}) \left(\tfrac{1}{z}\right)^k. \end{eqnarray*} We now express the splitting measure coefficient $\alpha_n^k(C_{\lambda})$ in terms of the character value $\chi_n^k(\lambda)$.

###### Theorem 5.1.
Let $n\geq 2$ and $\lambda$ be a partition of $n,$ then \begin{eqnarray*} \nu^*_{n,z}(C_\lambda) = \frac{1}{z_\lambda} \sum_{k=0}^{n-1}{(-1)^k \chi_n^k(\lambda) \left(\tfrac{1}{z}\right)^k}, \end{eqnarray*} where $\chi_n^k$ is the character of the $S_n$-representation $A_n^k$ defined in Proposition 4.2. Thus$,$ \begin{eqnarray*} \alpha_n^k(C_{\lambda}) = \frac{1}{z_\lambda}(-1)^k \chi_n^k(\lambda). \end{eqnarray*}

###### Proof.
In Theorem 3.2 we showed \begin{eqnarray*} N_\lambda(z) = \frac{1}{z_\lambda}\sum_{k=0}^{n}{(-1)^k h_n^k(\lambda)z^{n-k}}, \end{eqnarray*} where $h_n^k$ is the character of $H^k(P_n,\QQ)$. The $S_n$-representations $A_n^k$ were defined in Proposition 4.2 where we showed that

 \begin{eqnarray}\label{dirsum} H^k(P_n,\QQ) \cong A_n^{k-1} \oplus A_n^k. \end{eqnarray} (5.1)
Taking characters in (5.1) gives \begin{eqnarray*} h_n^k = \chi_n^{k-1} + \chi_n^k. \end{eqnarray*} We compute \begin{eqnarray*} \frac{N_\lambda(z)}{z^n} &=& \frac{1}{z_\lambda}\sum_{k=0}^n{(-1)^kh_n^k(\lambda) \left(\tfrac{1}{z}\right)^k}\\ &=& \frac{1}{z_\lambda}\sum_{k=0}^n{(-1)^k\left(\chi_n^{k-1}(\lambda) + \chi_n^k(\lambda)\right)\left(\tfrac{1}{z}\right)^k}\\ &=& \left(1 - \tfrac{1}{z}\right)\frac{1}{z_\lambda}\sum_{k=0}^{n-1}{(-1)^k \chi_n^k(\lambda)\left(\tfrac{1}{z}\right)^k}. \end{eqnarray*} Dividing both sides by $\left(1 - \frac{1}{z}\right)$ yields \begin{eqnarray*} \nu_{n,z}^\ast(C_\lambda) = \frac{N_\lambda(z)}{\left(1 - \tfrac{1}{z}\right)z^n} = \frac{1}{z_\lambda}\sum_{k=0}^{n-1}{(-1)^k \chi_n^k(\lambda)\left(\tfrac{1}{z}\right)^k}. \end{eqnarray*} Comparing coefficients in the two expressions for $\nu_{n,z}^\ast(C_\lambda)$ we find \begin{eqnarray*} \alpha_n^k(C_{\lambda}) = \frac{1}{z_\lambda}(-1)^k \chi_n^k(\lambda). \end{eqnarray*}

### 5.2. Cycle Polynomial and Splitting Measure as Equivariant Poincar[U+00B4]e Polynomials.

Given a complex manifold $X$, the Poincaré polynomial of $X$ is defined by \begin{eqnarray*} P(X, t) = \sum_{k\geq 0}{\dim H^k(X,\QQ) t^k}. \end{eqnarray*} If a finite group $G$ acts on $X$, then the cohomology $H^k(X,\QQ)$ is a $\QQ$- representation of $G$ with character $h_X^k$, and the equivariant Poincaré polynomial of $X$ at $g\in G$ is defined by \begin{eqnarray*} P_g(X,t) = \sum_{k\geq 0}\tr(g, H^k(X, \QQ) t^k = \sum_{k \ge 0} h_X^k(g) t^k. \end{eqnarray*} Note that if $g = 1$ is the identity of $G$, then $h_X^k(1) = \dim H^k(X,\QQ)$ and $P_1(X,t) = P(X,t)$.

Under the change of variables $z = -\frac{1}{t}$, the work of [19], Theorem 5.5 identifies (rescaled) cycle polynomials with equivariant Poincaré polynomials of $\pconf_n(\CC)$, for $g \in S_n$, as \begin{eqnarray*} \frac{1}{z^n}N_{[g]}(z) = \frac{|C_{\lambda}|}{n!}\sum_{k\geq 0}{h_n^k(g) t^k} = \frac{1}{z_\lambda}P_{g}(\pconf_n(\CC), t) \end{eqnarray*}

Using the result of Sect. 4.3 we obtain a similar interpretation of the splitting measure values.

###### Theorem 5.2.
Let $Y_n= \pconf_n(\CC)/\CC^{\times}$. Setting $\wakaka = -\frac{1}{z},$ for each $g \in S_n$ the $z$-splitting measure is given by the scaled equivariant Poincaré polynomial \begin{eqnarray*} \nu^*_{n,z}(g) = \frac{1}{n!} \sum_{k=0}^{n-1}{ \tr(g: H^k( Y_n, \QQ)) \wakaka^k}, \end{eqnarray*} attached to the complex manifold $Y_n,$ where $g$ acts as a permutation of the coordinates.

###### Proof.
This formula follows from Theorem 5.1, using also the identification of $A_n^k = H^k(Y_n, \QQ)$ as an $S_n$-module in Theorem 4.3. Since we evaluate the character on a single element $g \in S_n$, the prefactor becomes $\frac{1}{z_{\lambda} c_{\lambda}} = \frac{1}{n!}.$

###### Remark.
In the theory of hyperplane arrangements treated in [27] the change of variable $z= -\frac{1}{t}$ appears as an involution converting the Poincaré polynomial of a hyperplane complement (such as $\pconf_n(\CC)$) to another invariant, the characteristic polynomial of an arrangement, given in [27], Defn. 2.52.

### 5.3. Splitting Measures for $z = \pm \frac{1}{m}$.

Representation-theoretic interpretations of the rescaled $z$-splitting measures for $z = \pm 1$ were studied in [17], Sec. 5. Theorem 5.3 below generalizes those results to give representation-theoretic interpretations for $z = \pm \frac{1}{m}$ when $m\geq 1$ is an integer.

###### Theorem 5.3.
Let $n\geq 2$ and $\lambda$ be a partition of $n,$ then
1. 1. For $z = -\frac{1}{m}$ with $m\geq1$ an integer$,$ we have \begin{eqnarray*} \nu_{n,-\frac{1}{m}}^\ast(C_\lambda) = \frac{1}{z_\lambda}\sum_{k=0}^{n-1}{\chi_n^k(\lambda)m^k}. \end{eqnarray*} The function $z_\lambda \nu_{n, -\frac{1}{m}}^\ast(C_\lambda)$ is therefore the character of the $S_n$-representation \begin{eqnarray*} B_{n,m} = \bigoplus_{k=0}^{n-1}{\left(A_n^k\right)^{\oplus m^k}}, \end{eqnarray*} with dimension \begin{eqnarray*} \dim B_{n,m} = \prod_{j=2}^{n-1}{(1 + jm)}. \end{eqnarray*}
2. 2. For $z = \frac{1}{m}$ with $m\geq 1$ an integer$,$ we have \begin{eqnarray*} \nu_{n, \frac{1}{m}}^\ast(C_\lambda) = \frac{1}{z_\lambda}\sum_{k=0}^{n-1}{(-1)^k\chi_n^k(\lambda) m^k}. \end{eqnarray*} The function $z_\lambda \nu_{n,\frac{1}{m}}^\ast(C_\lambda)$ is a virtual character$,$ the difference of characters of representations $B_{n,m}^+$ and $B_{n,m}^-,$ \begin{eqnarray*} B_{n,m}^+ \cong \bigoplus_{2j < n}{\left(A_n^{2j}\right)^{\oplus m^{2j}}} \hspace{2em} B_{n,m}^- \cong \bigoplus_{2j+1 < n}{\left(A_n^{2j+1}\right)^{\oplus m^{2j+1}}}. \end{eqnarray*} These representations have dimensions \begin{eqnarray*} \dim B_{n,m}^{\pm} = \frac{1}{2} \left( \prod_{j=2}^{n-1}(1+ jm) \pm \prod_{j=2}^{n-1} (1-jm) \right) \end{eqnarray*} respectively.

###### Proof.
1. (1) The formula for the $(-\frac{1}{m})$-splitting measure follows by substituting $z = -\frac{1}{m}$ in Theorem 5.1. [1], Cor. 2 shows the Poincaré polynomial $p(t)$ of the pure braid group $P_n$ has the product form \begin{eqnarray*} p(t) = \prod_{j=1}^{n-1}{(1 + jt)} = \sum_{k=0}^n{h_n^k\left((1^n)\right) t^k}. \end{eqnarray*} On the other hand, by Theorem 3.2 we have

 \begin{eqnarray}\label{poincare} n! (-1)^n t^n N_{(1^n)}(-t^{-1}) = \sum_{k=0}^n{h_n^k\left((1^n)\right) t^k}. \end{eqnarray} (5.2)
Dividing (5.2) by $1 + t$ we have

 \begin{eqnarray}\label{poincare2} \prod_{j=2}^{n-1}{(1 + jt)} = n! (-1)^n t^n \frac{N_{(1^n)}(-t^{-1})}{1 + t} = \sum_{k=0}^{n-1}{\chi_n^k\left((1^n)\right) t^k}. \end{eqnarray} (5.3)
Substituting $t = m$ gives the dimension formula.
2. (2) Substituting $z= \frac{1}{m}$ in Theorem 5.1 gives the formula for the $\frac{1}{m}$-splitting measure. Separating the even and odd parts we have \begin{eqnarray*} z_\lambda \nu_{n, \frac{1}{m}}^\ast(C_\lambda) = \sum_{2j < n}{\chi_n^{2j}(\lambda)m^{2j}} - \sum_{2j+1 < n}{\chi_n^{2j+1}(\lambda)m^{2j+1}}. \end{eqnarray*} Hence $z_\lambda \nu_{n,\frac{1}{m}}^\ast(C_\lambda) = \chi_{n,m}^+(\lambda) - \chi_{n,m}^-(\lambda)$, where $\chi_{n,m}^{\pm}$ are characters of $B_{n,m}^{\pm}$ respectively. The dimension formulas follow from decomposing (5.3) into even and odd parts.

###### Remark.
Other results in [17], Theorems 3.2, 5.2 and 6.1 determine the values of the rescaled splitting measures for $z = \pm 1$, showing they are supported on remarkably few conjugacy classes; for $z=1$ these were the Springer regular elements of $S_n$. Theorem 5.3 does not account for the small support of the characters for $z =\pm1$. The characters $h_n^k$ and $\chi_n^k$ have large support in general, hence cancellation must occur to explain the small support. It would be interesting to account for this phenomenon.

### 5.4. Cohomology of the Pure Braid Group and the Regular Representation.

We use Theorem 5.1 together with the splitting measure values at $z=-1$ computed in [17] to determine a relation between the $S_n$-representation structure of the pure braid group cohomology and the regular representation of $S_n$. Let $A_n^k$ be the $S_n$-subrepresentation constructed in Proposition 4.2, and define the $S_n$-representation \begin{eqnarray*} A_n := \bigoplus_{k=0}^{n-1} A_n^k. \end{eqnarray*}

###### Theorem 5.4.
Let $\triv_n,$ $\Sign_n,$ and $\QQ [S_n]$ denote the trivial$,$ sign$,$ and regular representations of $S_n$ respectively. Then there are isomorphisms of $S_n$-representations$,$ \begin{eqnarray*} \bigoplus_{k=0}^n{H^k(P_n,\QQ)\otimes \Sign_n^{\otimes k} }\cong\QQ[S_n]. \end{eqnarray*} and \begin{eqnarray*} A_n\otimes \left( \triv_n \oplus\Sign_n\right)\cong\QQ [S_n]. \end{eqnarray*}

###### Proof.
We showed in Proposition 4.2 that $H^k(P_n,\QQ) \cong A_n^{k-1} \oplus A_n^k$, with $A_n^{-1} = A_n^n = 0$. Therefore, summing over $0 \le k \le n$, \begin{eqnarray*} A_n \cong \bigoplus_{k \text{ even}}H^k(P_n,\QQ) \cong \bigoplus_{k \text{ odd}}H^k(P_n,\QQ). \end{eqnarray*} Since $\Sign_n^{\otimes 2} \cong \triv_n$, we have \begin{eqnarray*} \bigoplus_{k=0}^n{H^k(P_n,\QQ)\otimes \Sign_n^{\otimes k}} &\cong& \left(\bigoplus_{k \text{ even}}H^k(P_n,\QQ)\otimes \triv_n \right) \oplus \left(\bigoplus_{k \text{ odd}}H^k(P_n,\QQ)\otimes \Sign_n \right)\\ &\cong& (A_n \otimes \triv_n) \oplus (A_n \otimes \Sign_n)\\ &\cong& A_n \otimes (\triv_n \oplus \,\Sign_n). \end{eqnarray*} If $\chi_n$ is the character of $A_n$, then it follows from Theorem 1.4 that \begin{eqnarray*} \chi_n(\lambda) = \sum_{k=0}^{n-1}\chi_n^k(\lambda) = z_\lambda \nu_{n,-1}^\ast(C_\lambda), \end{eqnarray*} so the values of $\chi_n$ are given by the rescaled $(-1)$-splitting measure.

Theorem 6.1 of [17] shows \begin{eqnarray*} \nu_{n,-1}^\ast(C_\lambda) = \begin{cases} \frac{1}{2} & \lambda = (1^n) \text{ or } (1^{n-2}\, 2),\\ 0 & \text{otherwise.} \end{cases} \end{eqnarray*}

Now let $\rho = \chi_n \cdot (1_n + \sign_n)$ be the character of $A_n \otimes (\triv_n \oplus\, \Sign_n)$. If $\lambda = (1^n)$, we compute \begin{eqnarray*} \rho(\lambda)= \chi_n(\lambda)\left(1 + \sign_n(\lambda)\right) = n! \nu_{n,-1}^\ast(C_\lambda)(2) = n!. \end{eqnarray*} If $\lambda = (1^{n-2}\, 2)$, then $\left(1 + \sign_n(\lambda)\right) = 0$, hence $\rho(\lambda) = 0$. If $\lambda$ is any other partition, then $\nu_{n,-1}^\ast(C_\lambda) = 0$, hence $\rho(\lambda) = 0$. Therefore $\rho$ agrees with the character of the regular representation, proving \begin{eqnarray*} \bigoplus_{k=0}^n{H^k(P_n,\QQ)\otimes \Sign_n^{\otimes k}} \cong A_n \otimes (\triv_n \oplus\, \Sign_n) \cong \QQ[S_n]. \end{eqnarray*}

## 6. Other Interpretations of $A_n^k$

Theorem 4.3 interprets the $S_n$-representation $A_n^k$ geometrically as \begin{eqnarray*} A_n^k \cong H^k(\pconf_n(\CC)/\CC^\times,\QQ). \end{eqnarray*}

In this section we note two other interpretations of $A_n^k$, coming from combinatorial constructions previously studied in the literature. These interpretations imply that the $A_n^k$ for fixed $k$ exhibit representation stability in the sense of [5] as $n \to \infty$.

Proposition 4.2 gave the following direct sum decomposition of the pure braid group cohomology,

 \begin{eqnarray}\label{eqn:Aspsiso} H^k(P_n, \QQ) \cong A_n^{k-1} \oplus A_n^k. \end{eqnarray} (6.1)
The isomorphisms (6.1) uniquely determine the $A_n^k$ as $S_n$-representations up to isomorphism. Uniqueness holds since finite-dimensional representations are semisimple by Maschke's theorem, using the general result that if $0 = C^0, C^1, C^2,\ldots$ is any sequence of semisimple modules with submodules $B^k \subseteq C^k$, then isomorphisms \begin{eqnarray*} C^k \cong B^{k-1} \oplus B^k \end{eqnarray*} for each $k$ determine the $B^k$ up to isomorphism.

Let $\Pi_n$ denote the collection of partitions of a set with $n$ elements, partially ordered by refinement (see [32], Example 3.10.4)). [15], Sec. 2 describe two other sequences of $S_n$-representations giving direct sum decompositions of $H^k(P_n, \QQ)$ coming from the Whitney and simplicial homology of the lattice $\Pi_n$.

###### Proposition 6.1.
1. 1. There is an isomorphism of $S_n$-representations

 \begin{eqnarray}\label{whitney} H^k(P_n, \QQ) \cong WH_k(\Pi_n), \end{eqnarray} (6.2)
where $WH_k(\Pi_n)$ is the $k$th Whitney homology of the lattice $\Pi_n$.
2. 2. There is an isomorphism of $S_n$-representations \begin{eqnarray*} WH_k(\Pi_n) \cong \beta_{ [k-1]} (\Pi_n) \oplus \beta_{ [k]} (\Pi_n) \end{eqnarray*} where $\beta_{ [k]} (\Pi_n)$ is the $[k]= \{ 1, 2,\ldots, k\}$-rank selected homology of the lattice $\Pi_n$.
3. 3. There is an isomorphism of $S_n$-representations \begin{eqnarray*} \beta_{[k]}(\Pi_n) \cong \widetilde{H}_{k-1}\left(\Pi_n^k\right), \end{eqnarray*} where $\Pi_n^k$ is the sub-poset of $\lambda \in \Pi_n$ with $|\lambda| - \ell(\lambda) \leq k$ and $\widetilde{H}_{k-1}\left(\Pi_n^k\right)$ denotes its reduced simplicial homology.

###### Proof.
(1) This result is due to [34], Theorem 4.4 (iii), cf. [15], Thm. 2.11, Sec. 2.3. (See [15], Sec. 2.4 for more on the Whitney homology of $\Pi_n$.) (2) [33], Prop. 1.9 decomposes $WH_k(\Pi_n)$ as

 \begin{eqnarray}\label{rankspsselected} WH_k(\Pi_n) \cong \beta_{[k-1]}(\Pi_n) \oplus \beta_{[k]}(\Pi_n), \end{eqnarray} (6.3)
where $[k] = \{1,2,\ldots,k\}$ and $\beta_{[k]}(\Pi_n)$ is the $[k]$ -rank selected homology of the lattice $\Pi_n$ ([15], Prop. 2.17). (3) Because the lattice $\Pi_n$ is Cohen--Macaulay , ([15], Sec. 2.5) note the isomorphism

 \begin{eqnarray}\label{simplicial} \beta_{[k]}(\Pi_n) \cong \widetilde{H}_{k-1}\big(\Pi_n^k\big), \end{eqnarray} (6.4)
where $\Pi_n^k$ is the sub-poset of $\lambda \in \Pi_n$ with $|\lambda| - \ell(\lambda) \leq k$ and $\widetilde{H}_{k-1}\left(\Pi_n^k\right)$ is its reduced simplicial homology .

The following proposition relates $A_n^k$, $\beta_{[k]}(\Pi_n)$, and $\widetilde{H}_{k-1}\left(\Pi_n^k\right)$ using (6.1) .

###### Proposition 6.2.
Let $\Pi_n$ be the lattice of partitions of an $n$-element set$,$ and $\Pi_n^k \subseteq \Pi_n$ the sub-poset comprised of $\lambda \in \Pi_n$ with $|\lambda| - \ell(\lambda) \leq k$. Then we have the following isomorphisms of $S_n$-representations \begin{eqnarray*} A_n^{k} \cong \beta_{[k]}(\Pi_n) \cong \widetilde{H}_{k-1}\big(\Pi_n^k\big). \end{eqnarray*}

###### Proof.
The isomorphisms (6.2) and (6.3) in Proposition 6.1 give the direct sum decompositions \begin{eqnarray*} H^k(P_n, \QQ) \cong \beta_{[k-1]}(\Pi_n) \oplus \beta_{[k]}(\Pi_n) \end{eqnarray*} for $0 \le k \le n$. By (6.1) we have that \begin{eqnarray*} H^k(P_n, \QQ) \cong A_n^{k-1} \oplus A_n^{k}. \end{eqnarray*} Since for $k=0$, \begin{eqnarray*} \beta_{[-1]}(\Pi_n) \cong A_n^{-1} = \{0\}, \end{eqnarray*} we obtain by induction on $k \ge 1$ that \begin{eqnarray*} A_n^k \cong \beta_{[k]}(\Pi_n) \end{eqnarray*} Combining this isomorphism with (6.4) finishes the proof.

We deduce the representation stability of the characters $\chi_n^k$ from known results.

###### Proof of Theorem 1.5.
The $S_n$-representations of the rank-selected homology $\beta_{[k-1]}(\Pi_n)$ were shown by [15], Corollary 5.4 to exhibit representation-stability for fixed $k$ and varying $n$ and to stabilize sharply at $n=3k+1$. This fact combined with Proposition 6.2 proves Theorem 1.5.

The following tables for $A_n^1$ and $A_n^2$ exhibit representation stability and the sharp stability phenomenon at $n=3k+1$. We give irreducible decompositions, with multiplicities, of $H^k(P_n, \QQ)$ and $A_n^1$ in Table 5 and for $A_n^2$ in Table 6. To read the tables, for example, the entry $[4,1,1]$ denotes the isomorphism class of the irreducible representation of $S_6$ associated to the Specht module of the partition $[4, 1, 1]$ of $n=6$, in the notation of [29], Sec. 2.3, who gives a construction of the Specht module representatives of the irreducible isomorphism classes.

Table 5 Irreducible $S_n$-module decompositions for $H^1(P_n, \QQ)$ and $A_n^1$.
 $n$ $\dim{H^1}$ $H^1(P_n, \QQ)$ $\dim A_n^1$ $A_n^1$ $2$ $1$ $[2]$ $0$ $0$ $3$ $3$ $[3] \oplus [2,1]$ $2$ $[2,1]$ $4$ $6$ $[4]\oplus [3,1] \oplus [2,2]$ $5$ $[3,1] \oplus [2,2]$ $5$ $10$ $[5] \oplus [4,1] \oplus [3,2]$ $9$ $[4, 1] \oplus [3, 2]$ $n \ge 4$ ${n \brack n-1}$ $[n] \oplus [n-1,1] \oplus [n-2,2]$ ${n \brack n-1} -1$ $[n-1,1] \oplus [n-2,2]$
Here $\lambda$ abbreviates the irreducible representation $\sS^{\lambda}$
Table 6 Irreducible $S_n$-module decomposition for $A_n^2$.
 $n$ $\dim A_n^2$ $A_n^2$ $3$ $0$ $0$ $4$ $6$ $[3, 1] \oplus [2,1,1]$ $5$ $26$ $[4,1] \oplus [3,2] \oplus 2[3,1,1] \oplus [2,2,1]$ $6$ $71$ $[5,1] \oplus [4,2] \oplus 2[4,1,1] \oplus [3,3] \oplus 2[3,2,1]$ $7$ $155$ $[6,1] \oplus [5,2] \oplus 2[5,1,1] \oplus [4,3] \oplus 2[4,2,1] \oplus [3,3,1]$ $8$ $295$ $[7,1] \oplus [6,2] \oplus 2[6,1,1] \oplus [5,3] \oplus 2[5,2,1] \oplus [4,3,1]$ $n \ge 7$ ${n \brack n-2} - {n \brack n-1} +1$ $[n-1,1] \oplus [n-2,2] \oplus 2[n-2,1,1] \oplus [n-3,3]$ $\oplus 2[n-3,2,1] \oplus [n-4,3,1]$
###### Acknowledgements.
We thank Richard Stanley for raising a question about the relation of the braid group cohomology to the regular representation, answered by Theorem 1.4. We thank Weiyan Chen for pointing out to us that Theorem 1.1 is shown in [19] and for subsequently bringing the work of [12] to our attention. We thank Philip Tosteson and John Wiltshire-Gordon for helpful conversations. We thank the reviewers for helpful comments. Work of J. C. Lagarias was partially supported by NSF Grant DMS-1401224.

## References

[1] Arnol'd, V.I.: The cohomology ring of the colored braid group. Math. Notes 5 , 138--140 (1969) [English translation of: Mat. Zametki 5 , 227--231 (1969)]
[2] Bhargava, M.: Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants. Int. Math. Res. Not. 17 , Art. ID rnm052, 20 pp. (2007)
[3] Callegaro, F., Gaiffi, G.: On models of the braid arrangement and their hidden symmetries. Int. Math. Res. Not. 21 , 11117--11149 (2015)
[4] Chen, W.: Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting (2016). eprint: arXiv:1603.03931
[5] Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245 , 250--314 (2013)
[6] Church, T., Ellenberg, J.S., Farb, B.: Representation stability in cohomology and asymptotics for families of varieties over finite fields. In: Algebraic Topology: Applications and New Directions, pp. 1--54. Contemporary Mathematics, vol. 620. American Mathematical Society, Providence (2014)
[7] Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164 (9), 1833--1910 (2015)
[8] Dedekind, R.: [U+00A8]Uber Zusammenhang zwischen der Theorie der Ideale und der Theorie der h[U+00A8]ohere Kongruenzen, Abh. K[U+00A8]onig. Ges. der Wissen. zu G[U+00A8]ottingen 23 , 1--23 (1878)
[9] Dimca, A., Yuzvinsky, S.: Lectures on Orlik--Solomon algebras. In: Arrangements, Local Systems and Singularities. Progress in Mathematics, vol. 283, pp. 83--110. Birkh[U+00A8]auser, Basel (2010)
[10] Dołega, M., Féray, V., [U+00B4]Sniady, P.: Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations. Adv. Math. 225 , 81--120 (2010)
[11] Farb, B.: Representation stability. In: Proceedings of the 2014 ICM, Seoul, Korea. eprint: arXiv:1404.4065
[12] Gaiffi, G.: The actions of $S_{n+1}$ and $S_n$ on the cohomology ring of a Coxeter arrangement of type $A_{n-1}$. Manuscr. math. 91 , 83--94 (1996)
[13] Getzler, E.: Operads and moduli spaces of genus $0$ Riemann surfaces. In: The Moduli Space of Curves (Texel Island 1994), pp. 199--230. Progress in Mathematics, vol. 129. Birkh[U+00A8]auser, Boston (1995)
[14] Grothendieck, A.: Rev^etements étales et groupe fondamental. Fasc. II: Exposés 6, 8 `a11, Volume 1960/61 of Séminaire de Géomeétrie Albebrique. IHES, Paris (1963)
[15] Hersh, P., Reiner, V.: Representation stability for cohomology of configuration spaces in ${\mathbb{R}}^d$ (Appendix joint with Steven Sam). In: International Mathematics Research Notices (2015). doi: . eprint: arXiv:1505.04196v3
[16] Kisin, M., Lehrer, G.I.: Equivariant Poincaré polynomials and counting points over finite fields. J. Algebra 247 (2), 435--451 (2002)
[17] Lagarias, J.C.: A family of measures on symmetric groups and the field with one element. J. Number Theory 161 , 311--342 (2016)
[18] Lagarias, J.C., Weiss, B.L.: Splitting behavior of $S_n$ polynomials. Res. Number Theory 1 , paper 9, 30 pp. (2015)
[19] Lehrer, G.I.: On the Poincaré series associated with Coxeter group actions on complements of hyperplanes. J. Lond. Math. Soc. 36 (2), 275--294 (1987)
[20] Lehrer, G.I.: The $\ell$-adic cohomology of hyperplane complements. Bull. Lond. Math. Soc. 24 , 76--82 (1992)
[21] Lehrer, G.I., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104 (2), 410--424 (1986)
[22] Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)
[23] Mathieu, O.: Hidden $\Sigma _{n+1}$-actions. Commun. Math. Phys. 176 , 467--474 (1996)
[24] Metropolis, N., Rota, G.-C.: Witt vectors and the algebra of necklaces. Adv. Math. 50 , 95--125 (1983)
[25] Moreau, C.: Sur les permutations circulaires distinctes. Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2 (11), 309--314 (1872)
[26] Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56 , 57--89 (1980)
[27] Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der math. Wiss, vol. 300. Springer, Berlin (1992)
[28] Rosen, M.: Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210. Springer, New York (2002)
[29] Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, vol. 203. Springer Science and Business Media, Berlin (2013)
[30] [U+00B4]Sniady, P.: Stanley character polynomials. In: The Mathematical Legacy of Richard P. Stanley, vol. 100, p. 323 (2016)
[31] Stanley, R.P.: Some aspects of groups acting on finite posets. J. Comb. Theory Ser. A 32 , 132--161 (1982)
[32] Stanley, R.P.: Enumerative combinatorics, vol. 1. In: Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997) [Corrected reprint of the 1986 original]
[33] Sundaram, S.: The homology representations of the symmetric group on Cohen--Macaulay subposets of the partition lattice. Adv. Math. 104 , 225--296 (1994)
[34] Sundaram, S., Welker, V.: Group actions on arrangements of linear subspaces and applications to configuration spaces. Trans. Am. Math. Soc. 349 (4), 1389--1420 (1997)
[35] Weiss, B.L.: Probabilistic Galois theory over $p$-adic fields. J. Number Theory 133 , 1537--1563 (2013)
[36] Yuzvinsky, S.: Orlik-Solomon algebras in algebra, topology and geometry. Russ. Math. Surv. 56 , 294--364 (2001)