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

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 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$.

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$.

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]$.

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.

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 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].

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. 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.

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).

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.

Proof.

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$.

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$.

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.

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.

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.

Proof.

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.

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].

Proof.

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)$.

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

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]).

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.

Proof.

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$.

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.

Proof.

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)$.

Proof.

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.

Proof.

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.

Proof.

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*}

Proof.