2 Hall sets

Let X be a nonempty set, considered as an alphabet. Let again $X^*$ be the free monoid on X. We consider its elements as associative words. It is equipped with the binary operation $(u,v)\mapsto uv$ of associative concatenation. Let $\mathcal {M}_X$ be the free magma on X (see [Reference Serre35, Part I, Ch. IV, §1], [Reference Efrat8, §2]). Thus the elements of $\mathcal {M}_X$ are the nonempty nonassociative words in the alphabet X, and it is equipped with the binary operation $(u,v)\mapsto (uv)$ of nonassociative concatenation. There is a natural foliage (brackets-dropping) map $f\colon \mathcal {M}_X\to X^*$ , which is the identity on X (considered as a subset of both $\mathcal {M}_X$ and $X^*$ ) and which commutes with the concatenation maps.

We fix a total order on X. It induces on $X^*$ the alphabetic order $\leq _{\text {alp}}$ , which is also total. We denote the length of a word $w\in X^*$ by $\lvert w\rvert $ .

Let $\mathcal {H}$ be a subset of words in $\mathcal {M}_X$ and $\leq $ any total order on $\mathcal {H}$ . We say that $(\mathcal {H},\leq )$ is a Hall set in $\mathcal {M}_X$ if the following conditions hold [Reference Reutenauer33, §4.1]:

1. (1) $X\subseteq \mathcal {H}$ as ordered sets.

2. (2) If $h=(h'h")\in \mathcal {H}\setminus X$ , then $h"\in \mathcal {H}$ and $h<h"$ .

3. (3) For $h=(h'h")\in \mathcal {M}_X \setminus X$ , one has $h\in \mathcal {H}$ if and only if

   • • $h',h"\in \mathcal {H}$ and $h'<h"$ , and

   • • either $h'\in X$ or $h'=(h_1h_2)$ with $h_2\geq h"$ .

In this case we say that $H=f(\mathcal {H})$ is a Hall set in $X^*$ .

Every $w\in H$ can be written as $w=f(h)$ for a unique $h\in \mathcal {H}$ [Reference Reutenauer33, Cor. 4.5]. If $w\in H\setminus X$ , then we can uniquely write $h=(h'h")$ with $h',h"\in \mathcal {H}$ [Reference Reutenauer33, p. 89]. Setting $w'=f(h'),w"=f(h")\in H$ , we call $w=w'w"$ the standard factorization of w.

The standard factorization of Lyndon words is explicitly given as follows:

We order $\mathbb {Z}_{\geq 0}\times X^*$ lexicographically with respect to the usual order on $\mathbb {Z}_{\geq 0}$ and $\leq _{\textrm {alp}}$ . We then define a second total order $\preceq $ on $X^*$ by setting

(2.1) $$ \begin{align} w_1\preceq w_2\quad \Longleftrightarrow\quad (\lvert w_1\rvert,w_1)\leq(\lvert w_2\rvert,w_2) \end{align} $$

with respect to the latter order on $\mathbb {Z}_{\geq 0}\times X^*$ .

3 Lie algebras

Let R be a unital commutative ring. We write $R\langle X\rangle $ for the free associative R-algebra over the set X. We view its elements as polynomials in the set X of noncommuting variables and with coefficients in R. Alternatively, it is the free R-module on the basis $X^*$ with multiplication induced by concatenation. The algebra $R\langle X\rangle $ is graded with respect to total degree.

We write $R\langle \langle X\rangle \rangle $ for the R-algebra of formal power series in the set X of noncommuting variables and with coefficients in R.

Let k be a field. For an associative k-algebra A, let $A_{\text {Lie}}$ be the Lie algebra on A with Lie bracket $[a,b]=ab-ba$ .

We now assume that X is a nonempty totally ordered set, and fix a Hall set H in $X^*$ .

Let $L(X)$ be the free Lie k-algebra on the set X. The universal enveloping algebra of $L(X)$ is $k\langle X\rangle $ [Reference Serre35, Part I, Ch. IV, Th. 4.2].

Let L be a Lie k-algebra containing X. Define a map $P_L=P_L^H\colon H\to L$ by $P_L(x)=x$ for $x\in X$ , and $P_L(w)=[P_L(u),P_L(v)]$ , if $w=uv$ is the standard factorization of w, as in §2. This construction is functorial in L in the natural sense.

By Proposition 3.1(a) and the Poincaré–Birkhoff–Witt theorem [Reference Serre35, Part I, Ch. III, §4], the products $\prod _{i=1}^mP_{L(X)}(w_i)$ , with $w_1\geq _{\text {alp}}\dotsb \geq _{\text {alp}}w_m$ in H, form a k-linear basis of the universal enveloping algebra $k\langle X\rangle $ of $L(X)$ .

Next assume that $\operatorname {\mathrm {char}}\,k=p>0$ . A restricted Lie k-algebra L is a Lie k-algebra with an additional unary operation $a\mapsto a^{[p]}$ for which there is an associative k-algebra A and a Lie k-algebra monomorphism $\theta \colon L\to A_{\text {Lie}}$ such that $\theta \left (a^{[p]}\right )=\theta (a)^p$ for every $a\in L$ ([Reference Dixon, Du Sautoy, Mann and Segal5, §12.1]; see also [Reference Jacobson19] for an alternative equivalent definition). A morphism of restricted Lie k-algebras is a morphism of Lie k-algebras which commutes with the $(\cdot )^{[p]}$ -maps.

Every associative k-algebra A is endowed with the structure of a restricted Lie algebra $A_{\text {res.Lie}}$ , where we set $[a,b]=ab-ba$ and $a^{[p]}=a^p$ . Every restricted Lie k-algebra L has a unique restricted universal enveloping algebra $\mathcal {U}_{\text {res}}(L)$ . This means that $\mathcal {U}_{\text {res}}(L)$ is an associative k-algebra, and the functor $A\mapsto A_{\text {res.Lie}}$ from the category of associative k-algebras to the category of restricted Lie k-algebras and the functor $L\mapsto \mathcal {U}_{\text {res}}(L)$ from the category of restricted Lie k-algebras to the category of associative k-algebras are adjoint ([Reference Dixon, Du Sautoy, Mann and Segal5, §12.1], [Reference Jacobson19, Ch. V, Th. 12]).

Given a restricted Lie k-algebra L containing X, we define a map $\widehat {P}_L=\widehat {P}_L^H\colon \mathbb {Z}_{\geq 0}\times H\to L$ by $\widehat {P}_L(j,w)=P_L(w)^{[p]^j}$ , where $(\cdot )^{[p]^j}$ denotes applying j times the operation $(\cdot )^{[p]}$ . In analogy with Proposition 3.1(b) we have the following:

Proof. Let $\widehat {L}_0$ be the k-linear subspace of L spanned by $\operatorname {\mathrm {Im}}\left (\widehat {P}_L\right )$ . Let $L_0$ be the k-linear subspace of L spanned by $\operatorname {\mathrm {Im}}(P_L)$ . Clearly, $X\subseteq L_0\subseteq \widehat {L}_0$ . By Proposition 3.1(b), $L_0$ is the Lie k-subalgebra of L generated by X.

Since $\operatorname {\mathrm {char}}\,k=p$ , the binomial formula implies that the subspace $\widehat {L}_0$ is closed under $(\cdot )^{[p]}$ .

If $w,u\in H$ , then $[P_L(w),P_L(u)]\in L_0$ . It follows from the k-bilinearity of the Lie bracket that for every $\alpha ,\beta \in L_0$ , also $[\alpha ,\beta ]\in L_0$ . By induction on $m\geq 1$ , the m-times iterated Lie brackets

$$ \begin{align*} [\alpha,_m,\beta]=[\alpha,[\alpha,[\dotsc [\alpha,\beta]\dotsc]]] \quad \text{and} \quad [\alpha,\beta,_m]=[\dotsc[[\alpha,\beta],\beta],\dotsc,\beta] \end{align*} $$

are also contained in $L_0$ . Using the identities $\left [\alpha ,\beta ^{[p]}\right ]=\left [\alpha ,_p,\beta \right ]$ and $\left [\alpha ^{[p]},\beta \right ]=\left [\alpha ,\beta ,_p\right ]$ (see [Reference Dixon, Du Sautoy, Mann and Segal5, p. 297]), we deduce that $\left [\alpha ^{[p]^j},\beta ^{[p]^r}\right ]\in L_0$ for every $j,r\geq 0$ . By the bilinearity again, $\widehat {L}_0$ is therefore closed under the Lie bracket.

Hence $\widehat {L}_0$ is the restricted Lie k-subalgebra of L generated by X.

There is a free restricted k-algebra $\widehat {L}(X)$ on the generating set X, with the standard universal property. It is the restricted Lie k-subalgebra of $k\langle X\rangle _{\text {res.Lie}}$ generated by X, and its restricted universal enveloping algebra is $k\langle X\rangle $ [Reference Gärtner15, Prop. 1.2.7]. We note that in the algebra $k\langle X\rangle $ one has

$$ \begin{align*} \widehat{P}_{\widehat{L}(X)}(j,w)=P_{L(X)}(w)^{p^j} \end{align*} $$

for every $j\geq 0$ and $w\in H$ . The following analogue of Proposition 3.1(a) generalizes a result of Gärtner [Reference Gärtner15, Th. 1.2.11] (who considers a specific Hall family H):

We grade $L(X)$ and $\widehat {L}(X)$ by total degree, and write $L(X)_n$ , $\widehat {L}(X)_n$ for their homogenous components of degree n.

4 The p-Zassenhaus filtration

We fix as before a prime number p. For an integer $1\leq i \leq n$ , let $j_n(i)=\lceil \log _p(n/i)\rceil $ – that is, $j_n(i)$ is the least integer j such that $ip^j\geq n$ .

Proof. Set $i_k=\left \lceil n/p^k\right \rceil $ . Thus $i_0=n$ , and the sequence $i_k$ is weakly decreasing to $1$ . We may restrict ourselves to k such that $p^k\leq n$ . Then $\left (n/p^k\right )+1\leq n/p^{k-1}$ , so $n/p^k\leq i_k<n/p^{k-1}$ . Thus $j_n(i_k)=k$ .

Since $n/p^k\leq \left \lceil n/p^{k+1}\right \rceil p$ , one has $i_kp^k\leq i_{k+1}p^{k+1}$ – that is, the sequence $i_kp^{j_n\left (i_k\right )}$ is weakly increasing in the above range.

We also observe that if $i< i_{k-1}$ , then $i<n/p^{k-1}$ – that is, $j_n(i)\geq k$ .

(a) $\Rightarrow $ (b): Since (b) certainly holds for $i=n$ , we may assume that $i< n$ , so there is k in the above range such that $i_k\leq i< i_{k-1}$ . By the previous observation, $j_n(i)\geq k$ . We take in (a) $i'=i_k$ to obtain

$$ \begin{align*} i_kp^{j_n\left(i_k\right)}\geq ip^{j_n(i)}\geq i_kp^k=i_kp^{j_n\left(i_k\right)}. \end{align*} $$

Hence $i=i_k$ .

(b) $\Rightarrow $ (a): Suppose that $1\leq i'<i_k$ . There exists l in the above range such that $i_l\leq i'<i_{l-1}$ . Necessarily, $l>k$ , so $i_lp^l\geq i_kp^k$ . As we have observed, $j_n(i')\geq l$ . Hence

$$ \begin{align*} i'p^{j_n\left(i'\right)}\geq i_lp^l\geq i_kp^k=i_kp^{j_n\left(i_k\right)}. \\[-38pt]\end{align*} $$

We define $J(n)$ to be the set of all $1\leq i\leq n$ such that the equivalent conditions of Lemma 4.1 hold.

Now let G be a profinite group. Given closed subgroups $K,K'$ of G and a positive integer m, we write $[K,K']$ (resp., $K^m$ ) for the closed subgroup of G generated by all commutators $[k,k']=k^{-1} (k')^{-1} kk'$ (resp., powers $k^m$ ) with $k\in K$ and $k'\in K'$ .

Recall that the (profinite) lower central series $G^{(i)}$ , $i=1,2,\dotsc ,$ of $G$ is defined inductively by $G^{(1)}=G$ , $G^{(i+1)}=\left [G,G^{(i)}\right ]$ . As in the Introduction, we denote the p-Zassenhaus filtration of G by $G_{\left (n,p\right )}$ , $n=1,2,\dotsc $ . Since $G^{(i)}\leq G^{(n)}$ for $i>n$ ,

$$ \begin{align*} G_{\left(n,p\right)}=\prod_{ip^j\geq n}\left(G^{(i)}\right)^{p^j}=\prod_{i=1}^n\left(G^{(i)}\right)^{p^{j_n(i)}}. \end{align*} $$

The subgroups $G_{\left (n,p\right )}$ of G are characteristic, hence normal. We note that $G^{(n)}\leq G_{\left (n,p\right )}$ .

The Zassenhaus filtration can also be defined inductively by

(4.1) $$ \begin{align} G_{\left(1,p\right)}=G, \quad G_{\left(n,p\right)}=\left(G_{\left(\left\lceil n/p\right\rceil,p\right)}\right)^p\prod_{i+j=n}\left[G_{\left(i,p\right)},G_{\left(j,p\right)}\right], \end{align} $$

for $n\geq 2$ . Indeed, this follows from a theorem of Lazard in the case of discrete groups ([Reference Dixon, Du Sautoy, Mann and Segal5, Th. 11.2], [Reference Lazard25, p. 209, Equation (3.14.5)]), and the profinite analog follows by a density argument. It follows from definition (4.1) that for $n\geq 2$ ,

(4.2) $$ \begin{align} G_{\left(np,p\right)}\leq \left(G_{\left(n,p\right)}\right)^p\left[G,G_{\left(n,p\right)}\right]. \end{align} $$

Let $r\geq 0$ . The following identity was proved in the discrete case by Shalev [Reference Shalev36, Prop. 1.2]; the profinite analog follows again by a density argument:

$$ \begin{align*} \prod_{ip^j\geq n}\left(G^{(i+r+1)}\right)^{p^j}=\left[G,\prod_{ip^j\geq n}\left(G^{(i+r)}\right)^{p^j}\right]. \end{align*} $$

In particular,

(4.3) $$ \begin{align} \prod_{ip^j\geq n}\left(G^{(i+1)}\right)^{p^j}=\left[G,G_{\left(n,p\right)}\right]. \end{align} $$

Proposition 4.3. Let $1\leq i\leq n$ be an integer such that $i\not \in J(n)$ . Then

$$ \begin{align*} \left(G^{(i)}\right)^{p^{j_n(i)}}\leq \left(G_{\left(n,p\right)}\right)^p\left[G,G_{\left(n,p\right)}\right]. \end{align*} $$

Proof. As $i\not \in J(n)$ , there exists $1\leq i'<i$ such that $ip^{j_n(i)}>i'p^{j_n\left (i'\right )}$ . We abbreviate $j=j_n(i)$ and $j'=j_n(i')$ , so $ip^j,i'p^{j'}\geq n$ .

If $j>j'$ , then

$$ \begin{align*} \left(G^{(i)}\right)^{p^j}\leq\left(\left(G^{\left(i'\right)}\right)^{p^{j'}}\right)^{p^{j-j'}}\leq \left(G_{\left(n,p\right)}\right)^p. \end{align*} $$

If $j\leq j'$ , then the inequality $i>i'p^{j'-j}$ and equation (4.3) give

$$ \begin{align*} \left(G^{(i)}\right)^{p^j}\leq \left(G^{\left(i'p^{j'-j}+1\right)}\right)^{p^j}\leq \left[G,G_{\left(n,p\right)}\right]. \\[-40pt]\end{align*} $$

It follows from definition (4.1) that for every n the quotient $G_{\left (n,p\right )}/G_{\left (n+1,p\right )}$ is abelian of exponent dividing p. Consider the graded $\mathbb {F}_p$ -module

$$ \begin{align*} \mathrm{gr} G=\bigoplus_{n\geq0} G_{\left(n,p\right)}/G_{\left(n+1,p\right)}. \end{align*} $$

The commutator map and the p-power map induce on $\mathrm {gr} G$ the structure of a p-restricted Lie $\mathbb {F}_p$ -algebra (see [Reference Dixon, Du Sautoy, Mann and Segal5, §12.2], [Reference Gärtner15, Prop. 1.2.14]).

We now specialize to the case where S is a free profinite group on the basis X, in the sense of [Reference Fried and Jarden14, §17.4]. It is the inverse limit of the free profinite groups on finite subsets of X [Reference Fried and Jarden14, Lemma 17.4.9], so in our following results one may assume whenever convenient that X is actually finite, and use limit arguments for the general case.

By [Reference Gärtner15, Th. 1.3.8], there is a well-defined isomorphism $\mathrm {gr} S\xrightarrow {\sim }\widehat {L}(X)$ of graded restricted Lie algebras. Specifically, the coset of $x\in X$ in $\mathrm {gr}_1 S=S/S_{\left (2,p\right )}$ maps to x.

Let H be, as before, a fixed Hall set in $X^*$ . For every word $w\in H$ we associate an element $\tau _w\in S$ as in [Reference Efrat8]. Thus $\tau _{(x)}=x$ for $x\in X$ , and for a word $w\in H$ of length $i>1$ with standard factorization $w=uv$ , where $u,v\in H$ (see §2), we set $\tau _w=[\tau _u,\tau _v]$ . Then $\tau _w\in S^{(i)}$ . Hence if $ip^j\geq n$ , then $\tau _w^{p^j}\in \left (S^{(i)}\right )^{p^j}\leq S_{\left (n,p\right )}$ .

Let $n\geq 1$ . For a word $w\in H$ of length $1\leq i\leq n$ we abbreviate

$$ \begin{align*} \sigma_w=\tau_w^{p^{j_n(i)}}. \end{align*} $$

Thus $\sigma _w\in S_{\left (n,p\right )}$ .

5 The fundamental matrix

For a profinite ring R, let $R\langle \langle X\rangle \rangle ^\times $ be the group of invertible elements in $R\langle \langle X\rangle \rangle $ (see §3). As before, let S be the free profinite group over the basis X. The continuous Magnus homomorphism

$$ \begin{align*} \Lambda=\Lambda_R\colon S\to R\langle\langle X\rangle\rangle^\times \end{align*} $$

is defined on the (profinite) generators $x\in X$ of S by $\Lambda (x)=1+x$ (see [Reference Efrat7, §5] for details, and note that $1+x$ is invertible by the geometric progression formula). For an arbitrary $\sigma \in S$ we write

$$ \begin{align*} \Lambda(\sigma)=\sum_{w\in X^*}\epsilon_{w,R}(\sigma)w, \end{align*} $$

with $\epsilon _{w,R}(\sigma )\in R$ . The map $\epsilon _{w,R}\colon S\to R$ is continuous, and $\epsilon _{\emptyset ,R}(\sigma )=1$ for every $\sigma $ (where $\emptyset $ denotes the empty word).

Let $\mathbb {U}_i(R)$ be the profinite group of all unitriangular $(i+1)\times (i+1)$ -matrices over R. Given a word $w=(x_1\dotsm x_i)\in X^*$ of length i, we define a continuous map $\rho _w\colon S\to \mathbb {U}_i(R)$ by

$$ \begin{align*} \rho_w(\sigma)=\left(\epsilon_{\left(x_kx_{k+1}\dotsm x_{l-1}\right),R}(\sigma)\right)_{1\leq k\leq l\leq i+1}. \end{align*} $$

The fact that $\Lambda $ is a homomorphism implies that $\rho _w$ is a homomorphism of profinite groups [Reference Efrat7, Lemma 7.5]. We call it the Magnus representation of S corresponding to w.

The subgroup $S^{(n)}$ of S is characterized in terms of the Magnus map as the set of all $\sigma \in S$ such that $\epsilon _{w,\mathbb {Z}_p}(\sigma )=0$ for every word w of length $1\leq i<n$ [Reference Efrat8, Prop. 4.1(a)]. The following result gives similar restrictions on the Magnus coefficients of elements of $S_{\left (n,p\right )}$ . In the discrete case it was proved in [Reference Chapman and Efrat2, Example 4.6], where it was further shown that these restrictions in fact characterize $S_{\left (n,p\right )}$ . While it is possible to derive the proposition from the discrete case using a density argument, we provide a direct proof.

Proof. Consider the subset

$$ \begin{align*} I=\sum_{i\geq1}\sum_{\lvert w\rvert=i}p^{j_n(i)}\mathbb{Z}_pw =\sum_{1\leq i\leq n}\sum_{\lvert w\rvert=i}p^{j_n(i)}\mathbb{Z}_pw+\sum_{i>n}\sum_{\lvert w\rvert=i}\mathbb{Z}_pw \end{align*} $$

of $\mathbb {Z}_p\langle \langle X\rangle \rangle $ . It is an ideal in $\mathbb {Z}_p\langle \langle X\rangle \rangle $ , and therefore $1+I$ is closed under multiplication. Moreover, the identity $\alpha ^{-1}=1-\alpha ^{-1}(\alpha -1)$ shows that $1+I$ is in fact a subgroup of $\mathbb {Z}_p\langle \langle X\rangle \rangle ^\times $ .

As $S_{\left (n,p\right )}=\prod _{i=1}^n\left (S^{(i)}\right )^{p^{j_n(i)}}$ , it therefore suffices to show that $\Lambda _{\mathbb {Z}_p}\left (\tau ^{p^{j_n(i)}}\right )\in 1+I$ for every $\tau \in S^{(i)}$ with $1\leq i\leq n$ . We abbreviate $j=j_n(i)$ . Then $\Lambda _{\mathbb {Z}_p}(\tau )=1+\sum _{\lvert w\rvert \geq i}\epsilon _{w,\mathbb {Z}_p}(\tau )w$ , by [Reference Efrat8, Prop. 4.1(a)]. For every $1\leq l\leq p^j$ such that $il\leq n$ , one has $p^{j_n(il)}\mid \binom {p^j}l$ [Reference Chapman and Efrat2, Example 3.9]. Hence

$$ \begin{align*} \Lambda_{\mathbb{Z}_p}\left(\tau^{p^j}\right)&=\sum_{0\leq l\leq p^j}\binom{p^j}l\left(\sum_{\lvert w\rvert\geq i}\epsilon_{w,\mathbb{Z}_p}(\tau)w\right)^l \subseteq 1+\sum_{1\leq l\leq p^j}\binom{p^j}l\left(\sum_{\lvert w\rvert\geq i}\mathbb{Z}_pw\right)^l\\ & \quad\subseteq 1+\sum_{1\leq l\leq p^j, il\leq n}p^{j_n(il)}\left(\sum_{\lvert w\rvert\geq i}\mathbb{Z}_pw\right)^l+\sum_{1\leq l\leq p^j, il>n}\left(\sum_{\lvert w\rvert\geq i}\mathbb{Z}_pw\right)^l \\ & \quad\subseteq1+I, \end{align*} $$

as desired.

As before, let H be a Hall set in $X^*$ .

For an integer $1\leq i\leq n$ , let

$$ \begin{align*} \pi_i\colon \mathbb{Z}_p\to\mathbb{Z}/p^{j_n(i)+1} \end{align*} $$

be the natural epimorphism. For words $w,w'$ of lengths i, $i'$ , respectively, with $w'\in H$ , we define

$$ \begin{align*} \langle w,w'\rangle_n=\pi_i\left(\epsilon_{w,\mathbb{Z}_p}\left(\sigma_{w'}\right)\right). \end{align*} $$

By Corollary 5.2, $\langle w,w'\rangle _n\in p^{j_n(i)}\mathbb {Z}_p/p^{j_n(i)+1}\mathbb {Z}_p$ . We identify the latter group with $\mathbb {Z}/p$ , and thus view $\langle w,w'\rangle _n$ as an element of $\mathbb {Z}/p$ .

Consider the (possibly infinite) transposed matrix

$$ \begin{align*} \Bigl[\langle w,w'\rangle_n\Bigr]_{w,w'}^T \end{align*} $$

over $\mathbb {Z}/p$ , where $w,w'$ range over all words in H of lengths in $J(n)$ , and indexed with respect to the total order $\preceq $ on $X^*$ defined in §2. We call it the fundamental matrix of level n of H.

We now focus on the Hall set of Lyndon words (see the Introduction). We record the following fundamental triangularity property of H [Reference Reutenauer33, Th. 5.1]: For every Lyndon word $w\in X^*$ , one has

(5.1) $$ \begin{align} \Lambda_{\mathbb{Z}_p}(\tau_w)=1+w+\text{ a combination of words strictly larger than}\ w\ \text{in}\ \preceq. \end{align} $$

Proof. Let w be a Lyndon word of length $i\leq n$ . By (5.1),

$$ \begin{align*} \Lambda_{\mathbb{Z}_p}(\sigma_w)=(1+w+\dotsb)^{p^{j_n(i)}}=1+p^{j_n(i)}w+\dotsb, \end{align*} $$

where the remaining terms are multiples of words strictly larger than w in $\preceq $ . Therefore $\langle w,w\rangle _n=\pi _i\left (p^{j_n(i)}\right )=1$ in $\mathbb {Z}/p$ .

Furthermore, for Lyndon words $w\prec w'$ we get $\epsilon _{w,\mathbb {Z}_p}\left (\sigma _{w'}\right )=0$ , whence $\langle w,w'\rangle _n=0$ (note that the empty word is not Lyndon).

Consequently, the matrix $[\langle w,w'\rangle _n]_{w,w'}$ is unipotent lower-triangular, and therefore its transpose is unitriangular.

Example 5.5. Suppose that $n=3$ . Then $J(n)=\{1,3\}$ for $p\geq 3$ and $J(3)=\{1,2,3\}$ for $p=2$ .

The Lyndon words w of length $3$ are of the forms

$$ \begin{align*} (xxy), (xyy), (xyz), (xzy), \end{align*} $$

where $x,y,z\in X$ and $x<y<z$ . For these words we have

$$ \begin{align*} \sigma_{(xxy)}=[x,[x,y]], \quad \sigma_{(xyy)}=[[x,y],y], \quad \sigma_{(xyz)}=[x,[y,z]], \quad \sigma_{(xzy)}=[[x,z],y], \end{align*} $$

respectively. We recall that $\langle w,w\rangle _3=1$ for every w, and $\langle w,w'\rangle _3=0$ when $w\prec w'$ . It remains to compute $\langle w,w'\rangle _3$ when $w'\prec w$ .

If $\lvert w\rvert ,\lvert w'\rvert \leq 2$ , then by Example 5.4, $\langle w,w'\rangle _3=0$ . We may therefore assume that $\lvert w'\rvert \leq \lvert w\rvert =3$ .

If w contains a letter which does not appear in $w'$ , then $\epsilon _{w,\mathbb {Z}_p}\left (\sigma _{w'}\right )=0$ , whence $\langle w,w'\rangle _3=0$ . Thus we may assume that every letter in w appears in $w'$ .

When $w=(xyy)$ and $w'=(xxy)$ , where $x<y$ , the proof of [Reference Efrat8, Prop. 11.2] gives

$$ \begin{align*} \langle w,w'\rangle_3=\epsilon_{(xyy)}([x,[x,y]])=0. \end{align*} $$

Similarly, when $w=(xzy)$ and $w'=(xyz)$ , where $x<y<z$ , the proof of [Reference Efrat8, Prop. 11.2] gives

$$ \begin{align*} \langle w,w'\rangle_3=\epsilon_{(xzy),\mathbb{Z}_p}([x,[y,z]])=-1. \end{align*} $$

This covers all possible cases when $p\geq 3$ . When $p=2$ we also need to consider Lyndon words $w'=(xy)$ of length $2$ , where $x<y$ . Then $w=(xxy)$ or $w=(xyy)$ . An explicit computation gives

$$ \begin{align*} \Lambda_{\mathbb{Z}_2}([x,y])=1+xy-yx+xyx-yxy-x^2y+y^2x+\dotsb, \end{align*} $$

where the remaining terms are of degree $\geq 4$ . The square of this series has no terms $(xxy)$ and $(xyy)$ , so $\epsilon _{(xxy),\mathbb {Z}_2}\left ([x,y]^2\right )=\epsilon _{(xyy),\mathbb {Z}_2}\left ([x,y]^2\right )=0$ . Therefore $\langle (xxy),(xy)\rangle _3=\langle (xyy),(xy)\rangle _3=0$ .

Altogether, we have shown that

$$ \begin{align*} \langle w,w'\rangle_3= \begin{cases} 1 & \text{if } w=w',\\ -1& \text{if}\ w=(xzy), w'=(xyz), \text{ where } x,y,z\in X, \ x<y<z,\\ 0& \text{otherwise}. \end{cases} \end{align*} $$

In particular, the fundamental matrix need not be the identity matrix.

6 Unitriangular matrices

Let $i\geq 1$ and $j\geq 0$ be integers and consider the ring $R=\mathbb {Z}/p^{j+1}$ . In this section we study the p-Zassenhaus filtration of the group $\mathbb {U}=\mathbb {U}_i(R)$ of all unitriangular $(i+1)\times (i+1)$ -matrices over R, and in particular characterize the values of $i,j$ for which $\mathbb {U}_{\left (n,p\right )}\cong \mathbb {Z}/p$ (see §5 for the notation).

We denote the unit matrix in $\mathbb {U}$ by I, and write $E_{1,i+1}$ for the matrix which is $1$ at entry $(1,i+1)$ and is $0$ elsewhere. For $i'\geq 1$ , the subgroup $\mathbb {U}^{\left (i'\right )}$ of $\mathbb {U}$ consists of all matrices in $\mathbb {U}$ which are zero on the first $i'-1$ diagonals above the main diagonal [Reference Bier and Hołubowski1, Th. 1.5(i)].

We record the following fact about binomial coefficients:

Proof. Let N be an $(i+1)\times (i+1)$ -matrix over $\mathbb {Z}/p^{j+1}$ such that $I+N\in \mathbb {U}^{\left (i'\right )}$ . Then $N^l=0$ for every integer l with $i/i'<l$ . Hence

$$ \begin{align*} (I+N)^{p^{j'}}=\sum_{l=0}^{p^{j'}}\binom{p^{j'}}lN^l =\sum_{l=0}^{\min\left(p^{j'},\lfloor i/i'\rfloor\right)}\binom{p^{j'}}lN^l. \end{align*} $$

Further, if $i'\mid i$ , then $N^{i/i'}\in \mathbb {Z} E_{1,i+1}$ .

In particular, let M be the $(i+1)\times (i+1)$ -matrix over $\mathbb {Z}/p^{j+1}$ which is $1$ on the (first) super-diagonal and is $0$ elsewhere. Then the matrix $M^{i'l}$ is $1$ on the $i'l$ th diagonal above the main one and is $0$ elsewhere. In particular, $I+M^{i'}\in \mathbb {U}^{\left (i'\right )}$ . By what we have justed noted,

$$ \begin{align*} \left(1+M^{i'}\right)^{p^{j'}}=\sum_{l=0}^{\min\left(p^{j'}, \lfloor i/i'\rfloor\right)}\binom{p^{j'}}lM^{i'l}. \end{align*} $$

This matrix is $\binom {p^{j'}}l$ on the $i'l$ th diagonals above the main one and is $0$ elsewhere.

(a) By the previous observations, $\left (\mathbb {U}^{\left (i'\right )}\right )^{p^{j'}}=\{I\}$ holds if and only if

$$ \begin{align*} p^{j+1}\mid\binom{p^{j'}}l, \quad l=1,2,\dotsc,\min\left(p^{j'},\lfloor i/i'\rfloor\right). \end{align*} $$

In light of Lemma 6.1, this is equivalent to $j'\geq j+1+\min \left (j',\left \lfloor \log _p\lfloor i/i'\rfloor \right \rfloor \right )$ , and it remains to note that $\left \lfloor \log _p\lfloor i/i'\rfloor \right \rfloor =\left \lfloor \log _p(i/i')\right \rfloor $ .

(b) First assume that $i=i'$ . Then $\mathbb {U}^{\left (i'\right )}=I+\mathbb {Z} E_{1,i+1}$ . Hence $\left (\mathbb {U}^{\left (i'\right )}\right )^{p^{j'}}=I+\mathbb {Z} p^{j'}E_{1,i+1}$ , and the equality $\left (\mathbb {U}^{\left (i'\right )}\right )^{p^{j'}}=I+\mathbb {Z} p^jE_{1,i+1}$ means that $j'=j$ , as desired.

Next we assume that $i>i'$ . By the previous observations, $\left (\mathbb {U}^{\left (i'\right )}\right )^{p^{j'}}=I+\mathbb {Z} p^jE_{1,i+1}$ holds if and only if the following conditions hold:

1. (i) $i/i'$ is an integer $\leq p^{j'}$ ;

2. (ii) $p^{j+1}\mid \binom {p^{j'}}l$ , $l=1,2,\dotsc , (i/i')-1$ ;

3. (iii) $p^j\mid \binom {p^{j'}}{i/i'}$ , $p^{j+1}{\,\not |\,}\binom {p^{j'}}{i/i'}$ .

By Lemma 6.1 again, (i)–(iii) mean that $i/i'$ is an integer $\leq p^{j'}$ , and

$$ \begin{align*} j'&\geq j+1+\left\lfloor\log_p((i/i')-1)\right\rfloor\\ j'&\geq j+\left\lfloor\log_p(i/i')\right\rfloor\\ j'&<j+1+\left\lfloor\log_p(i/i'))\right\rfloor. \end{align*} $$

This amounts to saying that $j'=j+\log _p(i/i')$ .

(c) This follows from (a) and (b).

The case $i'=1$ of Proposition 6.2(a) was shown by Sawin (see [Reference Efrat9, Prop. 2.3]).

The following corollary stands behind our definition of the sets $J(n)$ . In the case $i=n$ it was proved by Mináč, Rogelstad and Tân [Reference Mináč, Rogelstad and Tân26, Cor. 3.7].

Thus, for $i\in J(n)$ and $\mathbb {U}=\mathbb {U}_i\left (\mathbb {Z}/p^{j_n(i)+1}\right )$ there is a central extension

(6.1) $$ \begin{align} 0\to\mathbb{U}_{\left(n,p\right)}(\cong\mathbb{Z}/p)\to\mathbb{U}\to\overline{\mathbb{U}}:=\mathbb{U}/\mathbb{U}_{\left(n,p\right)}\to1, \end{align} $$

where the isomorphism is the projection on the $(1,i+1)$ -entry composed with the isomorphism $p^{j(i)}\mathbb {Z}/p^{j(i)+1}\mathbb {Z}\cong \mathbb {Z}/p$ .

7 The cohomology elements $\alpha _{w,n}$

Let S be again a free profinite group on the basis X, and let $n\geq 2$ . Consider the transgression homomorphism $\operatorname {\mathrm {trg}}\colon H^1\left (S_{\left (n,p\right )}\right )^S\to H^2\left (S/S_{\left (n,p\right )}\right )$ (recall that the cohomology groups are with respect to the coefficient module $\mathbb {Z}/p$ with trivial action). It is the differential $d_2^{01}$ in the Lyndon–Hochschild–Serre spectral sequence corresponding to the closed normal subgroup $S_{\left (n,p\right )}$ of S [Reference Neukirch, Schmidt and Wingberg31, Th. 2.4.3]. From the five-term sequence in profinite cohomology [Reference Neukirch, Schmidt and Wingberg31, Prop. 1.6.7] and the fact that S has cohomological dimension $1$ , it follows that $\operatorname {\mathrm {trg}}$ is an isomorphism.

Now consider a word w of length $i\in J(n)$ . Consider the ring $R_i=\mathbb {Z}/p^{j_n(i)+1}$ , and set $\mathbb {U}=\mathbb {U}_i(R_i)$ . As before, let $\overline {\mathbb {U}}=\mathbb {U}/\mathbb {U}_{\left (n,p\right )}$ . By Corollary 6.3, the projection on the $(1,i+1)$ -entry gives an isomorphism

$$ \begin{align*} \mathbb{U}_{\left(n,p\right)}\xrightarrow{\sim}p^{j_n(i)}\mathbb{Z}/p^{j_n(i)+1}\mathbb{Z}. \end{align*} $$

The Magnus representation $\rho =\rho _w\colon S\to \mathbb {U}$ induces continuous homomorphisms

$$ \begin{align*} \bar\rho_{w}\colon S/S_{\left(n,p\right)}\to\overline{\mathbb{U}}, \quad \rho_{w}^0=\rho\rvert_{S_{\left(n,p\right)}}\colon S_{\left(n,p\right)}\to\mathbb{U}_{\left(n,p\right)}. \end{align*} $$

Let $\bar \rho _w^*\colon H^2\left (\overline {\mathbb {U}}\right )\to H^2\left (S/S_{\left (n,p\right )}\right )$ be the pullback of $\bar \rho _w$ .

Let $\gamma =\gamma _{n,R_i}\in H^2\left (\overline {\mathbb {U}}\right )$ correspond to the extension (6.1) under the Schreier correspondence [Reference Neukirch, Schmidt and Wingberg31, Th. 1.2.4]. We set

$$ \begin{align*} \alpha_{w,n}=\bar\rho_w^*(\gamma)\in H^2\left(S/S_{\left(n,p\right)}\right). \end{align*} $$

Example 7.1 $\alpha _{w,n}$ for a word $w=(x)$ of length 1. Let $j=j_n(1)=\lceil \log _pn\rceil $ , so $\mathbb {U}=\mathbb {U}_1\left (\mathbb {Z}/p^{j+1}\right )\cong \mathbb {Z}/p^{j+1}$ . As $1\in J(n)$ , we have $\mathbb {U}_{\left (n,p\right )}\cong \mathbb {Z}/p$ , and the central extension (6.1) becomes

(7.1) $$ \begin{align} 0\to\mathbb{Z}/p\to\mathbb{Z}/p^{j+1}\to\mathbb{Z}/p^j\to0. \end{align} $$

We consider this extension as a sequence of trivial $S/S_{\left (n,p\right )}$ -modules. The Bockstein homomorphism

$$ \begin{align*} \mathrm{Bock}_{p^j,S/S_{\left(n,p\right)}}\colon H^1\left(S/S_{\left(n,p\right)},\mathbb{Z}/p^j\right)\to H^2\left(S/S_{\left(n,p\right)}\right) \end{align*} $$

is the associated connecting homomorphism.

We may identify $\rho _{(x)}\colon S\to \mathbb {U}$ with $\epsilon _{(x),\mathbb {Z}/p^{j+1}}\colon S\to \mathbb {Z}/p^{j+1}$ , and $\bar \rho _{(x)}\colon S/S_{\left (n,p\right )}\to \overline {\mathbb {U}}$ with $\epsilon _{(x),\mathbb {Z}/p^j}\colon S/S_{\left (n,p\right )}\to \mathbb {Z}/p^j$ , which are both continuous homomorphisms. Thus $\alpha _{(x),n}$ corresponds to the pullback of the extension (7.1) under $\epsilon _{(x),\mathbb {Z}/p^j}$ . By [Reference Efrat8, Remark 7.3],

$$ \begin{align*} \alpha_{(x),n}=\mathrm{Bock}_{p^j,S/S_{\left(n,p\right)}}\left(\epsilon_{(x),\mathbb{Z}/p^j}\right). \end{align*} $$

For the next Example, we first recall a few facts about Massey products. While these products are defined in the general context of differential graded algebras, in the special case of the n-fold Massey product $H^1(G,R)^n\to H^2(G,R)$ in profinite (or discrete) group cohomology it can be alternatively described in terms of unitriangular representations. This was discovered by Dwyer [Reference Dwyer6] in the discrete case, and we refer to [Reference Efrat7, §8] for the profinite case, which is considered here. We assume as before that $n\geq 2$ and R is a finite commutative ring on which G acts trivially (see [Reference Wickelgren38] for the case of a nontrivial action).

Specifically, let $\mathbb {U}=\mathbb {U}_n(R)$ and let $\overline {\mathbb {U}}$ be again the quotient of $\mathbb {U}$ by the central subgroup $I+R E_{1,n+1}\left (\cong R^+\right )$ . The central extension

(7.2) $$ \begin{align} 0\to R^+\to\mathbb{U}\to \overline{\mathbb{U}}\to1 \end{align} $$

of trivial G-modules corresponds to a cohomology element $\gamma _R\in H^2(G,R^+)$ . Given $\psi _1,\dotsc ,\psi _n\in H^1(G,R^+)$ , we consider the continuous homomorphisms $\bar \rho \colon G\to \overline {\mathbb {U}}$ whose projection $\bar \rho _{k,k+1}\colon G\to R$ on the $(k,k+1)$ -entry is $\psi _k$ , for $k=1,2,\dotsc , n$ . As before, let $\bar \rho ^*\colon H^2\left (\overline {\mathbb {U}},R^+\right )\to H^2(G,R^+)$ be the pullback of $\bar \rho $ . Then $\bar \rho ^*(\gamma _R)$ corresponds to the central extension

$$ \begin{align*} 0\to R^+\to\mathbb{U}\times_{\overline{\mathbb{U}}}G\to G\to1, \end{align*} $$

where the fiber product is with respect to the natural projection $\mathbb {U}\to \overline {\mathbb {U}}$ and to $\bar \rho $ . The n-fold Massey product $\langle \psi _1,\dotsc ,\psi _n\rangle $ is the subset of $H^2(G,R^+)$ consisting of all pullbacks $\bar \rho ^*(\gamma _R)$ [Reference Efrat7, Prop. 8.3]. Thus the n-fold Massey product $\langle \cdot ,\dotsc ,\cdot \rangle \colon H^1(G,R^+)^n\to H^2(G,R^+)$ is a multivalued map. In the special case $n=2$ , one has $\langle \psi _1,\psi _2\rangle =\{\psi _1\cup \psi _2\}$ .

8 The Lyndon bases

We continue with the setup of §7. Identifying $H^1\left (S_{\left (n,p\right )}\right )=\operatorname {\mathrm {Hom}}\left (S_{\left (n,p\right )},\mathbb {Z}/p\right )$ , we obtain a nondegenerate bilinear map

$$ \begin{align*} S_{\left(n,p\right)}/\left(S_{\left(n,p\right)}\right)^p\left[S,S_{\left(n,p\right)}\right]\times H^1\left(S_{\left(n,p\right)}\right)^S\to\mathbb{Z}/p, \quad (\bar\sigma,\varphi)\mapsto\varphi(\sigma) \end{align*} $$

[Reference Efrat and Mináč11, Cor. 2.2]. It gives rise to the bilinear transgression pairing

(8.1) $$ \begin{align} \begin{gathered} (\cdot,\cdot)_n\colon S_{\left(n,p\right)}/\left(S_{\left(n,p\right)}\right)^p\left[S,S_{\left(n,p\right)}\right]\times H^2\left(S/S_{\left(n,p\right)}\right)\to\mathbb{Z}/p, \\ (\bar\sigma,\alpha)_n=-\left(\operatorname{\mathrm{trg}}^{-1}\alpha\right)(\sigma), \end{gathered} \end{align} $$

where $\bar \sigma $ denotes the coset of $\sigma \in S_{\left (n,p\right )}$ . It is therefore also nondegenerate.

By Proposition 5.1 and Corollary 6.3, for a word w of length $i\in J(n)$ there is a commutative diagram

(8.2)

where, as before, $\pi _i\colon {\mathbb {Z}}_p\to {\mathbb {Z}}/p^{j_n(i)+1}$ is the natural projection, and the lower isomorphism is the projection on the $(1,i+1)$ -entry. We deduce the following link between cohomology and the Magnus map. As before, we identify $p^{j_n(i)}\mathbb {Z}/p^{j_n(i)+1}\mathbb {Z}$ with $\mathbb {Z}/p$ .

Proof. The central extension (6.1) gives rise to a transgression homomorphism $\operatorname {\mathrm {trg}}\colon H^1\left (\mathbb {U}^{\left (n,p\right )}\right )^{\mathbb {U}}\to H^2\left (\overline {\mathbb {U}}\right )$ . Let $\iota \colon \mathbb {U}_{\left (n,p\right )}\xrightarrow {\sim }\mathbb {Z}/p$ be the composition of the lower row in diagram (8.2) with the isomorphism $p^{j_n(i)}\mathbb {Z}/p^{j_n(i)+1}\mathbb {Z}\cong \mathbb {Z}/p$ . By the results of [Reference Efrat8, §7],

$$ \begin{align*} \gamma=-\operatorname{\mathrm{trg}}(\iota). \end{align*} $$

The functoriality of transgression gives a commutative square

As $\sigma \in S_{(n,p)}$ , this square and diagram (8.2) give

$$ \begin{align*} \left(\bar\sigma,\alpha_{w,n}\right)_n&=\left(\bar\sigma,\bar\rho_w^*(\gamma)\right)_n =-\left(\bar\sigma,\bar\rho_w^*(\operatorname{\mathrm{trg}}(\iota))\right)_n =-\left(\bar\sigma,\operatorname{\mathrm{trg}}\left(\left(\rho_w^0\right)^*(\iota)\right)\right)_n \\ &=\left(\left(\rho_w^0\right)^*(\iota)\right)(\sigma) =\iota\left(\rho_w^0(\sigma)\right) =\pi_i\left(\epsilon_{w,\mathbb{Z}_p}(\sigma)\right). \\[-35pt] \end{align*} $$

Now consider words $w,w'\in X^*$ of lengths $i,i'\in J(n)$ , respectively, with $w'$ Lyndon. We deduce from Proposition 8.1 that

$$ \begin{align*} \left(\bar\sigma_{w'},\alpha_{w,n}\right)_n=\pi_i\left(\epsilon_{w,\mathbb{Z}_p}\left(\sigma_{w'}\right)\right)=\langle w,w'\rangle_n. \end{align*} $$

We can therefore restate Proposition 5.3 cohomologically:

We will need the following elementary fact in linear algebra [Reference Efrat8, Lemma 8.4]:

We now deduce our first main result. Note that part (a) of the theorem strengthens Theorem 4.6 in the special case where H is the Hall set of Lyndon words.

When $2\leq n\leq p$ we have $J(n)=\{1,n\}$ (Remark 4.2(1)), $j_n(1)=1$ , and $j_n(n)=0$ . In view of Examples 7.1 and 7.2, we deduce the following:

The number of words of a given length in a Hall set H can be expressed in terms of Witt’s necklace function, defined for integers $i,m\geq 1$ by

$$ \begin{align*} \varphi_i(m)=\frac1i\sum_{d\mid i}\mu(d)m^{i/d}. \end{align*} $$

Here $\mu $ is the Möbius function – that is, $\mu (d)=(-1)^k$ if d is a product of k distinct prime numbers, and $\mu (d)=0$ otherwise. We also set $\varphi _i(\infty )=\infty $ . Then the number of words of length i in H is $\varphi _i(\lvert X\rvert )$ [Reference Reutenauer33, Cor. 4.14]. We deduce the following:

9 Shuffle relations

Recall that the shuffle product of words u, v was defined in the Introduction. It extends naturally to a bilinear, commutative, and associative product map . The shuffle algebra $\operatorname {\mathrm {Sh}}(X)$ on X is the graded $\mathbb {Z}$ -algebra whose underlying module is the free module on $X^*$ (graded by the length of words), and its multiplication is .

We define the infiltration product $u\downarrow v$ of words $u=(x_1\dotsm x_r)$ , $v=(x_{r+1}\dotsm x_{r+t})$ in $X^*$ as follows (see [Reference Chen, Fox and Lyndon4], [Reference Reutenauer33, pp. 134–135]). Consider all maps $\sigma \colon \{1,2,\dotsc , r+t\}\to \{1,2,\dotsc , r+t\}$ with $\sigma (1)<\dotsb <\sigma (r)$ and $\sigma (r+1)<\dotsb <\sigma (r+t)$ , and which satisfy the following weak form of injectivity: If $\sigma (i)=\sigma (j)$ , then $x_i=x_j$ . Let the image of $\sigma $ consist of $l_1<\dotsb <l_{m(\sigma )}$ . Then we set

(9.1) $$ \begin{align} u\downarrow v=\sum_\sigma\left(x_{\sigma^{-1}\left(l_1\right)}\dotsm x_{\sigma^{-1}\left(l_{m(\sigma)}\right)}\right)\in\mathbb{Z}\langle X\rangle. \end{align} $$

By our assumption, $x_{\sigma ^{-1}\left (l_i\right )}$ does not depend on the choice of the preimages $\sigma ^{-1}(l_i)$ of $l_i$ . We also write $\operatorname {\mathrm {Infil}}(u,v)$ for the set of all such words $\left (x_{\sigma ^{-1}\left (l_1\right )}\dotsm x_{\sigma ^{-1}\left (l_{m(\sigma )}\right )}\right )$ . Thus is the part of $u\downarrow v$ of degree $r+t$ – that is, the partial sum corresponding to all such maps $\sigma $ which in addition are bijective. The product $\downarrow $ on words extends by linearity to an associative and commutative bilinear map on $\mathbb {Z}\langle X\rangle $ .

There is a well-defined $\mathbb {Z}_p$ -bilinear map

$$ \begin{align*} (\cdot,\cdot)\colon \mathbb{Z}_p\langle\langle X\rangle\rangle\times\mathbb{Z}_p\langle X\rangle\to\mathbb{Z}_p, \quad (f,g)=\sum_{w\in X^*}f_wg_w, \end{align*} $$

where $f_w,g_w$ are the coefficients of f, g, respectively, at w [Reference Reutenauer33, p. 17].

The following connection between the Magnus representation and the infiltration product is proved in the discrete case in [Reference Chen, Fox and Lyndon4, Th. 3.6]. We refer to [Reference Vogel37, Prop. 2.25] and [Reference Morishita30, Prop. 8.16] for the profinite case. Here we view the infiltration and shuffle products as elements of $\mathbb {Z}\langle X\rangle \subseteq \mathbb {Z}_p\langle X\rangle $ .

Proposition 9.1. For every $\emptyset \neq u,v\in X^*$ and every $\sigma \in S$ , one has

$$ \begin{align*} \epsilon_{u,\mathbb{Z}_p}(\sigma)\epsilon_{v,\mathbb{Z}_p}(\sigma)=\left(\Lambda_{\mathbb{Z}_p}(\sigma),u\downarrow v\right). \end{align*} $$

We obtain the following shuffle relations. Here $X^i$ stands for the set of words in $X^*$ of length i.

Theorem 9.3. Let $\emptyset \neq u,v\in X^*$ with $i=\lvert u\rvert +\lvert v\rvert \in J(n)$ . Then

Proof. As $2\leq i\in J(n)$ , we have $(i-1)p^{j_n(i-1)}\geq ip^{j_n(i)}$ , whence $j_n(i-1)>j_n(i)$ .

We recall that

is homogenous of degree i. For $\sigma \in S_{\left (n,p\right )}$ , Corollary 9.2 gives

Therefore, by Proposition 8.1,

Now use the fact that $(\cdot ,\cdot )_n$ is nondegenerate.

Given a graded R-algebra $A=\bigoplus _{i\geq 0} A_i$ , we denote $A_+=\bigoplus _{i\geq 1} A_i$ . Let $\text {WD}(A)$ be the R-submodule of A generated by all products $aa'$ , where $a,a'\in A_+$ . We call $\text {WD}(A)$ the submodule of weakly decomposable elements of A. It is also generated by all products $aa'$ , where $a,a'\in A_+$ are homogenous. Hence the quotient $A_{\mathrm {indec}}=A/\text {WD}(A)$ has the structure of a graded R-module, which we call the indecomposable quotient of A.

Note that $\text {WD}(A)_0=\text {WD}(A)_1=\{0\}$ , so the graded module morphism $A\to A_{{\mathrm {indec}}}$ is an isomorphism in degrees $0$ and $1$ . For example, when $A=R\langle X\rangle $ , one has $A_{{\mathrm {indec}},0}=R$ , $A_{{\mathrm {indec}},1}$ is the free R-module on the basis X, and $A_{{\mathrm {indec}},i}=0$ for all $i\geq 2$ .

When $A=\operatorname {\mathrm {Sh}}(X)$ is the shuffle algebra, we recover the module $\operatorname {\mathrm {Sh}}(X)_{{\mathrm {indec}},n}$ as defined in the Introduction. The following key fact was proved in [Reference Efrat9, Prop. 6.3]. It is based on a construction by Radford [Reference Radford32] and Perrin and Viennot of a basis of $\mathbb {Z}\langle X\rangle $ , which arises from the decomposition of words in $X^*$ into Lyndon words.

In fact, in [Reference Efrat9, Th. 7.3(b)] it is proved that these images form a linear basis of $\operatorname {\mathrm {Sh}}(X)_{{\mathrm {indec}},n}\otimes (\mathbb {Z}/p)$ , but we shall not use this stronger result.

Theorem 9.5. Suppose that $n\geq 2$ . The map $w\mapsto \alpha _{w,n}$ induces an epimorphism of $\mathbb {F}_p$ -linear spaces

$$ \begin{align*} \left(\bigoplus_{i\in J(n)}\operatorname{\mathrm{Sh}}(X)_{{\mathrm{indec}},i}\right)\otimes(\mathbb{Z}/p)\to H^2\left(S/S_{\left(n,p\right)}\right). \end{align*} $$

Proof. For $i\in J(n)$ , the map $X^i\to H^2\left (S/S_{\left (n,p\right )}\right )$ , $w\mapsto \alpha _{w,n}$ , extends by linearity to a $\mathbb {Z}$ -module homomorphism

$$ \begin{align*} \Phi_i\colon \mathbb{Z}\langle X\rangle_i=\bigoplus_{w\in X^i}\mathbb{Z} w\to H^2\left(S/S_{\left(n,p\right)}\right), \quad f=\sum_{w\in X^i}f_ww\mapsto\sum_{w\in X^i}f_w\alpha_{w,n}. \end{align*} $$

By Theorem 9.3, for any nonempty words $u,v\in X^*$ with $i=\lvert u\rvert +\lvert v\rvert $ . Consequently, $\Phi _i$ factors via $\operatorname {\mathrm {Sh}}(X)_{{\mathrm {indec}},i}$ , and induces an $\mathbb {F}_p$ -linear map

$$ \begin{align*} \bar\Phi_i\colon \operatorname{\mathrm{Sh}}(X)_{{\mathrm{indec}},i}\otimes(\mathbb{Z}/p)\to H^2\left(S/S_{\left(n,p\right)}\right), \end{align*} $$

where $\bar \Phi _i(\bar w)=\alpha _{w,n}$ for $w\in X^i$ . Since the $\alpha _{w,n}$ , where w ranges over all Lyndon words of an arbitrary length $i\in J(n)$ , form an $\mathbb {F}_p$ -linear basis of $H^2\left (S/S_{\left (n,p\right )}\right )$ (Theorem 8.5(b)), we obtain an epimorphism

$$ \begin{align*} \bigoplus_{i\in J(n)}\bar\Phi_i\colon\left(\bigoplus_{i\in J(n)}\operatorname{\mathrm{Sh}}(X)_{{\mathrm{indec}},i}\right)\otimes(\mathbb{Z}/p)\to H^2\left(S/S_{\left(n,p\right)}\right). \\[-40pt]\end{align*} $$

We now obtain the Main Theorem from the Introduction:

Theorem 9.6. Suppose that $2\leq n<p$ . Then there is an isomorphism of $\mathbb {F}_p$ -linear spaces

(9.2) $$ \begin{align} \left(\bigoplus_{x\in X}\mathbb{Z}/p\right)\oplus\Bigl(\operatorname{\mathrm{Sh}}(X)_{{\mathrm{indec}},n}\otimes(\mathbb{Z}/p)\Bigr)\xrightarrow{\sim} H^2\left(S/S_{\left(n,p\right)}\right). \end{align} $$

Specifically, this isomorphism maps a generator $1_x$ of the $\mathbb {Z}/p$ -summand at $x\in X$ to $\mathrm {Bock}_{p,S/S_{\left (n,p\right )}}\left (\epsilon _{(x),\mathbb {Z}/p}\right )$ , and maps the image $\bar w$ of a word $w\in X^*$ of length n to the n-fold Massey product element $\alpha _{w,n}$ .