2.1 Notations and conventions

For a non-negative integer j, let $[j]$ denote the set $\{1,\cdots ,j\}$ .

Let $n\ge 1$ . Let $F_n=\langle x_1,\cdots ,x_n\rangle $ . Let $H=H(n)=H_1(F_n,\mathbb {Q})=\bigoplus _{j=1}^n \mathbb {Q} e_j$ , where we fix the basis $\{e_j\}_{j=1}^{n}$ for H given by the images of the $x_j$ ’s in $H\cong (F_n)^{\operatorname {ab}}\otimes \mathbb {Q}$ . We have $H^{\ast }=H(n)^{\ast }=H^1(F_n,\mathbb {Q})=\bigoplus _{j=1}^n \mathbb {Q} e_j^{\ast }$ .

In what follows, we consider representations of $\operatorname {GL}(H,\mathbb {Q})=\operatorname {GL}(n,\mathbb {Q})$ over $\mathbb {Q}$ , which are the same as $\mathbb {Q}[\operatorname {GL}(n,\mathbb {Q})]$ -representations. Let $[A]\in \mathbb {Q}[\operatorname {GL}(n,\mathbb {Q})]$ denote the canonical image of $A\in \operatorname {GL}(n,\mathbb {Q})$ . Sometimes we simply write ‘representations’ to mean $\operatorname {GL}(n,\mathbb {Q})$ -representations.

In computations, we use the following matrices in $\operatorname {GL}(n,\mathbb {Q})$ . Let $\operatorname {id}\in \operatorname {GL}(n,\mathbb {Q})$ denote the identity matrix. For distinct elements $k,l\in [n]$ , let $E_{k,l}\in \operatorname {GL}(n,\mathbb {Q})$ denote the matrix that maps $e_l$ to $e_k+e_l$ and fixes $e_a$ for $a\neq l$ . Then $E_{k,l}$ maps the dual basis $e_k^{\ast }$ to $e_k^{\ast }-e_l^{\ast }$ and fixes $e_a^{\ast }$ for $a\neq k$ . Let $P_{k,l}\in \operatorname {GL}(n,\mathbb {Q})$ denote the matrix that exchanges $e_k$ and $e_l$ and fixes $e_a$ for $a\neq k,l$ . For $k=l$ , we have $P_{k,k}=\operatorname {id}$ .

2.2 Irreducible polynomial representations of $\operatorname {GL}(n,\mathbb {Q})$

Here we recall several notions from representation theory. See [Reference Fulton and Harris16] for details.

Let $n\ge 1$ . A partition $\lambda =(\lambda _1,\lambda _2,\dots ,\lambda _n)$ with at most n parts is a sequence of non-negative integers such that $\lambda _1\ge \lambda _2\ge \dots \ge \lambda _n$ . Let $l(\lambda )=\max (\{0\}\cup \{i\mid \lambda _{i}> 0\})$ denote the length of $\lambda $ and $|\lambda |=\lambda _1+\cdots +\lambda _{l(\lambda )}$ the size of $\lambda $ . We write $\lambda \vdash |\lambda |$ . A Young diagram for a partition $\lambda $ is a diagram with $\lambda _i$ boxes in the i-th row such that the rows of boxes are left-aligned. A tableau on a Young diagram is a numbering of the boxes by integers in $[|\lambda |]$ . A tableau is standard if the numbering of each row and each column is increasing. The canonical tableau on a Young diagram is a standard tableau whose numbering starts with the first row from left to right, followed by the second row from left to right, and so on.

Let

$$ \begin{align*} a_{\lambda}= \sum_{\sigma\in R} \sigma\in \mathbb{Q}[\mathfrak{S}_{|\lambda|}],\quad b_{\lambda}= \sum_{\tau \in C} \operatorname{sgn}(\tau) \tau\in \mathbb{Q}[\mathfrak{S}_{|\lambda|}], \end{align*} $$

where R (resp. C) is a subgroup of $\mathfrak {S}_{|\lambda |}$ preserving rows (resp. columns) of the canonical tableau on the Young diagram corresponding to $\lambda $ . The Young symmetrizer $c_{\lambda }$ is defined by

$$ \begin{align*} c_{\lambda}= b_{\lambda} a_{\lambda}\in \mathbb{Q}[\mathfrak{S}_{|\lambda|}]. \end{align*} $$

Let $S^\lambda =\mathbb {Q}[\mathfrak {S}_{|\lambda |}] c_{\lambda }$ denote the Specht module corresponding to the partition $\lambda $ , which is an irreducible representation of $\mathfrak {S}_{|\lambda |}$ .

We call a $\operatorname {GL}(n,\mathbb {Q})$ -representation V polynomial if after choosing a basis for V, the $(\dim V)^2$ coordinate functions of the action $\operatorname {GL}(n,\mathbb {Q})\to \operatorname {GL}(V)$ are polynomial functions of the $n^2$ variables. Consider $H\cong \mathbb {Q}^n$ as the standard representation of $\operatorname {GL}(n,\mathbb {Q})$ . For a partition $\lambda $ , let

$$ \begin{align*}V_{\lambda}= H^{\otimes |\lambda|}\otimes_{\mathbb{Q}[\mathfrak{S}_{|\lambda|}]}S^\lambda.\end{align*} $$

If $\lambda $ has at most n parts, then $V_{\lambda }$ is an irreducible polynomial $\operatorname {GL}(n,\mathbb {Q})$ -representation, and otherwise, we have $V_{\lambda }=0$ . For a non-negative integer p, by the Schur–Weyl duality, we have a decomposition of $H^{\otimes p}$ as $\operatorname {GL}(n,\mathbb {Q})\times \mathfrak {S}_{p}$ -modules

$$ \begin{align*}H^{\otimes p}= \bigoplus_{\lambda\vdash p \text{ with at most n parts}} V_{\lambda}\otimes S^{\lambda}. \end{align*} $$

It is well known that irreducible polynomial representations of $\operatorname {GL}(n,\mathbb {Q})$ are classified by partitions with at most n parts, that is, any irreducible polynomial $\operatorname {GL}(n,\mathbb {Q})$ -representation is isomorphic to $V_{\lambda }$ for a partition $\lambda $ with at most n parts.

We have the following irreducible decomposition of the tensor product of two irreducible polynomial $\operatorname {GL}(n,\mathbb {Q})$ -representations $V_{\lambda }$ and $V_{\mu }$

$$ \begin{align*}V_{\lambda}\otimes V_{\mu}=\bigoplus_{\nu} V_{\nu}^{\oplus N_{\lambda\mu}^{\nu}}, \end{align*} $$

where $N_{\lambda \mu }^{\nu }$ is the Littlewood–Richardson coefficient.

2.3 Irreducible algebraic representations of $\operatorname {GL}(n,\mathbb {Q})$

We call a $\operatorname {GL}(n,\mathbb {Q})$ -representation V algebraic if after choosing a basis for V, the $(\dim V)^2$ coordinate functions of the action $\operatorname {GL}(n,\mathbb {Q})\to \operatorname {GL}(V)$ are rational functions of the $n^2$ variables. Here we recall irreducible algebraic representations of $\operatorname {GL}(n,\mathbb {Q})$ , which generalize irreducible polynomial representations of $\operatorname {GL}(n,\mathbb {Q})$ . See [Reference Fulton and Harris16, Reference Koike.26, Reference Patzt36] for details.

Let p and q be positive integers. For a pair $(k,l)\in [p]\times [q]$ , the contraction map

$$ \begin{align*}c_{k,l}:H^{\otimes p}\otimes (H^{\ast})^{\otimes q}\rightarrow H^{\otimes p-1}\otimes (H^{\ast})^{\otimes q-1}\end{align*} $$

is defined for $v_1\otimes \cdots \otimes v_{p}\in H^{\otimes p}$ and $f_1\otimes \cdots \otimes f_{q} \in (H^{\ast })^{\otimes q}$ by

$$ \begin{align*}c_{k,l}((v_1\otimes\cdots\otimes v_{p}) \otimes(f_1\otimes\cdots\otimes f_{q})) = \langle v_k,f_l\rangle (v_1\otimes\cdots\hat{v}_{k}\cdots\otimes v_{p})\otimes (f_1\otimes\cdots\hat{f}_{l}\cdots\otimes f_{q}), \end{align*} $$

where $\langle -,-\rangle :H\otimes H^{\ast }\rightarrow \mathbb {Q}$ denotes the dual pairing, and where $\hat {v}_{k}$ (resp. $\hat {f}_{l}$ ) denotes the omission of $v_{k}$ (resp. $f_{l}$ ).

For $p,q\ge 0$ , let $T_{p,q}$ denote the traceless part of $H^{\otimes p}\otimes (H^{\ast })^{\otimes q}$ , which is defined by

$$ \begin{align*}T_{p,q}=\bigcap_{(k,l)\in [p]\times [q]} \ker c_{k,l}\subset H^{\otimes p}\otimes (H^{\ast})^{\otimes q}.\end{align*} $$

Note that we have $T_{0,q}=(H^{\ast })^{\otimes q}$ and $T_{p,0}=H^{\otimes p}$ .

A bipartition $\underline {\lambda }=(\lambda ^{+},\lambda ^{-})$ with at most n parts is a pair of two partitions $\lambda ^{+}$ and $\lambda ^{-}$ such that $l(\underline \lambda )=l(\lambda ^{+})+l(\lambda ^{-})\le n$ . The size $|\underline \lambda |$ of $\underline \lambda $ is defined by $|\lambda ^{+}|+|\lambda ^{-}|$ .

For a bipartition $\underline \lambda $ , set $p=|\lambda ^{+}|$ and $q=|\lambda ^{-}|$ . Let

$$ \begin{align*}V_{\underline\lambda}=T_{p,q}\otimes_{\mathbb{Q}[\mathfrak{S}_{p}\times\mathfrak{S}_{q}]} (S^{\lambda^{+}}\otimes S^{\lambda^{-}}). \end{align*} $$

If $\underline \lambda $ has at most n parts, then $V_{\underline \lambda }$ is an irreducible algebraic representation, and otherwise, we have $V_{\underline \lambda }=0$ . We have the following decomposition of $T_{p,q}$ as a $\operatorname {GL}(n,\mathbb {Q})\times (\mathfrak {S}_p\times \mathfrak {S}_q)$ -modules, which generalizes the Schur–Weyl duality for $H^{\otimes p}$

(2.3.1) $$ \begin{align} T_{p,q}=\bigoplus_{\substack{\underline\lambda: \text{bipartition with at most n parts} \\ |\lambda^{+}|=p,\;|\lambda^{-}|=q}} V_{\underline\lambda}\otimes (S^{\lambda^{+}}\otimes S^{\lambda^{-}}). \end{align} $$

(See [Reference Koike.26, Theorem 1.1].) It is well known that irreducible algebraic representations of $\operatorname {GL}(n,\mathbb {Q})$ are classified by bipartitions with at most n parts, that is, any irreducible algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representation is isomorphic to $V_{\underline \lambda }$ for a bipartition $\underline \lambda $ with at most n parts. (See also [Reference Patzt36] for the correspondence between irreducible algebraic representations and bipartitions.)

For a bipartition $\underline \lambda =(\lambda ^{+},\lambda ^{-})$ with at most n parts, we have $V_{\underline \lambda }=V_{\mu }\otimes \det ^{k}$ , where $\mu $ is a partition and k is an integer satisfying

$$ \begin{align*}(\lambda^{+}_1,\cdots, \lambda^{+}_{l(\lambda^{+})},0,\cdots,0,-\lambda^{-}_{l(\lambda^{-})},\cdots,-\lambda^{-}_1)= (\mu_1+k,\cdots, \mu_n+k),\end{align*} $$

and where $\det $ denotes the $1$ -dimensional determinant representation.

The dual of $\underline \lambda $ is defined by $\underline \lambda ^{\ast }=(\lambda ^{-},\lambda ^{+})$ . Note that we have a $\operatorname {GL}(n,\mathbb {Q})$ -isomorphism $(V_{\underline \lambda })^{\ast }\cong V_{\underline \lambda ^{\ast }}$ .

For two bipartitions $\underline \lambda , \underline \mu $ such that $n\ge l(\underline \lambda )+l(\underline \mu )$ , we have the following irreducible decomposition of the tensor product of $V_{\underline \lambda }$ and $V_{\underline \mu }$

$$ \begin{align*}V_{\underline\lambda}\otimes V_{\underline\mu}=\bigoplus_{\underline\nu} V_{\underline\nu}^{\oplus N_{\underline\lambda \underline\mu}^{\underline\nu}}, \end{align*} $$

where $N_{\underline \lambda \underline \mu }^{\underline \nu }= \sum _{\alpha \beta \theta \delta } (\sum _{\kappa }N_{\kappa \alpha }^{\lambda ^+}N_{\kappa \beta }^{\mu ^-}) (\sum _{\epsilon }N_{\epsilon \theta }^{\lambda ^-}N_{\epsilon \delta }^{\mu ^+}) N_{\alpha \delta }^{\nu ^+}N_{\beta \theta }^{\nu ^-}$ (see [Reference Koike.26] for details).

2.4 Generator of the traceless part $T_{p,q}$

Here we give a generator of the traceless part $T_{p,q}$ of $H^{\otimes p}\otimes (H^{\ast })^{\otimes q}$ .

We have the following explicit highest weight vectors of $T_{p,q}$ .

Lemma 2.1 (Theorems 2.7 and 2.11 of [Reference Benkart, Chakrabarti, Halverson, Leduc, Lee and Stroomer2])

Let $p,q\ge 0$ . For a bipartition $\underline \lambda $ with at most n parts such that $|\lambda ^{+}|=p$ and $|\lambda ^{-}|=q$ , let

$$ \begin{align*}e(\underline\lambda)=(e_1^{\otimes \lambda^{+}_1} \otimes \cdots \otimes e_{l(\lambda^{+})}^{\otimes \lambda^{+}_{l(\lambda^{+})}}) \otimes ((e_n^{\ast})^{\otimes \lambda^{-}_1} \otimes \cdots \otimes (e_{n-l(\lambda^{-})+1}^{\ast})^{\otimes \lambda^{-}_{l(\lambda^{-})}}) \in H^{\otimes p}\otimes (H^{\ast})^{\otimes q}.\end{align*} $$

Let $St_{\lambda ^+}$ (resp. $St_{\lambda ^-}$ ) denote the subset of $\mathfrak {S}_p$ (resp. $\mathfrak {S}_q$ ) consisting of permutations which send the canonical tableau to standard tableaux on the Young diagram corresponding to $\lambda ^+$ (resp. $\lambda ^-$ ). Then the set

$$ \begin{align*}\{(\pi b_{\lambda^{+}}\otimes \operatorname{id}) (\operatorname{id}\otimes \rho b_{\lambda^{-}}) e(\underline\lambda)\mid |\lambda^{+}|=p,\: |\lambda^{-}|=q,\: \pi\in St_{\lambda^+},\: \rho\in St_{\lambda^-}\}\end{align*} $$

is a basis for the space of all highest weight vectors of $T_{p,q}$ .

In particular, for each $\underline \lambda $ , $(\pi b_{\lambda ^{+}}\otimes \operatorname {id}) (\operatorname {id}\otimes \rho b_{\lambda ^{-}}) e(\underline \lambda )$ generates an irreducible subrepresentation of $T_{p,q}$ which is isomorphic to $V_{\underline \lambda }$ .

Lemma 2.2. Let $n\ge p+q$ . Then the traceless part $T_{p,q}$ of $H^{\otimes p}\otimes (H^{\ast })^{\otimes q}$ is generated by

$$ \begin{align*}e_{p,q}=e_1\otimes \cdots \otimes e_p \otimes e_n^{\ast}\otimes \cdots \otimes e_{n-q+1}^{\ast} \in H^{\otimes p}\otimes (H^{\ast})^{\otimes q}\end{align*} $$

as a $\operatorname {GL}(n,\mathbb {Q})$ -representation.

Proof. The proof is analogous to that of [Reference Lindell.30, Lemma 2.1]. By Lemma 2.1, the traceless part $T_{p,q}$ is generated by

$$ \begin{align*}\{(\pi b_{\lambda^{+}}\otimes \operatorname{id}) (\operatorname{id}\otimes \rho b_{\lambda^{-}}) e(\underline\lambda)\mid |\lambda^{+}|=p,\: |\lambda^{-}|=q,\: \pi\in St_{\lambda^+},\: \rho\in St_{\lambda^-}\}.\end{align*} $$

Therefore, it suffices to show that

$$ \begin{align*}e_J:= e_{j_1}\otimes \cdots \otimes e_{j_p} \otimes e_{j_{n}}^{\ast}\otimes \cdots \otimes e_{j_{n-q+1}}^{\ast} \in \mathbb{Q}[\operatorname{GL}(n,\mathbb{Q})] e_{p,q} \end{align*} $$

for $j_1, \cdots , j_p \in [p]$ , $j_{n},\cdots ,j_{n-q+1}\in \{n,\cdots ,n-q+1\}$ .

Let

$$ \begin{align*}K_p=\{k\in [p]\mid e_{j_k}=e_{j_l} \text{ for some }l<k\}\end{align*} $$

and

$$ \begin{align*}K^{\prime}_q=\{k\in [q]\mid e_{j_{n-k+1}}^{\ast}=e_{j_{n-l+1}}^{\ast} \text{ for some } l<k\}.\end{align*} $$

We can reorder the basis $\{e_1,\cdots ,e_n\}$ for H by using permutation matrices in such a way that $e_{j_k}=e_k$ for $k\in [p]\setminus K_p$ , that is, we may assume that $e_J$ coincides with $e_1\otimes \cdots \otimes e_p\otimes e_{j_{n}}^{\ast }\otimes \cdots \otimes e_{j_{n-q+1}}^{\ast }$ except for tensor factors where $e_J$ has repeating elements. We can also assume that $e_J$ coincides with $e_{p,q}$ except for tensor factors where $e_J$ has repeating elements by reordering the basis $\{e_1,\cdots ,e_n\}$ again in such a way that $e_{j_{n-k+1}}^{\ast }=e_{n-k+1}^{\ast }$ for $k\in [q]\setminus K^{\prime }_q$ .

Now, we have to check that $e_{J}\in \mathbb {Q}[\operatorname {GL}(n,\mathbb {Q})] e_{p,q}$ . Define

$$ \begin{align*}E_J:=\prod_{k\in K_p} ([E_{j_k,k}]-[\operatorname{id}])\prod_{k\in K_q}([\operatorname{id}]-[E_{n-k+1,j_{n-k+1}}])\in \mathbb{Q}[\operatorname{GL}(n,\mathbb{Q})].\end{align*} $$

Then we have

$$ \begin{align*} e_{J}=E_J(e_{p,q})\in \mathbb{Q}[\operatorname{GL}(n,\mathbb{Q})] e_{p,q}, \end{align*} $$

which completes the proof.

2.5 Traceless tensor products

Here we define traceless tensor products of algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations.

By Lemma 2.1, we have a subrepresentation $V_{\underline \lambda }\subset H^{\otimes |\lambda ^+|}\otimes (H^{\ast })^{\otimes |\lambda ^-|}$ which is generated by $(b_{\lambda ^{+}}\otimes \operatorname {id}) (\operatorname {id}\otimes b_{\lambda ^{-}}) e(\underline \lambda )$ for each bipartition $\underline \lambda $ .

Let $\underline \lambda $ and $\underline \mu $ be two bipartitions. Let $p=|\lambda ^{+}|, q=|\lambda ^{-}|, r=|\mu ^{+}|$ and $s=|\mu ^{-}|$ . Define the traceless tensor product $V_{\underline \lambda }\widetilde {\otimes } V_{\underline \mu }$ of $V_{\underline \lambda }$ and $V_{\underline \mu }$ as

$$ \begin{align*}V_{\underline\lambda}\widetilde{\otimes} V_{\underline\mu}=(V_{\underline\lambda}\otimes V_{\underline\mu})\cap T_{p+r,q+s}\subset H^{\otimes (p+r)}\otimes (H^{\ast})^{\otimes (q+s)}.\end{align*} $$

In other words, $V_{\underline \lambda }\widetilde {\otimes } V_{\underline \mu }$ is a subrepresentation of $V_{\underline \lambda }\otimes V_{\underline \mu }$ which vanishes under any contraction maps. Then we have for $n\ge l(\underline \lambda )+l(\underline \mu )$ ,

$$ \begin{align*}V_{\underline\lambda}\widetilde{\otimes} V_{\underline\mu}\cong \bigoplus_{|\underline\nu|= |\underline\lambda|+|\underline\mu|} V_{\underline\nu}^{\oplus N_{\underline\lambda \underline\mu}^{\underline\nu}}\subset \bigoplus_{\underline\nu} V_{\underline\nu}^{\oplus N_{\underline\lambda \underline\mu}^{\underline\nu}} \cong V_{\underline\lambda}\otimes V_{\underline\mu}.\end{align*} $$

Let M be an algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representation. For each bipartition $\underline \lambda $ , define a vector space

$$ \begin{align*}M_{\underline\lambda}=\operatorname{Hom}_{\operatorname{GL}(n,\mathbb{Q})}(V_{\underline\lambda},M).\end{align*} $$

Since the category of algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations is split semisimple, we have a natural isomorphism

$$ \begin{align*}\iota_M: M\xrightarrow{\cong} \bigoplus_{\underline\lambda}V_{\underline\lambda}\otimes M_{\underline\lambda}. \end{align*} $$

For two algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations M and N, we have

$$ \begin{align*}\iota_M\otimes \iota_N: M\otimes N \xrightarrow{\cong} \left(\bigoplus_{\underline\lambda}V_{\underline\lambda}\otimes M_{\underline\lambda}\right) \otimes \left(\bigoplus_{\underline\mu}V_{\underline\mu}\otimes N_{\underline\mu}\right) \cong \bigoplus_{\underline\lambda,\underline\mu}(V_{\underline\lambda}\otimes V_{\underline\mu})\otimes (M_{\underline\lambda}\otimes N_{\underline\mu}). \end{align*} $$

Define the traceless tensor product $M \widetilde {\otimes } N$ of M and N as

$$ \begin{align*}M \widetilde{\otimes} N=(\iota_{M}\otimes \iota_{N})^{-1} \left(\bigoplus_{\underline\lambda,\underline\mu}(V_{\underline\lambda}\widetilde{\otimes} V_{\underline\mu})\otimes (M_{\underline\lambda}\otimes N_{\underline\mu})\right)\subset M\otimes N. \end{align*} $$

Let $M^{\widetilde {\otimes } 0}=\mathbb {Q}$ and $M^{\widetilde {\otimes } 1}=M$ . For $n\ge 2$ , we define $M^{\widetilde {\otimes }n}=M^{\widetilde {\otimes }n-1}\widetilde {\otimes } M\subset M^{\otimes n}$ iteratively, and define the traceless part $\widetilde {T}^{\ast }M$ of the tensor algebra $T^{\ast }M$ as

$$ \begin{align*}\widetilde{T}^{\ast}M=\bigoplus_{n\ge 0} M^{\widetilde{\otimes}n}\subset T^{\ast}M.\end{align*} $$

Let $\bigwedge\nolimits ^{\ast } M$ denote the exterior algebra of M and $\operatorname {Sym}^{\ast } M$ the symmetric algebra of M. We have canonical projections $T^{\ast }M\twoheadrightarrow \bigwedge\nolimits ^{\ast } M$ and $T^{\ast }M \twoheadrightarrow \operatorname {Sym}^{\ast } M$ . We define the traceless part $\widetilde {\bigwedge\nolimits }^{\ast } M$ of $\bigwedge\nolimits ^{\ast } M$ as the image of $\widetilde {T}^{\ast }M$ under the above projection, and the traceless part $\widetilde {\operatorname {Sym}}^{\ast } M$ of $\operatorname {Sym}^{\ast } M$ in a similar way.

2.6 Traceless parts of graded-symmetric algebras

Let $M_{\ast }=\bigoplus _{i\ge 1} M_i$ be a graded algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representation. Let $S^{\ast }(M_{\ast })=\bigoplus _{i\ge 0}S^{\ast }(M_{\ast })_i$ , where

$$ \begin{align*} S^{\ast} (M_{\ast})_i =\bigoplus_{k_1+2k_2 +\cdots + p k_p=i} S^{k_1}(M_{1}) \otimes\cdots \otimes S^{k_p}(M_{p}), \end{align*} $$

denote the graded-symmetric algebra of $M_{\ast }$ . That is, we have the following graded-commutativity

$$ \begin{align*}y x =(-1)^{ij} x y \end{align*} $$

for $x\in M_i$ and $y\in M_j$ . Then we have $S^k (M_i)=\operatorname {Sym}^k(M_i)$ for even i and $S^k (M_i)=\bigwedge\nolimits ^k(M_i)$ for odd i. We have a canonical projection $T^{\ast }(M_{\ast })\twoheadrightarrow S^{\ast }(M_{\ast })$ . Define the traceless part $\widetilde {S}^{\ast } (M_{\ast })=\bigoplus _{i\ge 0} \widetilde {S}^{\ast } (M_{\ast })_i$ of the graded-symmetric algebra $S^{\ast }(M_{\ast })$ as the image of $\widetilde {T}^{\ast }(M_{\ast })$ under the projection. Then $\widetilde {S}^{\ast } (M_{\ast })$ is a graded $\operatorname {GL}(n,\mathbb {Q})$ -representation satisfying

$$ \begin{align*} \widetilde{S}^{\ast} (M_{\ast})_i =\bigoplus_{k_1+2k_2 +\cdots + p k_p=i} \left(\widetilde{S}^{k_1}(M_{1})\right) \widetilde{\otimes}\cdots \widetilde{\otimes} \left(\widetilde{S}^{k_p}(M_{p})\right). \end{align*} $$

The graded-symmetric algebra $S^{\ast }(M_{\ast })$ has a coalgebra structure defined as follows. For an element $x=x_1\cdots x_k\in S^{\ast }(M_{\ast })$ , where $x_j\in M_{i_j}$ , and for $\sigma \in \mathfrak {S}_k$ , let $\operatorname {sgn}(\sigma; x)\in \{1,-1\}$ denote the sign satisfying

$$ \begin{align*}x_{\sigma(1)}\cdots x_{\sigma(k)}= \operatorname{sgn}(\sigma; x)x_1\cdots x_k. \end{align*} $$

Then the comultiplication $\Delta $ is defined by

$$ \begin{align*}\Delta(x_1\cdots x_k)=\sum_{p=0}^k \sum_{\sigma\in \operatorname{Sh}(p,k-p)}\operatorname{sgn}(\sigma; x) x_{\sigma(1)}\cdots x_{\sigma(p)} \otimes x_{\sigma(p+1)}\cdots x_{\sigma(k)}, \end{align*} $$

where $\operatorname {Sh}(p,k-p)\subset \mathfrak {S}_k$ denotes the set of $(p,k-p)$ -shuffles. Then we can check that the coalgebra structure of $S^{\ast }(M_{\ast })$ induces a subcoalgebra structure of the traceless part $\widetilde {S}^{\ast }(M_{\ast })$ . We can also check that the primitive part of $S^{\ast }(M_{\ast })$ is $M_{\ast }$ .

2.7 $\operatorname {GL}(\infty ,\mathbb {Q})$ -representations

Let $n\ge 0$ . We have an inclusion $\operatorname {GL}(n,\mathbb {Q})\hookrightarrow \operatorname {GL}(n+1,\mathbb {Q})$ sending $A\in \operatorname {GL}(n,\mathbb {Q})$ to $A\oplus {1}\in \operatorname {GL}(n+1,\mathbb {Q})$ . Let $\operatorname {GL}(\infty ,\mathbb {Q})=\varinjlim _{n}\operatorname {GL}(n,\mathbb {Q})=\bigcup _{n\ge 1} \operatorname {GL}(n,\mathbb {Q})$ . Here we recall representation theory of $\operatorname {GL}(\infty ,\mathbb {Q})$ . See [Reference Sam and Snowden40] for details.

We have sequences of algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations

$$ \begin{align*} & H(0)\hookrightarrow H(1)\hookrightarrow\dots \hookrightarrow H(n)\hookrightarrow H(n+1)\hookrightarrow\dots,\\ & H(0)^{\ast}\hookrightarrow H(1)^{\ast}\hookrightarrow\dots \hookrightarrow H(n)^{\ast}\hookrightarrow H(n+1)^{\ast}\hookrightarrow\dots, \end{align*} $$

where the map $H(n)^{(*)}\hookrightarrow H(n+1)^{(*)}$ is the canonical inclusion map that sends the basis vector $e_j^{(*)}$ to $e_j^{(*)}$ for $j\in [n]$ , which is a $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism if $H(n+1)^{(*)}$ is considered as a $\operatorname {GL}(n,\mathbb {Q})$ -representation. Let $\mathbf {H}=\varinjlim _{n} H(n)=\bigcup _{n\ge 0} H(n)$ , and $\mathbf {H}^{\ast }=\varinjlim _{n} (H(n)^{\ast })=\bigcup _{n\ge 0}(H(n)^{\ast })$ . The group $\operatorname {GL}(\infty ,\mathbb {Q})$ acts on the tensor product $\mathbf {H}^{\otimes p}\otimes (\mathbf {H}^{\ast })^{\otimes q}$ for any $p,q\ge 0$ . We call a $\operatorname {GL}(\infty ,\mathbb {Q})$ -representation algebraic if it is a subquotient of a finite direct sum of tensor products $\mathbf {H}^{\otimes p}\otimes (\mathbf {H}^{\ast })^{\otimes q}$ for $p,q\ge 0$ .

We define the contraction maps $c_{k,l}$ and the traceless part $T_{p,q}$ for $\operatorname {GL}(\infty ,\mathbb {Q})$ as in the case of $\operatorname {GL}(n,\mathbb {Q})$ . For any bipartition $\underline \lambda $ such that $|\lambda ^{+}|=p$ and $|\lambda ^{-}|=q$ , define a $\operatorname {GL}(\infty ,\mathbb {Q})$ -representation $V_{\underline \lambda }$ as

$$ \begin{align*}V_{\underline\lambda}=T_{p,q}\otimes_{\mathbb{Q}[\mathfrak{S}_p\times\mathfrak{S}_q]}(S^{\lambda^{+}}\otimes S^{\lambda^{-}}). \end{align*} $$

Then as in (2.3.1), we have the following decomposition of the traceless part $T_{p,q}$ as $\operatorname {GL}(\infty ,\mathbb {Q})$ -representations

$$ \begin{align*}T_{p,q}=\bigoplus_{\substack{\underline\lambda=(\lambda^{+},\lambda^{-}) \\ |\lambda^{+}|=p,\;|\lambda^{-}|=q}} V_{\underline\lambda}\otimes (S^{\lambda^{+}}\otimes S^{\lambda^{-}}). \end{align*} $$

Moreover, as in the case of $\operatorname {GL}(n,\mathbb {Q})$ , it is known that irreducible algebraic representations of $\operatorname {GL}(\infty ,\mathbb {Q})$ are classified by bipartitions. (See Proposition 3.14 in [Reference Sam and Snowden40].)

Remark 2.5. Note that the category of algebraic $\operatorname {GL}(\infty ,\mathbb {Q})$ -representations is not split semisimple. For example, we do not have a decomposition $\mathbf {H}\otimes \mathbf {H}^{\ast }\cong T_{1,1}\oplus \mathbb {Q}$ as a $\operatorname {GL}(\infty ,\mathbb {Q})$ -representation. For each $n\ge 0$ , we have a split short exact sequence of $\operatorname {GL}(n,\mathbb {Q})$ -representations

$$ \begin{align*} 0\to T_{1,1}(=V_{1,1})\to H(n)\otimes H(n)^{\ast}\overset{c_{1,1}}{\underset{s_n}{\rightleftarrows}} \mathbb{Q}\to 0, \end{align*} $$

where $s_n(1)=\sum _{j=1}^{n}e_j\otimes e_j^{\ast }.$ Since the $\operatorname {GL}(n,\mathbb {Q})$ -module maps $s_n$ do not give rise to a morphism between sequences of $\operatorname {GL}(n,\mathbb {Q})$ -representations, however, we do not have a section in the case of $\operatorname {GL}(\infty ,\mathbb {Q})$ -representations.

3.1 Definition of abelian cycles

Let $i\ge 1$ and $\mathbb {Z}^i=\bigoplus _{k=1}^i \mathbb {Z} z_k$ , where $\{z_k\}_k$ is the canonical basis for $\mathbb {Z}^i$ . Let $(\phi _1,\cdots ,\phi _i)$ be an i-tuple of mutually commuting elements of $\operatorname {IA}_n$ . Then we have a group homomorphism $\mathbb {Z}^i\to \operatorname {IA}_n$ mapping $z_k$ to $\phi _k$ for $1\le k\le i$ , which induces a $\mathbb {Q}$ -linear map

$$ \begin{align*}H_i(\mathbb{Z}^i,\mathbb{Q})\to H_i(\operatorname{IA}_n,\mathbb{Q}).\end{align*} $$

We have $H_i(\mathbb {Z}^i,\mathbb {Q})\cong H_i(T^i,\mathbb {Q})\cong \mathbb {Q}$ , where $T^i$ is the i-dimensional torus. Let $A(\phi _1,\cdots ,\phi _i)\in H_i(\operatorname {IA}_n,\mathbb {Q})$ denote the image of the fundamental class of $H_i(T^i,\mathbb {Q})$ and call it the abelian cycle determined by $(\phi _1,\cdots ,\phi _i)$ .

3.2 Abelian cycles in $H_i(\operatorname {IA}_n,\mathbb {Q})$

Magnus’s set of generators of $\operatorname {IA}_n$ is

$$ \begin{align*}\{g_{a,b}\mid 1\le a,b \le n,\; a\ne b \}\cup\{f_{a,b,c}\mid 1\le a,b,c\le n,\; a< b,\; a\ne c\ne b\}, \end{align*} $$

where $g_{a,b}$ and $f_{a,b,c}$ are defined by

$$ \begin{align*} g_{a,b}(x_b) & =x_a x_b x_a^{-1}, \quad g_{a,b}(x_d)=x_d \text{ for }d\neq b,\\ f_{a,b,c}(x_c) & =x_c[x_a,x_b], \quad f_{a,b,c}(x_d)=x_d \text{ for }d\neq c. \end{align*} $$

For $k>l$ , let

$$ \begin{align*}h_{k,l}:=g_{k,l} g_{k,l+1}\cdots g_{k,k-1}\in\operatorname{IA}_n. \end{align*} $$

Let $\operatorname {Aut}(F_n)(k,l)$ denote the image of the canonical injective map

$$ \begin{align*}\operatorname{Aut}(F_{k-l})\hookrightarrow \operatorname{Aut}(F_n),\quad x_j\mapsto x_{j+l-1}.\end{align*} $$

Since $h_{k,l}$ fixes $x_j$ for $j\in \{1,\cdots ,l-1,k,\cdots ,n\}$ and maps $x_j$ to $x_k x_j x_k^{-1}$ for $j\in \{l,l+1,\cdots ,k-1\}$ , it follows that $h_{k,l}$ commutes with any element of $\operatorname {Aut}(F_n)(k,l)$ .

For positive integers a and r, set

$$ \begin{align*}\mathbf{g}_{r,a}=(g_{a+1,a}=h_{a+1,a}, h_{a+2,a},\cdots, h_{a+r,a})\end{align*} $$

and

$$ \begin{align*}\mathbf{f}_{r,a}=(f_{a,a+1,a+2}, h_{a+3,a},h_{a+4,a},\cdots, h_{a+r+1,a}).\end{align*} $$

Let $i\ge 0$ . We call $(\mu ,\nu )$ a pair of partitions of total size i, denoted by $(\mu ,\nu )\vdash i$ , if $\mu $ and $\nu $ are partitions with $|\mu |+|\nu |=i$ . (Note that pairs of partitions are the same as bipartitions, but we regard them as different notions.) For $n\ge i+2l(\mu )+l(\nu )$ , define an i-tuple $\mathbf {h}_{(\mu ,\nu )}$ of elements of $\operatorname {IA}_n$ by

$$ \begin{align*}\mathbf{h}_{(\mu,\nu)}=(\mathbf{f}_{\mu_1,s(1)},\cdots, \mathbf{f}_{\mu_{l(\mu)},s(l(\mu))}, \mathbf{g}_{\nu_1,t(1)}, \cdots, \mathbf{g}_{\nu_{l(\nu)},t(l(\nu))}), \end{align*} $$

where

$$ \begin{align*}s(j)=1+\sum_{p=1}^{j-1}(\mu_p +2),\quad t(k)=1+|\mu|+2l(\mu)+\sum_{p=1}^{k-1}(\nu_p +1).\end{align*} $$

Then we can check that $\mathbf {h}_{(\mu ,\nu )}$ consists of mutually commuting elements. For example, we have

$$ \begin{align*} \mathbf{h}_{(2, 0)} & =\mathbf{f}_{2,1}=(f_{1,2,3},\; h_{4,1}),\\ \mathbf{h}_{(0,2)} & =\mathbf{g}_{2,1}=(h_{2,1},\; h_{3,1}),\\ \mathbf{h}_{(1^2,0)} & =(\mathbf{f}_{1,1},\mathbf{f}_{1,4})=(f_{1,2,3},\; f_{4,5,6}),\\ \mathbf{h}_{(0,1^2)} & =(\mathbf{g}_{1,1},\mathbf{g}_{1,3})=(h_{2,1},\; h_{4,3}),\\ \mathbf{h}_{(1,1)} & =(\mathbf{f}_{1,1},\mathbf{g}_{1,4})=(f_{1,2,3},\; h_{5,4}). \end{align*} $$

Then we obtain an abelian cycle corresponding to each pair of partitions.

4.1 Contraction maps

For $i\ge 1$ , let

$$ \begin{align*}U_i=\operatorname{Hom}\left(H,\bigwedge\nolimits ^{i+1} H\right)\cong \left(\bigwedge\nolimits ^{i+1} H\right)\otimes H^{\ast}.\end{align*} $$

Then we have $U=U_1$ . Note that $U_i$ vanishes for $n\le i$ . For $n\ge i+1$ , we have a direct sum decomposition

$$ \begin{align*}U_i=U_i^{\mathrm{tree}}\oplus U_i^{\mathrm{wheel}},\end{align*} $$

where $U_i^{\mathrm {tree}}$ denotes the subrepresentation of $U_i$ that is isomorphic to $V_{1^{i+1},1}$ , and where $U_i^{\mathrm {wheel}}$ denotes the other subrepresentation that is isomorphic to $V_{1^i,0}$ . In what follows, we assume $n\ge i+1$ and we identify $U_i^{\mathrm {wheel}}$ with $V_{1^i,0}$ .

We will construct two contraction maps

$$ \begin{align*} c^{\mathrm{tree}}_{i}: \bigwedge\nolimits ^{i} U\to U_i^{\mathrm{tree}}, \quad c^{\mathrm{wheel}}_{i}: \bigwedge\nolimits ^{i} U\to U_i^{\mathrm{wheel}} \end{align*} $$

as follows. Let

$$ \begin{align*}M_i=(H^{\otimes 2}\otimes H^{\ast})^{\otimes i}\cong H^{\otimes 2i}\otimes (H^{\ast})^{\otimes i}.\end{align*} $$

Let $\iota _{i}: \bigwedge\nolimits ^{i} U \hookrightarrow M_i$ be the canonical injective map defined by

$$ \begin{align*} \iota_i(u_1\wedge\cdots\wedge u_i)= \sum_{\sigma\in \mathfrak{S}_i} \operatorname{sgn}(\sigma) \varphi(u_{\sigma(1)})\otimes\cdots\otimes \varphi(u_{\sigma(i)}), \end{align*} $$

where

$$ \begin{align*}\varphi:U \hookrightarrow H^{\otimes 2}\otimes H^{\ast},\quad (a_1\wedge a_2)\otimes b^{\ast} \mapsto (a_1\otimes a_2- a_2\otimes a_1)\otimes b^{\ast}. \end{align*} $$

Define a $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*} \tilde{c}_{i}: M_i\to U_i \end{align*} $$

by

$$ \begin{align*}\tilde{c}_{i}\left(\bigotimes_{j=1}^i (a_{2j-1}\otimes a_{2j} \otimes b_j^{\ast})\right)=\left(\prod_{j=2}^i b_j^{\ast}(a_{2(j-1)})\right) a_1\wedge a_3\wedge\cdots\wedge a_{2i-1}\wedge a_{2i} \otimes b_1^{\ast}\end{align*} $$

and let

$$ \begin{align*} \tilde{c}^{\mathrm{tree}}_{i}: M_i\xrightarrow{\tilde{c}_i} U_i\overset{\operatorname{pr}^{\mathrm{tree}}_i}{\twoheadrightarrow} U_i^{\mathrm{tree}} \end{align*} $$

be the composition, where the map $\operatorname {pr}^{\mathrm {tree}}_i$ is the canonical projection. Then set

$$ \begin{align*} c^{\mathrm{tree}}_{i}: \bigwedge\nolimits ^{i} U \xrightarrow{\iota_i} M_i\xrightarrow{\tilde{c}^{\mathrm{tree}}_i} U_i^{\mathrm{tree}}. \end{align*} $$

Explicitly, we have

(4.1.1) $$ \begin{align} c^{\mathrm{tree}}_{i}\Bigg(\bigwedge\nolimits _{j=1}^{i} ((a_{2j-1}\wedge & a_{2j})\otimes b_j^{\ast})\Bigg)\\ =\operatorname{pr}^{\mathrm{tree}}_i \Bigg(\sum_{\substack{\sigma\in \mathfrak{S}_i \\ \rho\in \prod_{j=1}^{i}\mathfrak{S}_2}} &\operatorname{sgn}(\sigma)\operatorname{sgn}(\rho) \left(\prod_{j=2}^{i}b_{\sigma(j)}^{\ast}(a_{\rho(2\sigma(j-1))})\right) \nonumber \\ &a_{\rho(2\sigma(1)-1)}\wedge a_{\rho(2\sigma(2)-1)}\wedge \dots \wedge a_{\rho(2\sigma(i)-1)}\wedge a_{\rho(2\sigma(1))}\otimes b_{\sigma(1)}^{\ast} \Bigg).\nonumber \end{align} $$

Define another $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*} \tilde{c}^{\mathrm{wheel}}_{i}: M_i\to U_i^{\mathrm{wheel}} \cong V_{1^i,0} \end{align*} $$

by

$$ \begin{align*}\tilde{c}^{\mathrm{wheel}}_{i}\left(\bigotimes_{j=1}^i (a_{2j-1}\otimes a_{2j} \otimes b_j^{\ast})\right)= \left(\prod_{j=1}^i b_j^{\ast}(a_{2(j-1)})\right) a_1\wedge a_3\wedge\cdots\wedge a_{2i-1},\end{align*} $$

where $a_{0}$ means $a_{2i}$ , and let

$$ \begin{align*} c^{\mathrm{wheel}}_{i}: \bigwedge\nolimits ^i U \xrightarrow{\iota_i} M_i\xrightarrow{\tilde{c}^{\mathrm{wheel}}_{i}} U_i^{\mathrm{wheel}}. \end{align*} $$

Explicitly, we have

(4.1.2) $$ \begin{align} \begin{aligned} &c^{\mathrm{wheel}}_{i}\left(\bigwedge\nolimits _{j=1}^{i} ((a_{2j-1}\wedge a_{2j})\otimes b_j^{\ast})\right)\\ &=\sum_{\substack{\sigma\in \mathfrak{S}_i \\ \rho\in \prod_{j=1}^{i}\mathfrak{S}_2}} \operatorname{sgn}(\sigma)\operatorname{sgn}(\rho) \left(\prod_{j=1}^{i}b_{\sigma(j)}^{\ast}(a_{\rho(2\sigma(j-1))})\right) \bigwedge\nolimits _{j=1}^{i} a_{\rho(2\sigma(j)-1)}, \end{aligned} \end{align} $$

where $a_{\rho (2\sigma (0))}=a_{\rho (2\sigma (i))}$ .

4.2 Non-trivial quotient representation of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$

Here we detect two irreducible quotient representations of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ by using $c^{\mathrm {tree}}_{i}$ and $c^{\mathrm {wheel}}_{i}$ .

Recall that in Definition 3.2, we defined the abelian cycle $\alpha _{(0,i)}\in H_i(\operatorname {IA}_n,\mathbb {Q})$ corresponding to the pair of partitions $(0,i)$ for $n\ge i+1$ , and that we have the following $\operatorname {GL}(n,\mathbb {Z})$ -homomorphisms

where $\tau _{\ast } $ is induced by the Johnson homomorphism.

Lemma 4.2. For $n\ge i+1$ , we have

$$ \begin{align*} c^{\mathrm{wheel}}_{i} \tau_{\ast} (\alpha_{(0,i)}) = i!\: e_2\wedge e_3\wedge\cdots \wedge e_{i+1}\in U_i^{\mathrm{wheel}}\setminus \{0\}, \end{align*} $$

and

$$ \begin{align*} \tilde{c}_{i} \iota_{i} \tau_{\ast} (\alpha_{(0,i)})= (-1)^i (i+1)!\: e_1\wedge e_2\wedge\cdots \wedge e_{i+1}\otimes e_{1}^{\ast}. \end{align*} $$

Therefore, for $n\ge i+2$ , we have

$$ \begin{align*} ([\operatorname{id}]- [E_{1,n}])c^{\mathrm{tree}}_{i} \tau_{\ast} (\alpha_{(0,i)}) =(-1)^i (i+1)!\: e_1\wedge e_2\wedge \cdots \wedge e_{i+1}\otimes e_{n}^{\ast} \in U_i^{\mathrm{tree}}\setminus \{0\}. \end{align*} $$

Proof. We have

(4.2.1) $$ \begin{align} \tau_{\ast} (\alpha_{(0,i)})= e_{2,1}^{1} \wedge (e_{3,1}^{1}+e_{3,2}^2) \wedge \cdots \wedge (\sum_{j=1}^i e_{i+1,j}^{j}) \in \bigwedge\nolimits ^{i} U. \end{align} $$

First, we consider the image under $c^{\mathrm {wheel}}_{i}$ . By (4.2.1) and (4.1.2), we have

$$ \begin{align*} c^{\mathrm{wheel}}_{i} \tau_{\ast} (\alpha_{(0,i)}) &= c^{\mathrm{wheel}}_{i} ( e_{2,1}^{1} \wedge e_{3,1}^{1}\wedge \cdots \wedge e_{i+1,1}^{1}) \\ &= i!\: e_2\wedge e_3\wedge\cdots \wedge e_{i+1}\in U_i^{\mathrm{wheel}}\setminus \{0\} \end{align*} $$

since the product $\prod _{j=1}^{i}b_{\sigma (j)}^{\ast }(a_{\rho (2\sigma (j-1))})$ in (4.1.2) vanishes for any other wedges with at least one component $e_{k,j}^{j} \; (j\neq 1)$ and for $\rho \in \prod _{j=1}^{i}\mathfrak {S}_2\setminus \{\operatorname {id}\}$ .

Next, we consider the image under the contraction map $c^{\mathrm {tree}}_{i}$ . For $K\in [n]^{3i}$ regarded as a map $[3i]\to [n]$ , let $e_{K}=\bigotimes _{j=1}^i (e_{K(j)}\otimes e_{K(2i+1-j)}\otimes e_{K(2i+j)}^{\ast })$ . Then $\{e_{K} \mid K\in [n]^{3i} \}$ forms a basis for $M_i$ . We write

$$ \begin{align*} \iota_{i} \tau_{\ast} (\alpha_{(0,i)})=\sum_{K\in [n]^{3i}} a_K e_K \in M_i, \end{align*} $$

where $a_K\in \mathbb {Z}$ . Then by (4.2.1), we can easily check the following properties of the coefficient $a_K$

• • for each $K\in [n]^{3i}$ , we have $a_K\in \{0,1,-1\}$ .

• • for $K\notin [i+1]^{3i}$ , we have $a_K=0$ .

• • for $K\in [i+1]^{3i}$ such that $K(2i+j)\neq \min \{K(j),K(2i+1-j)\}$ for some $1\le j\le i$ , we have $a_K=0$ .

By the second property of $a_K$ and the definition of $\tilde {c}_i$ , if we have $\tilde {c}_{i}(a_K e_K)\in U_i\setminus \{0\}$ , then there exists a permutation $\sigma \in \mathfrak {S}_{i+1}$ such that for any $j\in [i+1]$ , we have $K(j)=\sigma (j)$ .

Claim 1. For each $\sigma \in \mathfrak {S}_{i+1}$ , there uniquely exists $K\in [i+1]^{3i}$ such that

• • $K(j)=\sigma (j)$ for $j\in [i+1]$

• • $\tilde {c}_{i}(a_K e_K)\neq 0$ .

In particular, we have

$$ \begin{align*}\tilde{c}_{i}(a_K e_K)= a_K (e_{\sigma(1)}\wedge e_{\sigma(2)}\wedge \cdots\wedge e_{\sigma(i+1)})\otimes e_1^{\ast}.\end{align*} $$

Proof of Claim 1

Define $K\in [i+1]^{3i}$ by

$$ \begin{align*} K(j)= \begin{cases} \sigma(j) & (1\le j\le i+1)\\ \min \{K(2i+2-j), \cdots , K(i+1)\} & (i+2\le j \le 2i)\\ \min \{K(j-2i),\cdots , K(i+1) \} & (2i+1\le j\le 3i). \end{cases} \end{align*} $$

Then we can check that

$$ \begin{align*} \tilde{c}_{i}(a_K e_K)=a_K (e_{\sigma(1)}\wedge e_{\sigma(2)}\wedge \cdots\wedge e_{\sigma(i+1)})\otimes e_1^{\ast} \end{align*} $$

since we have $K(2i+1)=\min \{K(1),\cdots , K(i+1)\}=1$ . The uniqueness follows from the third property of $a_{K}$ and the construction of K.

Claim 2. For each $\sigma \in \mathfrak {S}_{i+1}$ , let $K(\sigma )$ denote the unique element of $[i+1]^{3i}$ that we constructed in the proof of Claim 1. Then we have $a_{K(\sigma )}=(-1)^i \operatorname {sgn} (\sigma )$ .

Proof of Claim 2

We use induction on i. For $i=1$ , we have

$$ \begin{align*}a_{K(\operatorname{id})}=-1=(-1)^1 \operatorname{sgn} (\operatorname{id}), \quad a_{K((12))}=1=(-1)^1 \operatorname{sgn}((12)). \end{align*} $$

Assume that the statement holds for $i\ge 1$ . For any $\sigma \in \mathfrak {S}_{i+1}$ , define $\tau \in \mathfrak {S}_{i+1}$ as

$$ \begin{align*} \tau(j)= \begin{cases} \sigma(1) & (j=1)\\ j-1 & (2\le j\le \sigma(1))\\ j &(\sigma(1)+1\le j\le i+1). \end{cases} \end{align*} $$

Let $\tilde {\tau }=\tau ^{-1}\sigma $ . Then $\tilde {\tau }$ maps $1$ to $1$ , so we regard $\tilde {\tau }\in \mathfrak {S}_i$ by identifying $\{2,\cdots ,i+1\}$ with $\{1,\cdots ,i\}$ via the order preserving bijection. We will prove that

(4.2.2) $$ \begin{align} a_{K(\sigma)}=(-1)^{\sigma(1)}a_{K(\tilde{\tau})}. \end{align} $$

If $\sigma (1)=1$ , then we have $K(1)=1, K(2i)=2, K(2i+1)=1$ by the definition of K in Claim 1. Therefore, by (4.2.1), we have $a_{K(\sigma )}=(-1)a_{K(\tilde {\tau })}$ . In a similar way, if $\sigma (1)\neq 1$ , then we have $K(1)=\sigma (1), K(2i)=1, K(2i+1)=1$ . Therefore, we have $a_{K(\sigma )}=(-1)^{(\sigma (1)-2)}a_{K(\tilde {\tau })}$ , where the sign $(-1)^{(\sigma (1)-2)}$ comes from the permutation which brings the factor $e_{\sigma (1),1}^1$ in a summand of (4.2.1) to the first factor. Hence, we have (4.2.2).

Since we have $\operatorname {sgn}(\tau )=(-1)^{\sigma (1)-1}$ , by induction hypothesis, we have

$$ \begin{align*} (-1)^{\sigma(1)} a_{K(\tilde{\tau})} =(-1)^{\sigma(1)}(-1)^i \operatorname{sgn}(\tilde{\tau}) =(-1)^{i+1} \operatorname{sgn}(\tau) \operatorname{sgn}(\tilde{\tau}) =(-1)^{i+1}\operatorname{sgn}(\sigma), \end{align*} $$

which completes the proof.

By using Claims 1 and 2, we have

$$ \begin{align*} \tilde{c}_{i} \iota_{i} \tau_{\ast} (\alpha_{(0,i)}) &=\sum_{K\in [n]^{3i}} \tilde{c}_{i} (a_K e_K)\\ &=\sum_{\sigma\in \mathfrak{S}_{i+1}} a_{K(\sigma)} (e_{\sigma(1)}\wedge e_{\sigma(2)}\wedge \cdots\wedge e_{\sigma(i+1)})\otimes e_1^{\ast}\\ &=\sum_{\sigma\in \mathfrak{S}_{i+1}} a_{K(\sigma)} \operatorname{sgn}(\sigma) (e_1\wedge e_2\wedge\cdots\wedge e_{i+1})\otimes e_1^{\ast}\\ &=(-1)^i (i+1)!\: e_1\wedge e_2\wedge\cdots \wedge e_{i+1}\otimes e_{1}^{\ast}. \end{align*} $$

Since for $n\ge i+2$ , we have

$$ \begin{align*}([\operatorname{id}]- [E_{1,n}])\tilde{c}_{i} \iota_{i} \tau_{\ast} (\alpha_{(0,i)}) =(-1)^i (i+1)!\: e_1\wedge e_2\wedge \cdots \wedge e_{i+1}\otimes e_{n}^{\ast} \in U_i^{\mathrm{tree}}\setminus \{0\},\end{align*} $$

the proof is complete.

Theorem 4.3. For $n\ge i+1$ , we have a surjective $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*}H^A_i(\operatorname{IA}_n,\mathbb{Q}) \twoheadrightarrow \operatorname{Hom}\left(H,\bigwedge\nolimits ^{i+1} H\right)\cong V_{1^i,0} \oplus V_{1^{i+1},1}.\end{align*} $$

Therefore, $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ includes a $\operatorname {GL}(n,\mathbb {Q})$ -subrepresentation which is isomorphic to $\operatorname {Hom}(H,\bigwedge\nolimits ^{i+1} H)$ .

4.3 A representation-stable quotient of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$

Here we study the sequence $H_i^A(\operatorname {IA}_{\ast },\mathbb {Q})$ of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ .

We have a canonical inclusion map $\operatorname {IA}_n\hookrightarrow \operatorname {IA}_{n+1}$ sending $f\in \operatorname {IA}_n$ to $f\in \operatorname {IA}_{n+1}$ which is the same as f on $F_n\subset F_{n+1}$ and fixes $x_{n+1}$ . Let $H^A_i(\operatorname {IA}_{\ast },\mathbb {Q})$ denote the sequence of group homomorphisms $H^A_i(\operatorname {IA}_n,\mathbb {Q})\to H^A_i(\operatorname {IA}_{n+1},\mathbb {Q})$ induced by the inclusion maps. Since the inclusion map $\operatorname {IA}_n\hookrightarrow \operatorname {IA}_{n+1}$ sends $g_{a,b}\in \operatorname {IA}_n$ to $g_{a,b}\in \operatorname {IA}_{n+1}$ and $f_{a,b,c}\in \operatorname {IA}_n$ to $f_{a,b,c}\in \operatorname {IA}_{n+1}$ , the abelian cycle $\alpha _{(0,i)}\in H_i(\operatorname {IA}_n,\mathbb {Q})$ is sent to $\alpha _{(0,i)}\in H_i(\operatorname {IA}_{n+1},\mathbb {Q})$ . Therefore, the element $\tau _{\ast }(\alpha _{(0,i)})\in H^A_i(\operatorname {IA}_n,\mathbb {Q})$ is sent to $\tau _{\ast }(\alpha _{(0,i)})\in H^A_i(\operatorname {IA}_{n+1},\mathbb {Q})$ .

The canonical inclusion maps $H(n)\hookrightarrow H(n+1)$ and $H(n)^{\ast }\hookrightarrow H(n+1)^{\ast }$ that we have observed in Section 2.7 send $e_1\wedge e_2\wedge \cdots \wedge e_{i+1}\otimes e_1^{\ast }\in \bigwedge\nolimits ^{i+1} H(n)\otimes H(n)^{\ast }$ to $e_1\wedge e_2\wedge \cdots \wedge e_{i+1}\otimes e_1^{\ast }\in \bigwedge\nolimits ^{i+1} H(n+1) \otimes H(n+1)^{\ast }$ . Let $\left (\bigwedge\nolimits ^{i+1} H\right )\otimes H^{\ast }$ denote the sequence of group homomorphisms $\left (\bigwedge\nolimits ^{i+1} H(n)\right )\otimes H(n)^{\ast }\hookrightarrow \left (\bigwedge\nolimits ^{i+1} H(n+1)\right )\otimes H(n+1)^{\ast }$ . Then it is obvious that $\left (\bigwedge\nolimits ^{i+1} H\right )\otimes H^{\ast }$ is representation stable.

Let $\{V_n,\phi _n:V_n\to V_{n+1}\}$ and $\{W_n,\psi _n:W_n\to W_{n+1}\}$ be sequences of algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations. By a morphism from $\{V_n,\phi _n\}$ to $\{W_n,\psi _n\}$ , we mean a family $\{f_n\}$ of $\operatorname {GL}(n,\mathbb {Q})$ -module maps $f_n:V_n\to W_n$ satisfying $\psi _n f_n=f_{n+1}\phi _n$ . Then we have the following proposition by Lemma 4.2 and Theorem 4.3.

Proposition 4.4. We have a surjective morphism of sequences of algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representations

$$ \begin{align*}H^A_i(\operatorname{IA}_{\ast},\mathbb{Q})\twoheadrightarrow\left(\bigwedge\nolimits ^{i+1} H\right) \otimes H^{\ast},\end{align*} $$

where $\left (\bigwedge\nolimits ^{i+1} H\right ) \otimes H^{\ast }$ is representation stable.

By the representation stability, we can consider an analogue of Theorem 4.3 in the category of algebraic $\operatorname {GL}(\infty ,\mathbb {Q})$ -representations. Let $\operatorname {IA}_{\infty }=\varinjlim _{n}\operatorname {IA}_n$ denote the direct limit. Both group homology and Albanese homology preserve direct limits (see Proposition A.1), that is, we have

$$ \begin{align*}H_i(\operatorname{IA}_{\infty},\mathbb{Q})\cong \varinjlim_{n}H_i(\operatorname{IA}_n,\mathbb{Q}),\quad H^A_i(\operatorname{IA}_{\infty},\mathbb{Q})\cong \varinjlim_{n}H^A_i(\operatorname{IA}_n,\mathbb{Q}). \end{align*} $$

Since $\alpha _{(0,i)}$ is preserved under $H_i(\operatorname {IA}_n,\mathbb {Q})\to H_i(\operatorname {IA}_{n+1},\mathbb {Q})$ , we obtain a cycle $\alpha _{(0,i)}$ in $H_i(\operatorname {IA}_{\infty },\mathbb {Q})$ and an element $\tau _{\ast }(\alpha _{(0,i)})\in H^A_i(\operatorname {IA}_{\infty },\mathbb {Q})$ . Therefore, by Lemma 4.2, we have a $\operatorname {GL}(\infty ,\mathbb {Q})$ -homomorphism

$$ \begin{align*}\tilde{c}_{i} \iota_{i}: H^A_i(\operatorname{IA}_{\infty},\mathbb{Q}) \to \left(\bigwedge\nolimits ^{i+1} \mathbf{H}\right)\otimes\mathbf{H}^{\ast}\end{align*} $$

such that

$$ \begin{align*} \tilde{c}_{i} \iota_{i} \tau_{\ast} (\alpha_{(0,i)})= (-1)^i (i+1)!\: e_1\wedge e_2\wedge\cdots \wedge e_{i+1}\otimes e_{1}^{\ast}. \end{align*} $$

Corollary 4.5. The $\operatorname {GL}(\infty ,\mathbb {Q})$ -homomorphism

$$ \begin{align*}\tilde{c}_{i} \iota_{i}: H^A_i(\operatorname{IA}_{\infty},\mathbb{Q})\to\left(\bigwedge\nolimits ^{i+1} \mathbf{H}\right)\otimes\mathbf{H}^{\ast} \end{align*} $$

is surjective. In particular, for each $i\ge 1$ , $H_i(\operatorname {IA}_{\infty },\mathbb {Q})$ is infinite-dimensional.

Remark 4.6. We have a non-split exact sequence

$$ \begin{align*}0\to V_{1^{i+1},1}\to \left(\bigwedge\nolimits ^{i+1} \mathbf{H}\right)\otimes\mathbf{H}^{\ast}\to V_{1^i,0}\to 0,\end{align*} $$

which generalizes the observation in Section 2.7 that $\mathbf {H}\otimes \mathbf {H}^{\ast }\to \mathbb {Q}$ does not split. Therefore, $H^A_i(\operatorname {IA}_{\infty },\mathbb {Q})$ is not semisimple.

6.1 The subquotient representation $W_i$ of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$

Recall that for $n\ge i+1$ , we have

$$ \begin{align*}U_i=\operatorname{Hom}\left(H,\bigwedge\nolimits ^{i+1} H\right)=U_i^{\mathrm{tree}}\oplus U_i^{\mathrm{wheel}}\cong V_{1^{i+1},1}\oplus V_{1^i,0}.\end{align*} $$

Let $U_{\ast }=\bigoplus _{i\ge 1} U_i$ , which is a graded algebraic $\operatorname {GL}(n,\mathbb {Q})$ -representation. Define $W_{\ast }=\widetilde {S}^{\ast } (U_{\ast })$ as the traceless part of the graded-symmetric algebra $S^{\ast } (U_{\ast })$ of $U_{\ast }$ , which we defined in Section 2.5. We can also construct $W_{\ast }$ by using an operad $\mathcal {C} om$ of non-unital commutative algebras as we explain in Section 11.

For $i\ge 0$ , let $(\mu ,\nu )\vdash i$ be a pair of partitions. If we write

$$ \begin{align*}\mu=(\mu_1^{k_1'},\cdots,\mu_r^{k_r'}),\quad \nu=(\nu_1^{k_1"},\cdots,\nu_s^{k_s"}),\end{align*} $$

then it means that $\mu _1>\mu _2>\cdots >\mu _r$ and $\mu _j$ appears $k_j'>0$ times in $\mu $ and that $\nu _1>\nu _2>\cdots >\nu _s$ and $\nu _j$ appears $k_j">0$ times in $\nu $ . Then the length of $\mu $ (resp. $\nu $ ) is $l(\mu )=\sum _{j=1}^r k_j'$ (resp. $l(\nu )=\sum _{j=1}^s k_j"$ ). Let

$$ \begin{align*}U^{\mathrm{tree}}_{\mu}=\bigotimes_{j=1}^r S^{k_j'} (U_{\mu_j}^{\mathrm{tree}}),\quad U^{\mathrm{wheel}}_{\nu}=\bigotimes_{j=1}^s S^{k_j"} (U_{\nu_j}^{\mathrm{wheel}}).\end{align*} $$

Then we have

$$ \begin{align*}S^{\ast}(U_{\ast})= \bigoplus_{i\ge 0} \bigoplus_{(\mu,\nu)\vdash i} U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu}.\end{align*} $$

Since $W_i=\widetilde {S}^{\ast } (U_{\ast })_i$ by the definition of $W_{\ast }$ , we have

$$ \begin{align*}W_i= \bigoplus_{(\mu,\nu)\vdash i} W(\mu,\nu)\subset S^{\ast}(U^{\ast})_i,\end{align*} $$

where

$$ \begin{align*}W(\mu,\nu)= \widetilde{\bigotimes}_{j=1}^r \widetilde{S}^{k_j'}\left(U_{\mu_j}^{\mathrm{tree}}\right) \widetilde{\otimes}\; \widetilde{\bigotimes}_{j=1}^s \widetilde{S}^{k_j"}\left(U_{\nu_j}^{\mathrm{wheel}}\right)\subset U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu}.\end{align*} $$

For example, we have for $n\ge 3$ ,

$$ \begin{align*}W_1=W(1,0)\oplus W(0,1)\cong V_{1^2,1}\oplus V_{1,0},\end{align*} $$

and we have for $n\ge 6$ ,

$$ \begin{align*} W_2&=W(2,0)\oplus W(0,2)\oplus W(1^2,0)\oplus W(1,1)\oplus W(0,1^2)\\ &\cong V_{1^3,1}\oplus V_{1^2,0} \oplus (V_{1^4,1^2}\oplus V_{2 1^2,2}\oplus V_{2^2,1^2})\oplus (V_{21,1}\oplus V_{1^3,1})\oplus V_{1^2,0}. \end{align*} $$

Here, we observe that the subrepresentation $W(\mu ,\nu )$ of $W_i$ is an image of the traceless part $T_{i+l(\mu ),l(\mu )}$ under a projection. Let

$$ \begin{align*}\pi_j^{\mu}: H^{\otimes k_j'(1+\mu_j)}\otimes (H^{\ast})^{\otimes k_j'}=(H^{\otimes (1+\mu_j)}\otimes H^{\ast})^{\otimes k_j'}\twoheadrightarrow (V_{1^{1+\mu_j}, 1})^{\otimes k_j'}\twoheadrightarrow S^{k_j'} (V_{1^{1+\mu_j}, 1}) \end{align*} $$

and

$$ \begin{align*}\pi_j^{\nu}: H^{\otimes k_j"\nu_j}= (H^{\otimes \nu_j})^{\otimes k_j"} \twoheadrightarrow (V_{1^{\nu_j}})^{\otimes k_j"}\twoheadrightarrow S^{k_j"} (V_{1^{\nu_j}}) \end{align*} $$

be the composition of two canonical projections, and let

$$ \begin{align*} \pi^{(\mu,\nu)}=\left(\bigotimes_{j=1}^r \pi_j^{\mu} \right) \otimes \left(\bigotimes_{j=1}^s \pi_j^{\nu}\right): H^{\otimes (i+l(\mu))} \otimes (H^{\ast})^{\otimes l(\mu)}\to U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu}. \end{align*} $$

Then we have

$$ \begin{align*}W(\mu,\nu)=\pi^{(\mu,\nu)}(T_{i+l(\mu), l(\mu)}).\end{align*} $$

Define

$$ \begin{align*} F_{(\mu,\nu)}: H_i(U,\mathbb{Q})=\bigwedge\nolimits ^i U \to U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu} \end{align*} $$

to be the composition of the inclusion $\iota _{i}: \bigwedge\nolimits ^i U\hookrightarrow M_i$ and

$$ \begin{align*}\left(\bigotimes_{j=1}^r \frac{1}{k_j'!} \prod^{k_j'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \right) \otimes \left(\bigotimes_{j=1}^s \frac{1}{k_j"!} \prod^{k_j"}\tilde{c}^{\mathrm{wheel}}_{\nu_j} \right) : M_i \to U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu}.\end{align*} $$

Then we have a $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*} F_i:= \bigoplus_{(\mu,\nu)\vdash i} F_{(\mu,\nu)} : H_i(U,\mathbb{Q})\to S^{\ast}(U_{\ast})_i= \bigoplus_{(\mu,\nu)\vdash i} U^{\mathrm{tree}}_{\mu}\otimes U^{\mathrm{wheel}}_{\nu}. \end{align*} $$

Theorem 6.1. For $n\ge 3i$ , we have

$$ \begin{align*}F_i (H^A_i(\operatorname{IA}_n,\mathbb{Q})) \supset W_i.\end{align*} $$

We will prove Theorem 6.1 in the rest of this section by using the abelian cycles $\alpha _{(\mu ,\nu )}$ .

It seems natural to make the following conjecture, which holds for $i=1$ [Reference Cohen and Pakianathan9, Reference Farb14, Reference Kawazumi.22], for $i=2$ [Reference Pettet37] and for $i=3$ as we will observe in Theorem 10.2.

Conjecture 6.2. For $n\ge 3i$ , $F_i$ restricts to a $\operatorname {GL}(n,\mathbb {Q})$ -isomorphism

$$ \begin{align*}F_i : H^A_i(\operatorname{IA}_n,\mathbb{Q}) \xrightarrow{\cong} W_i. \end{align*} $$

Conjecture 6.2 implies that as a $\operatorname {GL}(n,\mathbb {Q})$ -representation, the Albanese homology $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ is generated by the images of abelian cycles of $H_i(\operatorname {IA}_n,\mathbb {Q})$ under $\tau _{\ast }$ .

We will introduce the following relaxed version of Conjecture 6.2.

Conjecture 6.3. We have a $\operatorname {GL}(n,\mathbb {Q})$ -isomorphism

$$ \begin{align*}H^A_i(\operatorname{IA}_n,\mathbb{Q}) \cong W_i \end{align*} $$

for sufficiently large n with respect to i.

We would like to realize $W_i$ as a subrepresentation of $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ .

Problem 6.4. Construct a lift

$$ \begin{align*}W_i\to H^A_i(\operatorname{IA}_n,\mathbb{Q}) \end{align*} $$

of the inclusion map $W_i\hookrightarrow F_i(H^A_i(\operatorname {IA}_n,\mathbb {Q}))$ along $H^A_i(\operatorname {IA}_n,\mathbb {Q}){\overset {F_i}{\longrightarrow }} F_i(H^A_i(\operatorname {IA}_n,\mathbb {Q}))$ as $\operatorname {GL}(n,\mathbb {Q})$ -representations for sufficiently large n with respect to i.

We would like to realize the Albanese homology $H^A_i(\operatorname {IA}_n,\mathbb {Q})$ as a subrepresentation of $H_i(\operatorname {IA}_n,\mathbb {Q})$ .

Problem 6.5. Construct a section of

$$ \begin{align*}H_i(\operatorname{IA}_n,\mathbb{Q})\overset{\tau_{\ast}}{\twoheadrightarrow} H^A_i(\operatorname{IA}_n,\mathbb{Q}) \end{align*} $$

as $\operatorname {GL}(n,\mathbb {Z})$ -representations for sufficiently large n with respect to i.

6.2 Computation of the contraction maps

In a way similar to Lemma 4.2, we obtain the following lemma.

Lemma 6.6. For $n\ge i+2$ , we have

$$ \begin{align*} c^{\mathrm{wheel}}_{i} \tau_{\ast} (\alpha_{(i,0)}) = 0, \end{align*} $$

and

$$ \begin{align*} c^{\mathrm{tree}}_{i} \tau_{\ast} (\alpha_{(i,0)}) = (-1)^{i-1}(i+1)! \: e_1\wedge e_2\wedge e_4\wedge \cdots \wedge e_{i+2}\otimes e_{3}^{\ast} \in U_i^{\mathrm{tree}}\setminus \{0\}. \end{align*} $$

Proof. For the abelian cycle $\alpha _{(i,0)}$ , we have

$$ \begin{align*} \tau_{\ast} (\alpha_{(i,0)})= e_{1,2}^{3} \wedge \left(\sum_{j=1}^{3} e_{4,j}^{j}\right) \wedge \cdots \wedge \left(\sum_{j=1}^{i+1} e_{i+2,j}^{j}\right). \end{align*} $$

By using (4.1.2), we can check that

(6.2.1) $$ \begin{align} c^{\mathrm{wheel}}_{i}(e_{a_1,b_1}^{d_1} \wedge e_{a_2,b_2}^{b_2}\wedge\cdots \wedge e_{a_i,b_i}^{b_i})=0 \end{align} $$

if $d_1\notin \{a_1,\cdots , a_i, b_1\}$ . Therefore, we have $c^{\mathrm {wheel}}_{i} \tau _{\ast } (\alpha _{(i,0)}) = 0$ .

We will compute $c^{\mathrm {tree}}_3\tau _{\ast }(\alpha _{(3,0)})$ for example. We have

$$ \begin{align*} c^{\mathrm{tree}}_3(e_{1,2}^{3} \wedge e_{4,i}^{i}\wedge e_{5,j}^{j}) =\begin{cases} 0 & (i,j)=(1,2),(2,1),(3,4),\\ 4 e_1\wedge e_2\wedge e_4\wedge e_5 \otimes e_3^{\ast},& (i,j)=(1,1),(2,2),(3,3),\\ 2 e_1\wedge e_2\wedge e_4\wedge e_5 \otimes e_3^{\ast},& (\text{otherwise}) \end{cases} \end{align*} $$

and thus we have

$$ \begin{align*} c^{\mathrm{tree}}_3\tau_{\ast}(\alpha_{(3,0)}) &=c^{\mathrm{tree}}_3(e_{1,2}^{3} \wedge (e_{4,1}^{1}+e_{4,2}^2+ e_{4,3}^3)\wedge (e_{5,1}^{1}+e_{5,2}^2+e_{5,3}^3+e_{5,4}^{4}))\\ &=24 e_1\wedge e_2\wedge e_4\wedge e_5 \otimes e_3^{\ast}. \end{align*} $$

In general, it is easy to see that

$$ \begin{align*}c^{\mathrm{tree}}_{i} (e_{1,2}^{3} \wedge e_{4,b_2}^{b_2}\wedge\cdots \wedge e_{i+2,b_i}^{b_i}) \in \mathbb{Z} e_1\wedge e_2\wedge e_4\wedge \cdots \wedge e_{i+2}\otimes e_{3}^{\ast}\end{align*} $$

by (4.1.1). The computation of the coefficient in $c^{\mathrm {tree}}_{i}(\tau _{\ast } (\alpha _{(i,0)}))$ is similar to that of the coefficient in $c^{\mathrm {tree}}_{i}(\tau _{\ast } (\alpha _{(0,i)}))$ in Lemma 4.2.

We call elements of the form

$$ \begin{align*}e_{a_1,b_1}^{d_1}\wedge e_{a_2,b_2}^{d_2}\wedge\cdots \wedge e_{a_i,b_i}^{d_i}\in \bigwedge\nolimits ^i U \end{align*} $$

the basis elements of $\bigwedge\nolimits ^i U$ . Two basis elements $x=e_{a_1,b_1}^{d_1}\wedge e_{a_2,b_2}^{d_2}\wedge \cdots \wedge e_{a_i,b_i}^{d_i}$ of $\bigwedge\nolimits ^i U$ and $y=e_{p_1,q_1}^{r_1}\wedge e_{p_2,q_2}^{r_2}\wedge \cdots \wedge e_{p_j,q_j}^{r_j}$ of $\bigwedge\nolimits ^j U$ are said to be disjoint if $\{a_k,b_k,d_k\}_{k=1}^{i}\cap \{p_l,q_l,r_l\}_{l=1}^{j}=\emptyset $ . We also call linear sums $x=\sum _{k=1}^{m} a_k x_k\in \bigwedge\nolimits ^i U$ and $y=\sum _{l=1}^{n} b_l y_l\in \bigwedge\nolimits ^j U$ of basis elements disjoint if any pair $(x_k,y_l)$ of basis elements is disjoint. If basis elements $x\in \bigwedge\nolimits ^i U$ and $y\in \bigwedge\nolimits ^j U$ are disjoint, then it is easy to see that

(6.2.2) $$ \begin{align} c^{\mathrm{wheel}}_{i+j}(x\wedge y)=c^{\mathrm{tree}}_{i+j}(x\wedge y)=0. \end{align} $$

Note that we have

$$ \begin{align*} c^{\mathrm{wheel}}_{i}(\tau_{\ast}(\alpha_{(\mu,\nu)}))=c^{\mathrm{tree}}_{i}(\tau_{\ast}(\alpha_{(\mu,\nu)}))=0 \end{align*} $$

for $(\mu ,\nu )\vdash i$ with $l(\mu )+l(\nu )\ge 2$ by (6.2.2).

Let $P_i$ denote the set of pairs of partitions of total size i. For $l\in \{0,\cdots ,i\}$ , let $P_i^l$ denote the subset of $P_i$ consisting of elements $(\mu ,\nu )$ with $l(\mu )=l$ . In what follows, we consider a sufficient condition for $(\mu ,\nu ), (\xi ,\eta )\in P_i^l$ that $F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\xi ,\eta )}))$ vanishes. In order to state the condition, we will put a partial order on $P_i^l$ as follows. For $(\xi ,\eta ), (\mu ,\nu ) \in P_i^l$ , we write $(\xi ,\eta )\ge (\mu ,\nu )$ if there exist a decomposition $L_1\sqcup \cdots \sqcup L_{l+l(\eta )} = \{1,\cdots ,l(\nu )\}$ and $\sigma \in \mathfrak {S}_l$ such that

(6.2.3) $$ \begin{align} \xi_j-\mu_{\sigma(j)}=\sum_{k\in L_j}\nu_k \quad(1\le j\le l), \end{align} $$

(6.2.4) $$ \begin{align} \eta_j=\sum_{k\in L_{l+j}}\nu_k\quad (1\le j\le l(\eta)). \end{align} $$

We can check that $(P_i^l,\ge )$ is a partially ordered set with the minimum element $(1^l, 1^{i-l})$ and with the maximal elements $(\mu ,0)$ for $\mu \vdash i$ with $l(\mu )=l$ .

Lemma 6.7. For $(\xi ,\eta ),(\mu ,\nu )\in P_i^l$ , we have

$$ \begin{align*} F_{(\mu,\nu)}(\tau_{\ast}(\alpha_{(\xi,\eta)}))=0 \quad \text{unless } (\xi,\eta)\geq (\mu,\nu). \end{align*} $$

Proof. We have

(6.2.5) $$ \begin{align} \tau_{\ast}(\alpha_{(\xi,\eta)})= \tau(\xi,1)\wedge\cdots\wedge \tau(\xi,l)\wedge \tau(\eta,1)\wedge\cdots\wedge \tau(\eta,l(\eta))\in \bigwedge\nolimits ^i U, \end{align} $$

where $\tau (\xi ,j)$ (resp. $\tau (\eta ,j)$ ) is obtained from $\tau _{\ast }(\alpha _{(\xi _j,0)})$ (resp. $\tau _{\ast }(\alpha _{(0,\eta _j)})$ ) by the shift that appears in the definition of $\alpha _{(\xi ,\eta )}$ . Note that $\{\tau (\xi ,j)\mid 1\le j\le l\}\cup \{\tau (\eta ,j)\mid 1\le j\le l(\eta )\}$ is a set of mutually disjoint elements. Therefore, by (6.2.2) and by the definition of $F_{(\mu ,\nu )}$ , we have $F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\xi ,\eta )}))=0$ unless there exist decompositions $L_1\sqcup \cdots \sqcup L_{l+l(\eta )} = \{1,\cdots ,l(\nu )\}$ and $M_1\sqcup \cdots \sqcup M_{l+l(\eta )} = \{1,\cdots ,l\}$ such that

$$ \begin{align*} \xi_j & =\sum_{k\in M_j}\mu_k+ \sum_{k\in L_j}\nu_k\quad (1\le j\le l),\\ \eta_j & =\sum_{k\in M_{l+j}}\mu_k+ \sum_{k\in L_{l+j}}\nu_k\quad (1\le j\le l(\eta)). \end{align*} $$

Moreover, it follows from Lemma 6.6 (more specifically from the vanishing property (6.2.1)) that we have $F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\xi ,\eta )}))=0$ unless $M_j\neq \emptyset $ for $1\le j\le l$ . Since a decomposition $M_1\sqcup \cdots \sqcup M_{l+l(\eta )} = \{1,\cdots ,l\}$ with $M_j\neq \emptyset $ for $1\le j\le l$ is equivalent to a permutation $\sigma \in \mathfrak {S}_l$ , we have $F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\xi ,\eta )}))=0$ unless $(\xi ,\eta )\geq (\mu ,\nu )$ .

6.3 Proof of Theorem 6.1

To prove Theorem 6.1, we use the following lemma, which is an analogue of [Reference Lindell.30, Theorem 1.5].

Lemma 6.8. Let $(\mu ,\nu )\vdash i$ . Then for $n\ge i+2l(\mu )+l(\nu )$ , we have

$$ \begin{align*}F_{(\mu,\nu)} (H^A_i(\operatorname{IA}_n,\mathbb{Q})) \supset W(\mu,\nu). \end{align*} $$

Proof. Our proof is analogous to the proof of [Reference Lindell.30, Theorem 1.5]. By Corollary 2.3, $W(\mu ,\nu )=\pi ^{(\mu ,\nu )}(T_{i+l(\mu ), l(\mu )})$ is generated by the element $\pi ^{(\mu ,\nu )}(e_{i+l(\mu ),l(\mu )})$ . Therefore, it suffices to show that there is an element $x \in \mathbb {Q}[\operatorname {GL}(n,\mathbb {Q})]$ such that $x (F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\mu ,\nu )})))=\pi ^{(\mu ,\nu )}(e_{i+l(\mu ),l(\mu )})$ .

Here, we write

$$ \begin{align*}\mu=(\mu_1,\cdots,\mu_{l(\mu)})=(\tilde{\mu}_1^{k_1'},\cdots, \tilde{\mu}_r^{k_r'}),\quad \nu=(\nu_1,\cdots, \nu_{l(\nu)})=(\tilde{\nu}_1^{k_1"},\cdots, \tilde{\nu}_s^{k_s"}).\end{align*} $$

As in (6.2.5), we have

$$ \begin{align*}\tau_{\ast}(\alpha_{(\mu,\nu)})= \tau(\mu,1)\wedge\cdots\wedge \tau(\mu,l(\mu))\wedge \tau(\nu,1)\wedge\cdots\wedge \tau(\nu,l(\nu))\in \bigwedge\nolimits ^i U. \end{align*} $$

For $1\le j\le r$ and $1\le m\le k_j'$ , let $\tau (\mu ,j,m)=\tau (\mu ,\phi _{\mu }(j,m))$ , where $\phi _{\mu }(j,m)=k_1'+\cdots + k_{j-1}'+m$ . In a similar way, we define $\phi _{\nu }(j,m)$ and $\tau (\nu ,j,m)$ . Then we have

$$ \begin{align*} &F_{(\mu,\nu)}(\tau_{\ast}(\alpha_{(\mu,\nu)}))\\ &=\left(\bigotimes_{j=1}^r \frac{1}{k_j'!} \prod_{m=1}^{k_j'}c^{\mathrm{tree}}_{\tilde{\mu}_j}(\tau(\mu,j,m)) \right)\otimes \left(\bigotimes_{j=1}^s \frac{1}{k_j"!} \prod_{m=1}^{k_j"}c^{\mathrm{wheel}}_{\tilde{\nu}_j} (\tau(\nu,j,m)) \right). \end{align*} $$

By Lemma 6.6, we have

$$ \begin{align*}c^{\mathrm{tree}}_{\tilde{\mu}_j} (\tau(\mu,j,m))=(-1)^{\tilde{\mu}_j-1}(\tilde{\mu}_j+1)! \: s(\phi_{\mu}(j,m))(e_{1}\wedge e_{2}\wedge e_{4}\wedge \cdots \wedge e_{\tilde{\mu}_j+2}\otimes e_{3}^{\ast}),\end{align*} $$

where $s(j)$ is the function that we used in Section 3.2 to define the abelian cycle $\alpha _{(\mu ,\nu )}$ , and where $s(\phi _{\mu }(j,m))$ denotes the shift homomorphism by $s(\phi _{\mu }(j,m))$ . By Lemma 4.2, we also have

$$ \begin{align*}c^{\mathrm{wheel}}_{\tilde{\nu}_j} (\tau(\nu,j,m))=(\tilde{\nu}_j)!\: t(\phi_{\tau}(j,m))(e_2\wedge e_3\wedge\cdots \wedge e_{\tilde{\nu}_j+1}).\end{align*} $$

Therefore, we can take an element $x \in \mathbb {Q}[\operatorname {GL}(n,\mathbb {Q})]$ to satisfy $x (F_{(\mu ,\nu )}(\tau _{\ast }(\alpha _{(\mu ,\nu )})))=\pi ^{(\mu ,\nu )}(e_{i+l(\mu ),l(\mu )})$ , which completes the proof.

Proof of Theorem 6.1

Let $F_{i}^{l}:=\bigoplus _{(\mu ,\nu )\in P_i^l} F_{(\mu ,\nu )}$ .

For $(\mu ,\nu )\in P_i^l$ and $(\xi ,\eta )\in P_i^{l'}$ with $l\neq l'$ , any irreducible component of $W(\mu ,\nu )$ and any irreducible component of $W(\xi ,\eta )$ are not isomorphic. Therefore, it suffices to show that we have

$$ \begin{align*}\bigoplus_{(\mu,\nu)\in P_i^l} W(\mu,\nu)\subset F_{i}^{l} (H^A_i(\operatorname{IA}_n,\mathbb{Q}))\end{align*} $$

for any $l\in \{0,\cdots ,i\}$ .

By Lemma 6.8, it suffices to show that for each $(\xi ,\eta )\in P_i^l$ , the element

$$ \begin{align*}F_{(\xi,\eta)}(\tau_{\ast}(\alpha_{(\xi,\eta)}))\in W(\xi,\eta)\subset \bigoplus_{(\mu,\nu)\in P_i^l} W(\mu,\nu)\end{align*} $$

is included in $F_{i}^{l} (H^A_i(\operatorname {IA}_n,\mathbb {Q}))$ . In what follows, we identify an element of $W(\xi ,\eta )$ with the image under the canonical inclusion $W(\xi ,\eta )\hookrightarrow \bigoplus _{(\mu ,\nu )\in P_i^l} W(\mu ,\nu )$ .

We use the induction with respect to the partial order $\ge $ of $P_i^l$ . For the minimum element $(1^l,1^{i-l})$ of $P_i^l$ , by Lemma 6.7, we have

$$ \begin{align*} F_{(1^l,1^{i-l})}(\tau_{\ast}(\alpha_{(1^l,1^{i-l})}))= F_{i}^{l}(\tau_{\ast}(\alpha_{(1^l,1^{i-l})}))\in F_{i}^{l} (H^A_i(\operatorname{IA}_n,\mathbb{Q})). \end{align*} $$

For $(\xi ,\eta )\in P_i^l$ , suppose that for any $(\zeta ,\epsilon )\le (\xi ,\eta )$ , we have

$$ \begin{align*}F_{(\zeta,\epsilon)}(\tau_{\ast}(\alpha_{(\zeta,\epsilon)}))\in F_{i}^{l} (H^A_i(\operatorname{IA}_n,\mathbb{Q})).\end{align*} $$

Then by Lemma 6.7, we have

$$ \begin{align*} F_{i}^{l} (\tau_{\ast} (\alpha_{(\xi,\eta)}))&=F_{(\xi,\eta)}(\tau_{\ast}(\alpha_{(\xi,\eta)}))+ X(\xi,\eta) \end{align*} $$

where $X(\xi ,\eta )$ is an element of $\bigoplus _{(\zeta ,\epsilon )\le (\xi ,\eta )} W(\zeta ,\epsilon )$ . Therefore, by induction hypothesis, we have

$$ \begin{align*}F_{(\xi,\eta)}(\tau_{\ast}(\alpha_{(\xi,\eta)}))\in F_{i}^{l} (H^A_i(\operatorname{IA}_n,\mathbb{Q})). \end{align*} $$

This completes the proof.

7.1 Coalgebra structure of $H_{\ast }(G,\mathbb {Q})$

Let G be a group. We briefly recall the graded-cocommutative coalgebra structure $(H_{\ast }(G,\mathbb {Q}), \Delta _{\ast }^G,\epsilon _{\ast })$ . (See [Reference Brown4] for details.)

The rational homology $H_{\ast }(G,\mathbb {Q})$ is defined by

$$ \begin{align*}H_{\ast}(G,\mathbb{Q})=H_{\ast}(F\otimes_{G}\mathbb{Q}), \end{align*} $$

where F is a projective resolution of $\mathbb {Z}$ over $\mathbb {Z}[G]$ . Here, we take the bar resolution. The diagonal map $\Delta ^G: G\to G\times G,\;g\mapsto (g,g)$ induces a homomorphism

$$ \begin{align*}\Delta_{\ast}^G: H_{\ast}(G,\mathbb{Q})\to H_{\ast}(G,\mathbb{Q})\otimes H_{\ast}(G,\mathbb{Q}),\end{align*} $$

which coincides with the map induced by the Alexander–Whitney map

$$ \begin{align*}\Delta: F\to F\otimes F,\quad (g_0,\cdots,g_n)\mapsto \sum_{p=0}^{n}(g_0,\cdots,g_p)\otimes (g_p,\cdots,g_n).\end{align*} $$

(See Brown [Reference Brown4, Section 1 of Chapter 5].) Then the induced map $\Delta _{\ast }^G$ can be written explicitly as follows:

$$ \begin{align*}\Delta_{\ast}^G([x_1\otimes \cdots \otimes x_i])=\sum_{p=0}^i [x_1\otimes \cdots \otimes x_p] \otimes [x_{p+1}\otimes \cdots \otimes x_i]\end{align*} $$

for $x_1,\cdots , x_i\in G$ . We also have a trivial map $\epsilon : G\to 1$ , which induces

$$ \begin{align*}\epsilon_{\ast}: H_{\ast}(G,\mathbb{Q})\to \mathbb{Q}.\end{align*} $$

The canonical projection $\pi ^G: G\twoheadrightarrow G^{\operatorname {ab}}$ induces a coalgebra map

$$ \begin{align*}\pi^G_{\ast}: H_{\ast}(G,\mathbb{Q})\to H_{\ast}(G^{\operatorname{ab}},\mathbb{Q}).\end{align*} $$

Therefore, the coalgebra structure of $H_{\ast }(G^{\operatorname {ab}},\mathbb {Q})$ induces a subcoalgebra structure on $H^A_{\ast }(G,\mathbb {Q})$ .

7.2 Coalgebra structure of $H^A_{\ast }(\operatorname {IA}_n,\mathbb {Q})$

As we saw in the previous subsection, we have a coalgebra structure of $H_{\ast }(U,\mathbb {Q})$ , which is compatible with the graded $\operatorname {GL}(n,\mathbb {Q})$ -representation structure. We consider the coalgebra structure on $S^{\ast }(U_{\ast })$ that we observed in Section 2.6. Then the two coalgebra structures are compatible in the sense of the following proposition.

Proposition 7.1. The graded $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*}F_{\ast}=\bigoplus_{i\ge 0} F_i: H_{\ast}(U,\mathbb{Q}) \to S^{\ast}(U_{\ast})\end{align*} $$

is a coalgebra map.

Proof. It suffices to show that we have

$$ \begin{align*}\left(\bigoplus_{p=0}^{i}(F_p\otimes F_{i-p})\right) \Delta_i^U(x)=\Delta F_i (x) \end{align*} $$

for $x=x_1\wedge \cdots \wedge x_i\in \bigwedge\nolimits ^i U\cong H_i(U,\mathbb {Q})$ .

We have

$$ \begin{align*}\Delta_i^U (x)=\sum_{p=0}^i\sum_{\sigma\in \operatorname{Sh}(p,i-p)} \operatorname{sgn}(\sigma) (x_{\sigma(1)}\wedge \cdots\wedge x_{\sigma(p)})\otimes (x_{\sigma(p+1)}\wedge \cdots\wedge x_{\sigma(i)}). \end{align*} $$

Let $p\in \{0,\cdots ,i\}$ . For pairs of partitions $(\xi ,\eta )\vdash p, (\zeta ,\epsilon )\vdash i-p$ , we write $\xi =(\xi _1^{k_1'},\cdots , \xi _r^{k_r'})$ , $\eta =(\eta _1^{k_1"},\cdots , \eta _s^{k_s"})$ and $\zeta =(\zeta _1^{m_1'},\cdots , \zeta _t^{m_t'})$ , $\epsilon =(\epsilon _1^{m_1"},\cdots , \epsilon _u^{m_u"})$ . Then we have

$$ \begin{align*} &F_{(\xi,\eta)}(x_{\sigma(1)}\wedge \cdots\wedge x_{\sigma(p)})\\ &= \left(\bigotimes_{j=1}^r \frac{1}{k_{j}'!}\prod^{k_{j}'}\tilde{c}^{\mathrm{tree}}_{\xi_j} \otimes \bigotimes_{j=1}^s \frac{1}{k_{j}"!} \prod^{k_{j}"}\tilde{c}^{\mathrm{wheel}}_{\eta_j}\right)\iota_p(x_{\sigma(1)}\wedge \cdots\wedge x_{\sigma(p)})\\ &=\sum_{\rho\in \mathfrak{S}_p}\operatorname{sgn}(\rho)\left(\bigotimes_{j=1}^r \frac{1}{k_{j}'!}\prod^{k_{j}'}\tilde{c}^{\mathrm{tree}}_{\xi_j} \otimes \bigotimes_{j=1}^s \frac{1}{k_{j}"!} \prod^{k_{j}"}\tilde{c}^{\mathrm{wheel}}_{\eta_j}\right)(\varphi(x_{\sigma\rho(1)})\otimes \cdots\otimes \varphi(x_{\sigma\rho(p)})). \end{align*} $$

In a similar way, we have

$$ \begin{align*} &F_{(\zeta,\epsilon)}(x_{\sigma(p+1)}\wedge \cdots\wedge x_{\sigma(i)})\\ &=\sum_{\theta\in \mathfrak{S}_{i-p}}\operatorname{sgn}(\theta)\left(\bigotimes_{j=1}^t \frac{1}{m_{j}'!}\prod^{m_{j}'}\tilde{c}^{\mathrm{tree}}_{\zeta_j} \otimes \bigotimes_{j=1}^u \frac{1}{m_{j}"!} \prod^{m_{j}"}\tilde{c}^{\mathrm{wheel}}_{\epsilon_j}\right)\\ &\quad\quad (\varphi(x_{\sigma(p+\theta(1))})\otimes \cdots\otimes \varphi(x_{\sigma(p+\theta(i-p))})). \end{align*} $$

Therefore, we have

$$ \begin{align*} &(\bigoplus_{p=0}^{i}(F_p\otimes F_{i-p}))\Delta_i^U(x)\\ &=\sum_{p=0}^{i}(F_p\otimes F_{i-p})(\sum_{\sigma\in \operatorname{Sh}(p,i-p)} \operatorname{sgn}(\sigma) (x_{\sigma(1)}\wedge \cdots\wedge x_{\sigma(p)})\otimes (x_{\sigma(p+1)}\wedge \cdots\wedge x_{\sigma(i)}))\\ &=\sum_{p=0}^{i}\sum_{\substack{(\xi,\eta)\vdash p\\ (\zeta,\epsilon)\vdash i-p}} \left(\sum_{\sigma\in \operatorname{Sh}(p,i-p)}\operatorname{sgn}(\sigma)\right) \left(\sum_{\rho\in \mathfrak{S}_p}\operatorname{sgn}(\rho)\right) \left(\sum_{\theta\in \mathfrak{S}_{i-p}}\operatorname{sgn}(\theta)\right)\\ &\quad \left(\bigotimes_{j=1}^r \frac{1}{k_{j}'!}\prod^{k_{j}'}\tilde{c}^{\mathrm{tree}}_{\xi_j} \otimes \bigotimes_{j=1}^s \frac{1}{k_{j}"!} \prod^{k_{j}"}\tilde{c}^{\mathrm{wheel}}_{\eta_j} \right)(\varphi(x_{\sigma\rho(1)})\otimes \cdots\otimes \varphi(x_{\sigma\rho(p)}))\\ &\quad\otimes \left( \bigotimes_{j=1}^t \frac{1}{m_{j}'!}\prod^{m_{j}'}\tilde{c}^{\mathrm{tree}}_{\zeta_j} \otimes \bigotimes_{j=1}^u \frac{1}{m_{j}"!} \prod^{m_{j}"}\tilde{c}^{\mathrm{wheel}}_{\epsilon_j} \right) (\varphi(x_{\sigma(p+\theta(1))})\otimes \cdots\otimes \varphi(x_{\sigma(p+\theta(i-p))}))\\ &=\sum_{p=0}^{i}\sum_{\substack{(\xi,\eta)\vdash p\\ (\zeta,\epsilon)\vdash i-p}} \sum_{\tau\in \mathfrak{S}_i}\operatorname{sgn}(\tau)\\ &\quad\left(\bigotimes_{j=1}^r \frac{1}{k_{j}'!}\prod^{k_{j}'}\tilde{c}^{\mathrm{tree}}_{\xi_j} \otimes \bigotimes_{j=1}^s \frac{1}{k_{j}"!} \prod^{k_{j}"}\tilde{c}^{\mathrm{wheel}}_{\eta_j} \right)(\varphi(x_{\tau(1)})\otimes \cdots\otimes \varphi(x_{\tau(p)}))\\ &\quad\otimes \left(\bigotimes_{j=1}^t \frac{1}{m_{j}'!}\prod^{m_{j}'}\tilde{c}^{\mathrm{tree}}_{\zeta_j} \otimes \bigotimes_{j=1}^u \frac{1}{m_{j}"!} \prod^{m_{j}"}\tilde{c}^{\mathrm{wheel}}_{\epsilon_j} \right)(\varphi(x_{\tau(p+1)})\otimes \cdots\otimes \varphi(x_{\tau(i)})). \end{align*} $$

On the other hand, we have

$$ \begin{align*} \Delta F_i(x) &=\Delta \left(\sum_{(\mu,\nu)\vdash i} \bigotimes_{j=1}^v \frac{1}{n_{j}'!}\prod^{n_{j}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \frac{1}{n_{j}"!}\prod^{n_{j}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)\iota_i(x_1\wedge \cdots \wedge x_i)\\ &=\sum_{\sigma\in \mathfrak{S}_i}\operatorname{sgn}(\sigma)\sum_{(\mu,\nu)\vdash i}\Delta\left(\bigotimes_{j=1}^v \frac{1}{n_{j}'!}\prod^{n_{j}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \frac{1}{n_{j}"!}\prod^{n_{j}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)\\ &\quad\quad(\varphi(x_{\sigma(1)})\otimes \cdots\otimes \varphi(x_{\sigma(i)})), \end{align*} $$

where we write $\mu =(\mu _1^{n_1'},\dots ,\mu _v^{n_v'})$ and $\nu =(\nu _1^{n_1"},\dots , \nu _w^{n_w"})$ . By the definition of $\Delta $ in Section 2.6, we have

$$ \begin{align*} &\Delta\left(\bigotimes_{j=1}^v \frac{1}{n_{j}'!}\prod^{n_{j}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \frac{1}{n_{j}"!}\prod^{n_{j}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)(\varphi(x_{\sigma(1)})\otimes \cdots\otimes \varphi(x_{\sigma(i)}))\\ &=\prod_{j=1}^v\frac{1}{n_j'!}\prod_{j=1}^w\frac{1}{n_j"!}\sum_{\substack{\mu=\mu(1)\cup \mu(2)\\\nu=\nu(1)\cup \nu(2)}} \sum_{\rho}\operatorname{sgn}(\rho;c\varphi(x))\\ &\quad \left(\bigotimes_{j=1}^v \prod^{n_{j,1}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \prod^{n_{j,1}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)(\varphi(x_{\sigma\rho(1)})\otimes \cdots\otimes \varphi(x_{\sigma\rho(|\mu(1)|+|\nu(1)|)}))\\ &\quad \otimes \left(\bigotimes_{j=1}^v \prod^{n_{j,2}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \prod^{n_{j,2}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)(\varphi(x_{\sigma\rho(|\mu(1)|+|\nu(1)|+1)})\otimes \cdots\otimes \varphi(x_{\sigma\rho(i)})), \end{align*} $$

where $\mu (p)=(\mu _1^{n_{1,p}'},\dots ,\mu _v^{n_{v,p}'})$ , $\nu (p)=(\nu _1^{n_{1,p}"},\dots ,\nu _w^{n_{v,p}"})$ ( $p=1,2$ ) are partitions such that $n_{j}'=n_{j,1}'+n_{j,2}'$ , $n_{j}"=n_{j,1}"+n_{j,2}"$ (note that $n_{j,p}', n_{j,p}"$ are possibly zero), and where $\rho $ runs over the elements of $\mathfrak {S}_i$ that are induced by combinations of $(n_{j,1}', n_{j,2}')$ -shuffles and $(n_{j,1}", n_{j,2}")$ -shuffles via the following correspondence

$$ \begin{align*} &(\mu_1^{n_1'},\dots, \mu_v^{n_v'},\nu_1^{n_1"},\dots, \nu_w^{n_w"})\\ &\mapsto (\mu_1^{n_{1,1}'},\dots, \mu_v^{n_{v,1}'},\nu_1^{n_{1,1}"},\dots, \nu_w^{n_{w,1}"}, \mu_1^{n_{1,2}'},\dots, \mu_v^{n_{v,2}'},\nu_1^{n_{1,2}"},\dots, \nu_w^{n_{w,2}"}), \end{align*} $$

and where $\operatorname {sgn}(\rho ;c\varphi (x))$ denotes the sign with regard to $\rho $ and $x_j$ as in the definition of $\Delta $ in Section 2.6. Since the number of $\rho $ satisfying the above condition is $\prod _{j=1}^{v}\binom {n_j'}{n_{j,1}'}\prod _{j=1}^{w}\binom {n_j"}{n_{j,1}"}$ , and since we have

$$ \begin{align*} \prod_{j=1}^{v}\frac{1}{n_j'!} \binom{n_j'}{n_{j,1}'} \prod_{j=1}^{w}\frac{1}{n_j"!}\binom{n_j"}{n_{j,1}"} =\prod_{j=1}^{v}\frac{1}{n_{j,1}'!\; n_{j,2}'!} \prod_{j=1}^{w}\frac{1}{n_{j,1}"!\; n_{j,2}"!}, \end{align*} $$

it follows that

$$ \begin{align*} \Delta F_i(x) &=\sum_{\sigma\in \mathfrak{S}_i}\sum_{\substack{(\mu,\nu)\vdash i\\ \mu=\mu(1)\cup\mu(2)\\\nu=\nu(1)\cup \nu(2)}}\operatorname{sgn}(\sigma)\prod_{j=1}^{v}\frac{1}{n_{j,1}'!\; n_{j,2}'!} \prod_{j=1}^{w}\frac{1}{n_{j,1}"!\; n_{j,2}"!}\\ & \left(\bigotimes_{j=1}^v \prod^{n_{j,1}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \prod^{n_{j,1}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)(\varphi(x_{\sigma(1)})\otimes \cdots\otimes \varphi(x_{\sigma(|\mu(1)|+|\nu(1)|)}))\\ & \otimes \left(\bigotimes_{j=1}^v \prod^{n_{j,2}'}\tilde{c}^{\mathrm{tree}}_{\mu_j} \otimes \bigotimes_{j=1}^w \prod^{n_{j,2}"}\tilde{c}^{\mathrm{wheel}}_{\nu_j}\right)(\varphi(x_{\sigma(|\mu(1)|+|\nu(1)|+1)})\otimes \cdots\otimes \varphi(x_{\sigma(i)})). \end{align*} $$

By letting $(\xi ,\eta ):=(\mu (1),\nu (1))$ and $(\zeta ,\epsilon ):=(\mu (2),\nu (2))$ , we obtain

$$ \begin{align*} \Delta F_i(x) &=\sum_{\sigma\in \mathfrak{S}_i}\sum_{p=0}^{i}\sum_{\substack{(\xi,\eta)\vdash p\\ (\zeta,\epsilon)\vdash i-p}} \operatorname{sgn}(\sigma)\\ &\quad\left(\bigotimes_{j=1}^r \frac{1}{k_{j}'!}\prod^{k_{j}'}\tilde{c}^{\mathrm{tree}}_{\xi_j} \otimes \bigotimes_{j=1}^s \frac{1}{k_{j}"!} \prod^{k_{j}"}\tilde{c}^{\mathrm{wheel}}_{\eta_j} \right)(\varphi(x_{\sigma(1)})\otimes \cdots\otimes \varphi(x_{\sigma(p)}))\\ &\quad\otimes \left(\bigotimes_{j=1}^t \frac{1}{m_{j}'!}\prod^{m_{j}'}\tilde{c}^{\mathrm{tree}}_{\zeta_j} \otimes \bigotimes_{j=1}^u \frac{1}{m_{j}"!} \prod^{m_{j}"}\tilde{c}^{\mathrm{wheel}}_{\epsilon_j} \right)(\varphi(x_{\sigma(p+1)})\otimes \cdots\otimes \varphi(x_{\sigma(i)}))\\ &=\left(\bigoplus_{p=0}^{i}(F_p\otimes F_{i-p})\right) \Delta_i^U(x).\\[-49pt] \end{align*} $$

By Proposition 7.1, the subcoalgebra $H^A_{\ast }(\operatorname {IA}_n,\mathbb {Q})\subset H_{\ast }(U,\mathbb {Q})$ is mapped to a subcoalgebra of $S^{\ast }(U_{\ast })$ , which includes $W_{\ast }$ as a subcoalgebra. For a coaugmented coalgebra A, let $\operatorname {Prim}(A)$ denote the primitive part of A. We can check that

$$ \begin{align*}\operatorname{Prim}(S^{\ast}(U_{\ast}))=U_{\ast}\subset W_{\ast}.\end{align*} $$

Since coalgebra maps preserve the primitive part, the graded $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism $F_{\ast }$ restricts to

$$ \begin{align*}F_{\ast}: \operatorname{Prim}(H^A_{\ast}(\operatorname{IA}_n,\mathbb{Q}))\to \operatorname{Prim}(S^{\ast}(U_{\ast}))=U_{\ast}.\end{align*} $$

If Conjecture 6.2 is true, then the following conjecture is true.

Conjecture 7.2. For $n\ge 3i$ , the $\operatorname {GL}(n,\mathbb {Q})$ -homomorphism

$$ \begin{align*}F_i:\operatorname{Prim}(H^A_{\ast}(\operatorname{IA}_n,\mathbb{Q}))_i\to U_i\end{align*} $$

is an isomorphism, where $\operatorname {Prim}(H^A_{\ast }(\operatorname {IA}_n,\mathbb {Q}))_i$ denotes the degree i part of $\operatorname {Prim}(H^A_{\ast }(\operatorname {IA}_n,\mathbb {Q}))$ .