The cohomology rings of the unordered configuration spaces of the torus

TL;DR

This paper investigates the cohomology ring of unordered configuration spaces on the torus, computing its mixed Hodge structure, mapping class group action, and establishing formality over the rationals.

Contribution

It provides explicit computations of the cohomology ring, mixed Hodge structure, and demonstrates formality, advancing understanding of configuration spaces on the torus.

Findings

Computed the cohomology ring structure

Determined the mixed Hodge structure

Proved formality over the rationals

Abstract

We study the cohomology ring of the configuration space of unordered points in the two dimensional torus. In particular, we compute the mixed Hodge structure on the cohomology, the action of the mapping class group, the structure of the cohomology ring and we prove the formality over the rationals.

Equations

$${\mathcal{F}}^n(E)\stackrel{\text{def}}{=}\{\underline{p}\in E^n\mid p_i\ne p_j\}.$$

$${\mathcal{C}}^n(E)\stackrel{\text{def}}{=}{\mathcal{F}}^n(E)/{\mathfrak{S}}_n.$$

$$\mathcal{M}^{n}(E)\stackrel{{\scriptstyle\textnormal{def}}}{{=}}\big{\{}\underline{p}\in\mathcal{F}^{n}(E)\mid\sum p_{i}=0\big{\}}.$$

$$\{x_i,y_i,{\omega }_{i,j}{\}}_{1\le i<j\le n}.$$

$$\mathrm{d}{\omega }_{i,j}\stackrel{\text{def}}{=}(x_i-x_j)(y_i-y_j).$$

$$(x_i-x_j){\omega }_{i,j}=0\text{and}(y_i-y_j){\omega }_{i,j}=0,$$

$${\omega }_{i,j}{\omega }_{j,k}-{\omega }_{i,j}{\omega }_{k,i}+{\omega }_{j,k}{\omega }_{k,i}=0.$$

$$A^{p,q}\cong E_q^{p+q}.$$

$$A^{to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil},to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil}}\cong B^{to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil},to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil}}{\otimes }_{\mathbb{Q}}{D}^{to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil},0}$$

$$E\times {\mathcal{M}}^n(E)$$

$$(q,\underline{p})$$

$${\sigma }^{-1}\cdot (p_1,\dots ,p_n)=(p_{\sigma (1)},\dots ,p_{\sigma (n)})$$

$${\sigma }^{-1}(x_i)=x_{\sigma (i)},$$

$${\sigma }^{-1}(y_i)=y_{\sigma (i)},$$

$${\sigma }^{-1}({\omega }_{i,j})={\omega }_{\sigma (i),\sigma (j)}$$

$$\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\cdot x_{i}=ax_{i}+cy_{i}\quad\quad\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\cdot y_{i}=bx_{i}+dy_{i}.$$

$${\phi }_{L_{\ast }}\stackrel{\text{def}}{=}{\mathrm{sgn}}_n{|}_{C_{L_{\ast }}}\cdot ({\phi }_{{\lambda }_1}⊠\cdots ⊠{\phi }_{{\lambda }_r}).$$

$${\alpha }_{L_{\ast },H_{\ast }}(\sigma )\stackrel{\text{def}}{=}(-1{)}^{\lambda +\mathrm{deg}(h)+1},$$

$$A^{to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil},to4.0833pt\text{hfil}\text{raise}0.0pt\text{scalebox}0.75\text{lower}0.0pt\bullet \text{hfil}}=\underset{\begin{array}{c}|L_{\ast }|=q\\ |H_{\ast }|=p\end{array}}{⨁}{A}_{L_{\ast },H_{\ast }}$$

$$A_{L_{\ast },H_{\ast }}{\otimes }_{\mathbb{Q}}\mathbb{C}\simeq {\mathrm{Ind}}_{Z_{L_{\ast },H_{\ast }}}^{{\mathfrak{S}}_n}{\xi }_{L_{\ast },H_{\ast }}.$$

$$(1,2,3,4,5)$$

$${\mathbb{V}}_k=\underset{a=0}{\overset{k}{⨁}}V(2a-k),$$

$${\mathbb{V}}_m\otimes {\mathbb{V}}_n\cong {\mathbb{V}}_{m+n}\oplus {\mathbb{V}}_{m+n-2}\oplus \cdots \oplus {\mathbb{V}}_{m-n}.$$

$$A_a^{p,q}=\underset{\begin{array}{c}|L_{\ast }|=q\\ |H_{\ast }|=p,\Vert H_{\ast }\Vert =a\end{array}}{⨁}{A}_{L_{\ast },H_{\ast }}$$

$$p_a(Y\cdot x^{a+b+2}{y}^b)=p_a((x+y{)}^{a+b+2}{y}^b)=(a+b+2)x^{a+b+1}{y}^{b+1}.$$

$$A^{p,q}\cong \underset{a=0}{\overset{p}{⨁}}\mathrm{coker}{\pi }_a⊠{\mathbb{V}}_a.$$

$$A^{p,q}=\underset{a=-p}{\overset{p}{⨁}}{A}_a^{p,q}$$

$$⟨\text{mathds}{1}_n,A^{p,q}{⟩}_{{\mathfrak{S}}_n}=0$$

$$⟨\text{mathds}{1}_n,{\mathrm{Ind}}_{Z_{L_{\ast },H_{\ast }}}^{{\mathfrak{S}}_n}{\xi }_{L_{\ast },H_{\ast }}{⟩}_{{\mathfrak{S}}_n}=0$$

$$⟨\text{mathds}{1}_n,{\mathrm{Ind}}_{Z_{L_{\ast },H_{\ast }}}^{{\mathfrak{S}}_n}{\xi }_{L_{\ast },H_{\ast }}{⟩}_{{\mathfrak{S}}_n}=⟨{\mathrm{Res}}_{Z_{L_{\ast },H_{\ast }}}^{{\mathfrak{S}}_n}\text{mathds}{1}_n,{\xi }_{L_{\ast },H_{\ast }}{⟩}_{Z_{L_{\ast },H_{\ast }}}$$

$$T(u,v)=\frac{1+u^3{v}^4}{(1-u^2{v}^3{)}^2}=1+2u^2{v}^3+u^3{v}^4+3u^4{v}^6+2u^5{v}^7+\dots$$

$$\sum_{i=0}^{2n}\sum_{k=i}^{2i}\Bigg{[}\faktor{W_{k}H^{i}(\mathcal{C}^{n}(E);\mathbb{Q})}{W_{k-1}H^{i}(\mathcal{C}^{n}(E);\mathbb{Q})}\Bigg{]}u^{i}v^{k},$$

Full text

Introduction

We fully describe the cohomology with rational coefficients of the configuration spaces of unordered points in an elliptic curve (frequently called torus).

Configuration spaces of points are related to physics (state spaces of non-colliding particles on a manifold), robotics (motion planning), knot theory, and topology. Configuration spaces give invariants of the homeomorphism type of the base space. In the algebraic setting, configuration spaces are open in the moduli spaces of points.

Since the literature is very extensive, we compare our work only with the main results on the (co-)homology of configuration spaces. The first computation of the cohomology algebra of configuration spaces is due to Arnol’d [Arn69, Arn14] in the case of $\mathbb{R}^{2}$. This result has been generalized by Cohen, Lada, and May [CLM76] to the configuration space of $\mathbb{R}^{n}$ and later by Goresky and Macpherson [GM88]. Partially additive results have been obtained: by Bödigheimer and Cohen [BC88] for once-punctured oriented surfaces, by the same authors and Taylor [BCT89] for odd dimensional manifolds, and by Drummond-Cole and Knudsen [DCK17] for surfaces in general. However there is no description of the ring structure; we provide it in the case of elliptic curves. The Betti numbers $\mathcal{C}^{n}(X)$ are described in the following cases: for $X=\mathbb{P}^{2}(\mathbb{R})$ by Wang [Wan02], for $X$ a sphere by Salvatore [Sal04], for $X=\mathbb{P}^{2}(\mathbb{C})$ by Felix and Tanré [FT05] and for elliptic curves by Maguire and Schiessl [Mag16, Sch16].

In this paper we improve the previous results on configuration spaces in an elliptic curve in three ways. We describe:

• •

the mixed Hodge structure on the cohomology (3.3),

• •

the action of the mapping class group (3.3),

• •

the ring structure (4.1).

The formality result over the rationals is proven in 4.3.

We prove these results using the Križ model [Kri94, Tot96, Bib16, Dup15] and the representation theory on it [AAB14, Aza15].

In Section 1 we recall the Križ model, then in Section 2 we improve the result on the decomposition of the Križ model into irreducible representations, see 2.7. Descriptions of the mixed Hodge structure and of the action of the mapping class group are obtained in Section 3 by computing the cohomology of the model. Finally, the ring structure is presented in the last section.

1. The Križ model

Let $E$ be an elliptic curve and consider the configuration space of $n$ ordered distinct points

[TABLE]

The symmetric group $\mathfrak{S}_{n}$ acts on $\mathcal{F}^{n}(E)$ by permuting the coordinates and the quotient is the configuration space of $n$ unordered points

[TABLE]

We also consider the space $\mathcal{M}^{n}(E)$, defined by

[TABLE]

Notice that there exists a non canonical isomorphism $\mathcal{F}^{n}(E)\cong E\times\mathcal{M}^{n}(E)$.

In this section we recall a rational model for the cohomology algebra of $\mathcal{F}^{n}(E)$. The model is a commutative differential bi-graded algebra (dga) that can be obtain in two different ways: as a specialization of the Križ model for the configuration spaces or as the second page of the Leray spectral sequence (also known as the Totaro spectral sequence) for elliptic arrangements. Our main references for the first approach are [Kri94, AAB14, Aza15] and for the second one are [Tot96, Dup15, Bib16]. In the following we define the models for the cohomology of $\mathcal{F}^{n}(E)$ and of $\mathcal{M}^{n}(E)$.

Let $\Lambda$ be the exterior algebra over $\mathbb{Q}$ with generators

[TABLE]

We set the degree of each $x_{i}$ and $y_{i}$ equal to $(1,0)$ and the degree of $\omega_{i,j}$ equal to $(0,1)$. Define the differential $\mathop{\mathrm{\mathstrut d}}\nolimits\!\colon\!\Lambda\to\Lambda$ of bi-degree $(2,-1)$ on generators as follows: $\mathop{\mathrm{\mathstrut d}}\nolimits\!x_{i}=0$ and $\mathop{\mathrm{\mathstrut d}}\nolimits\!y_{i}=0$ for $i=1,\dots,n$ and

[TABLE]

For the sake of notation we set $\omega_{i,j}:=\omega_{j,i}$ for $i>j$.

We define the dga A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} as the quotient of $\Lambda$ by the following relations:

[TABLE]

Notice that the ideal is preserved by the differential map, thus the differential \mathop{\mathrm{\mathstrut d}}\nolimits\!\colon\!A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}\to A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} is well defined.

Remark 1.1*.*

The model A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} coincides with the Križ model E_{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}} introduced in [Kri94] up to shifting the degrees, ie

[TABLE]

The dga E_{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}} is a rational model for $X$, as shown in [Kri94, Theorem 1.1].

In order to study the cohomology of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} we need to introduce the elements $u_{i,j}=x_{i}-x_{j}$, $v_{i,j}=y_{i}-y_{j}$ and $\gamma=\sum_{i=1}^{n}x_{i}$, $\overline{\gamma}=\sum_{i=1}^{n}y_{i}\in A^{1,0}$.

We define the dga B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} as the subalgebra of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} generated by $u_{i,j},v_{i,j}$ and $\omega_{i,j}$ for $1\leq i<j\leq n$. Let D^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},0} be the subalgebra of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} generated by $\gamma$ and $\overline{\gamma}$ endowed with the zero differential map. Notice that

[TABLE]

as differential algebras and that D^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},0} is the cohomology ring of the elliptic curve $E$.

The mixed Hodge structure on the cohomology of algebraic varieties defines a bigrading compatible with the algebra structure (see [Del75, p.81] or [Voi07, Theorem 8.35]). In our case the bigrading given by the mixed Hodge structure coincides with the one given by the Leray spectral sequence as shown by Totaro [Tot96, Theorem 3] and by Gorinov [Gor17]. Explicitly, the subspace $A^{p,q}$ has weight $p+2q$ and degree $p+q$.

The following result is a particular case of [Bib16, Theorem 3.3] and of [Dup15, Theorem 1.2].

Theorem 1.2**.**

The cohomology algebra of $\mathcal{F}^{n}(E)$ (or of $\mathcal{M}^{n}(E)$) with rational coefficients is isomorphic to the cohomology of the dga A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} (respectively of B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}). Moreover, the $n^{2}$-sheeted covering

[TABLE]

induces the isomorphism of eq. (1).

2. Representation theory on the Križ model

Now we study the action of the symmetric group $\mathfrak{S}_{n}$ and of $\operatorname{SL_{2}}(\mathbb{Q})$ on the algebras A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} and B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}. Those actions are given by a geometric action on $\mathcal{F}^{n}(E)$. For general reference about the representation theory of the Lie group and of the Lie algebra we refer to [Hal15] and to [FH91], respectively. The cases of $\operatorname{SL_{2}}(\mathbb{C})$ and of $\operatorname{\mathfrak{sl}_{2}(\mathbb{C})}$ can be found in [GW09].

2.1. Definition of the actions

Consider the action of $\mathfrak{S}_{n}$ on $\mathcal{F}^{n}(E)$ defined by

[TABLE]

for all $\sigma\in\mathfrak{S}_{n}$. This induces an action on A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} and on B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} defined by

[TABLE]

for all $1\leq i<j\leq n$ and all $\sigma\in\mathfrak{S}_{n}$.

The mapping class group $\operatorname{MCG}(E)$ acts naturally on $\mathcal{F}^{n}(E)$ and on $\mathcal{C}^{n}(E)$.

Theorem 2.1** (Theorem 2.5 [FM12]).**

The mapping class group $\operatorname{MCG}(E)$ of the torus is isomorphic to $\operatorname{SL_{2}}(\mathbb{Z})$ and the isomorphism is given by the natural action of $\operatorname{MCG}(E)$ on $H^{1}(E;\mathbb{Z})$.

Let $f$ be an automorphism of $E$, the map induces the following vertical morphisms

{\mathcal{F}^{n}(E)}$${E^{n}}$${\mathcal{F}^{n}(E)}$${E^{n}}$$\scriptstyle{f^{n}_{|\mathcal{F}^{n}(E)}}$$\scriptstyle{f^{n}}

and by functoriality of the Leray spectral sequence it induces the action of $\operatorname{SL_{2}}(\mathbb{Z})$ on A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}. We explicitly describe this action on the generators $\omega_{i,j}$, $x_{i}$, and $y_{i}$: since $f^{n}\colon\!E^{n}\to E^{n}$ fixes the divisor $\{p_{i}=p_{j}\}$, then $f\cdot\omega_{i,j}=\omega_{i,j}$. The other generators belongs to $A^{1,0}=H^{1}(E^{n})\cong H^{1}(E)^{\otimes n}$. Therefore the action of $\operatorname{MCG}(E)\cong\operatorname{SL_{2}}(\mathbb{Z})$ on $A^{1,0}$ is given by

[TABLE]

This action extends to $\operatorname{SL_{2}}(\mathbb{Q})$ and since the actions of $\mathfrak{S}_{n}$ and of $\operatorname{SL_{2}}(\mathbb{Q})$ commute, then A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}, B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} and D^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},0} become $\mathfrak{S}_{n}\times\operatorname{SL_{2}}(\mathbb{Q})$-modules.

2.2. Decomposition into $\mathfrak{S}_{n}$-representations

We recall a result of [AAB14, Theorem 3.15] on the decomposition of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} into $\mathfrak{S}_{n}$-modules. The notations used here follow the ones in [AAB14].

Let $L_{*}=(\lambda_{1},\dots,\lambda_{t})$ be a partition of the number $n$, ie $\lambda_{i}\in\mathbb{N}_{+}$ and $\sum_{i=1}^{t}\lambda_{i}=n$. We mark all blocks with labels in $\{1,x,y,xy\}$, an ordered basis of H^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}(E). The order is $1\prec x\prec y\prec xy$.

Definition 2.2**.**

A marked partition $(L_{*},H_{*})$ is a partition $L_{*}\vdash n$ together with marks $H_{*}=(h_{1},\dots,h_{t})$, $h_{i}\in\{1,x,y,xy\}$ such that: if $\lambda_{i}=\lambda_{i+1}$ then $h_{i}\succeq h_{i+1}$.

Let $C_{k}$ be the cyclic group of order $k$. For any partition $L_{*}\vdash n$ define $C_{L_{*}}$ as the product of the cyclic groups $C_{\lambda_{i}}$ for $i=1,\dots,t$. It acts on $\{1,\dots,n\}$ in the natural way. Consider a marked partition $(L_{*},H_{*})$ and define $N_{L_{*},H_{*}}$ as the group that permutes the blocks of $L_{*}$ with the same labels. The group $N_{L_{*},H_{*}}$ is a product of symmetric groups. Call $Z_{L_{*},H_{*}}$ the semidirect product $C_{L_{*}}\rtimes N_{L_{*},H_{*}}$.

Example 2.3**.**

Let $(L_{*},H_{*})$ be the marked partition $L_{*}=(5,5,5,5,1,1,1)\vdash 23$ and $H_{*}=(xy,xy,xy,1,x,x,x)$. The group $C_{L_{*}}\cong(\mathbb{Z}/5\mathbb{Z})^{4}<\mathfrak{S}_{23}$ is generated by $(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)$, and $(16,17,18,19,20)$. The subgroup $N_{L_{*},H_{*}}\cong\mathfrak{S}_{3}\times\mathfrak{S}_{3}$ is generated by the permutations $(1,6)(2,7)(3,8)(4,9)(5,10)$, $(1,11)(2,12)(3,13)(4,14)(5,15)$, $(21,22)$, and $(21,23)$. Finally, $Z_{L_{*},H_{*}}$ is a group isomorphic to $(\mathbb{Z}/5\mathbb{Z}\wr\mathfrak{S}_{3})\times\mathbb{Z}/5\mathbb{Z}\times\mathfrak{S}_{3}$.

Given two representations $V,W$ of two groups $G$ and $H$ respectively, define the tensor representation $V\boxtimes W$ of $G\times H$ by the vector space $V\otimes W$ with the action $(g,h)(v\otimes w)=g(v)\otimes h(w)$.

We define the following one-dimensional representations. Let $\varphi_{n}$ be a faithful character of the cyclic group and $\varphi_{L_{*}}$ the character of $C_{L_{*}}\cong\mathbb{Z}/\lambda_{1}\mathbb{Z}\times\cdots\times\mathbb{Z}/\lambda_{t}\mathbb{Z}$ given by

[TABLE]

Recall that the degree $\deg$ of $1,x,y,xy$ are respectively $0,1,1,2$. Let $\alpha_{L_{*},H_{*}}$ be the one dimensional representation of $N_{L_{*},H_{*}}\cong\mathfrak{S}_{\mu_{1}}\times\cdots\times\mathfrak{S}_{\mu_{l}}$ defined on generators by

[TABLE]

where $\sigma$ is the permutation that exchange two blocks of size $\lambda$ and label $h$. Set $\xi_{L_{*},H_{*}}$ to be the one dimensional representation of $Z_{L_{*},H_{*}}$ such that $\operatorname{Res}_{C_{L_{*}}}^{Z_{L_{*},H_{*}}}\xi_{L_{*},H_{*}}=\varphi_{L_{*}}$ and $\operatorname{Res}_{N_{L_{*},H_{*}}}^{Z_{L_{*},H_{*}}}\alpha_{L_{*},H_{*}}$ .

We define $|L_{*}|=n-t$ for a partition $L_{*}=(\lambda_{1},\dots,\lambda_{t})$ of $n$ and for a mark $H_{*}$ the numbers $|H_{*}|=\sum_{i=1}^{t}\deg(h_{i})$ and $\|H_{*}\|=|\{i\mid h_{i}=x\}|-|\{i\mid h_{i}=y\}|$.

Theorem 2.4** ([AAB14, Theorem 3.15]).**

There exist $\mathfrak{S}_{n}$-representations $A_{L_{*},H_{*}}\subset A^{p,q}$ such that

[TABLE]

as $\mathfrak{S}_{n}$-representation. Moreover:

[TABLE]

Example 2.5**.**

Consider the marked partition $(L_{*},H_{*})$ of Example 2.3, the characters are shown in the following table.

[TABLE]

2.3. Decomposition into $\mathfrak{S}_{n}\times\operatorname{SL_{2}}(\mathbb{Q})$-representations

Let $T=\{H_{t}\}\cong\mathbb{Q}^{*}$ be the maximal torus in $\operatorname{SL_{2}}(\mathbb{Q})$ generated by the diagonal matrices $H_{t}=\left(\begin{smallmatrix}t&0\\ 0&t^{-1}\end{smallmatrix}\right)$. Let $\mathbb{V}_{1}$ be the irreducible representation $\mathbb{Q}^{2}$ with the standard action of matrix-vector multiplication and let $\mathbb{V}_{k}=S^{k}\mathbb{V}_{1}$ be the irreducible representation given by the symmetric power of $\mathbb{V}_{1}$. The representation $\mathbb{V}_{k}$ has dimension $k+1$ and can be view as $\mathbb{Q}[x,y]_{k}$, ie the vector space of homogeneous polynomials in two variables. The action of $T$ on the monomials is given by $H_{t}\cdot x^{a}y^{k-a}=t^{2a-k}x^{a}y^{k-a}$, thus $\mathbb{V}_{k}$ decomposes, as representations of $T$

[TABLE]

where $V(2a-k)$ is the subspace where $H_{t}$ acts with character $t^{2a-k}$, ie the subspace generated by $x^{a}y^{k-a}$. Since the group $\operatorname{SL_{2}}(\mathbb{Q})$ is dense in $\operatorname{SL_{2}}(\mathbb{C})$, each irreducible regular representation of $\operatorname{SL_{2}}(\mathbb{Q})$ is isomorphic to $\mathbb{V}_{k}$ for some $k\in\mathbb{N}$. For a proof see [GW09, Proposition 2.3.5] and use a density reasoning.

As a consequence we can decompose a representation $V$ of $\operatorname{SL_{2}}(\mathbb{Q})$ using its decomposition $V=\oplus_{a\in\mathbb{Z}}V(a)^{\oplus n_{a}}$ as representation of $T$: indeed $V\cong\oplus_{k\in\mathbb{N}}\mathbb{V}_{k}^{\oplus m_{k}}$ as representation of $\operatorname{SL_{2}}(\mathbb{Q})$, where $m_{k}=n_{k}-n_{k+2}$. By setting $V=\mathbb{V}_{m}\otimes\mathbb{V}_{n}$, we obtain the following formula for $n\leq m$:

[TABLE]

As observed in Section 2.1, the group $\operatorname{SL_{2}}(\mathbb{Q})$ acts trivially on $\omega_{i,j}$ for all $1\leq i<j\leq n$ and the two dimensional subspace generated by $x_{i}$ and $y_{i}$ is isomorphic to $\mathbb{V}_{1}$ as representation of $\operatorname{SL_{2}}(\mathbb{Q})$.

We will use the decomposition of 2.4 to obtain a decomposition of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} into $\mathfrak{S}_{n}\times\operatorname{SL_{2}}(\mathbb{Q})$-modules. Let

[TABLE]

be a $\mathfrak{S}_{n}\times T$-stable subspace of $A^{p,q}$ and hence we have $A^{p,q}=\oplus_{a=-p}^{p}A^{p,q}_{a}$. Let $p_{a}\colon\!A^{p,q}\to A^{p,q}_{a}$ be the $\mathfrak{S}_{n}\times T$-equivariant projection and define $Y=\left(\begin{smallmatrix}1&0\\ 1&1\end{smallmatrix}\right)\in\operatorname{SL_{2}}(\mathbb{Q})$ and, for $a\geq 0$, define $\pi_{a}\colon\!A^{p,q}_{a+2}\to A^{p,q}_{a}$ by $v\mapsto p_{a}(Y\cdot v)$.

Notice that, $\pi_{a}$ is a morphism of $\mathfrak{S}_{n}$-representations and that $A^{p,q}_{a}$ is zero if $a\not\equiv p\mod 2$, or if $a>p$, or if $a>n-q$.

Lemma 2.6**.**

The map $\pi_{a}$ is injective.

Proof.

If $V\subseteq A^{p,q}$ is a $SL(\mathbb{Q})$-representation, then $p_{a}(V)\subseteq V$, thus it is enough to prove that $\pi_{a}\colon\!V(a+2)\to V(a)$ is injective for all irreducible representation $\mathbb{V}_{k}\subseteq A^{p,q}$. If $k\not\equiv a\mod 2$ or if $k<a+2$ then $V(a+2)=0$. Otherwise $k=a+2b+2$ for some $b\geq 0$ and $V(a+2)$ is a one dimensional vector space generated by the homogeneous monomial $x^{a+b+2}y^{b}$. The projection $p_{a}$ has kernel equal to $\oplus_{l\neq a}V(l)$, thus

[TABLE]

This last term is non-zero since $a,b\geq 0$ and therefore $\pi_{a}$ is injective. ∎

Theorem 2.7**.**

The algebra A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} decomposes as $\mathfrak{S}_{n}\times\operatorname{SL_{2}}(\mathbb{Q})$-representation in the following way:

[TABLE]

Proof.

Observe that the maximal torus $T$ of $\operatorname{SL_{2}}(\mathbb{Q})$ acts on $A^{p,q}_{a}$ with character $t^{a}$, thus by 2.4 we have

[TABLE]

as $\mathfrak{S}_{n}\times T$ representations. By using eq (2) we obtain that $\bigoplus_{a=0}^{p}\operatorname{coker}\pi_{a}\boxtimes\mathbb{V}_{a}$ is isomorphic to $\bigoplus_{a=-p}^{p}A^{p,q}_{a}$ as representation of $\mathfrak{S}_{n}\times T$. The representation theory of $\operatorname{SL_{2}}(\mathbb{Q})$ ensure that the representations $A^{p,q}$ and $\bigoplus_{a=0}^{p}\operatorname{coker}\pi_{a}\boxtimes\mathbb{V}_{a}$ are isomorphic as $\mathfrak{S}_{n}\times\operatorname{SL_{2}}(\mathbb{Q})$-representations. ∎

Define the $\mathfrak{S}_{n}$-invariant subalgebra of A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} by U\!A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} and of B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}} by U\!B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}. Obviously we have U\!A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}=U\!B^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}\otimes_{\mathbb{Q}}D^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}. We use the previous calculation to compute U\!A^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}

Corollary 2.8**.**

For $q>p+1$ we have $U\!A^{p,q}=0$.

Proof.

Let $\mathds{1}_{n}$ be the trivial representation of $\mathfrak{S}_{n}$. We use 2.4 to show that

[TABLE]

for $q>p+1$. Indeed, it is enough to prove that

[TABLE]

for all $(L_{*},H_{*})$ with $|L_{*}|=q$ and $|H_{*}|=p$. By Frobenius reciprocity we have

[TABLE]

Since the representations in the right hand side are one-dimensional the value of $\langle\mathds{1}_{n},\operatorname{Ind}_{Z_{L_{*},H_{*}}}^{\mathfrak{S}_{n}}\xi_{L_{*},H_{*}}\rangle_{\mathfrak{S}_{n}}$ is non zero if and only if $\xi_{L_{*},H_{*}}=\mathds{1}$.

By definition $\xi_{L_{*},H_{*}}=\mathds{1}$ is equivalent to $\varphi_{L_{*}}=\mathds{1}$ and $\alpha_{L_{*},H_{*}}=\mathds{1}$. From the fact that $\varphi_{k}=\operatorname{sgn}_{k}$ only for $k=1,2$, $\psi_{L_{*}}=\mathds{1}$ if and only if $\lambda_{i}=1,2$ for all $i=1,\dots,t$. The condition $\alpha_{L_{*},H_{*}}=\mathds{1}$ implies that the only marked blocks of $(L_{*},H_{*})$ that appear more than once are the ones with $\lambda_{i}=2$ and $\deg(h_{i})=1$ or the ones with $\lambda_{i}=1$ and $\deg(h_{i})\neq 1$.

Consequently, $\langle\mathds{1}_{n},\operatorname{Ind}_{Z_{L_{*},H_{*}}}^{\mathfrak{S}_{n}}\xi_{L_{*},H_{*}}\rangle_{\mathfrak{S}_{n}}\neq 0$ only if $L_{*}=(2^{q},1^{n-2q})$ and the degree of $h_{i}$ is $1$ for $i<q$, this implies $p\geq q-1$ contrary to our hypothesis. ∎

Corollary 2.9**.**

For $q>p+1$ we have $U\!B^{p,q}=0$. ∎

3. The additive structure of the cohomology

We compute the cohomology with rational coefficients of the unordered configuration spaces of $n$ points, taking care of the mixed Hodge structure and of the action of $\operatorname{SL_{2}}(\mathbb{Q})$. The integral cohomology groups are known only for small $n$ in [Nap03, Table 2], where a cellular decomposition of ordered configuration spaces is given. In this section, we use the calculation of the Betti numbers of $\mathcal{C}^{n}(E)$ to determine the Hodge polynomial in the Grothendieck ring of $\operatorname{SL_{2}}(\mathbb{Q})$.

Observe that H^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}(\mathcal{C}^{n}(E))=H^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}(\mathcal{F}^{n}(E))^{\mathfrak{S}_{n}} by the Transfer Theorem. Define the series

[TABLE]

and let $T_{n}(u,v)$ be its truncation at degree $n$ in the variable $u$.

The computation of the Betti numbers of unordered configuration space of $n$ points in an elliptic curve was done simultaneously by [DCK17], [Mag16], and [Sch16] in different generality. We point to the last reference because [Sch16, Theorem] fits exactly our generality.

Theorem 3.1**.**

The Poincaré polynomial of $\mathcal{C}^{n}(E)$ is $(1+t)^{2}T_{n-1}(t,1)$.

We use the notation $Vu^{k}v^{h}$ to denote a vector space $V$ in degree $k$ with a Hodge structure of weight $h$. The Grothendieck ring of $\operatorname{SL_{2}}(\mathbb{Q})$ is the free $\mathbb{Z}$-module with basis given by $[V]$ for all finite-dimensional irreducible representations $V$ of $\operatorname{SL_{2}}(\mathbb{Q})$ and product defined by the tensor product of representations.

Definition 3.2**.**

The Hodge polynomial of $\mathcal{C}^{n}(E)$ with coefficients in the Grothendieck ring of $\operatorname{SL_{2}}(\mathbb{Q})$ is

[TABLE]

where $W_{k}H^{i}(\mathcal{C}^{n}(E);\mathbb{Q})$ is the weight filtration on $H^{i}(\mathcal{C}^{n}(E);\mathbb{Q})$. The ordinary Hodge polynomial is

[TABLE]

We prove a stronger version of 3.1.

Theorem 3.3**.**

The Hodge polynomial of $\mathcal{C}^{n}(E)$ with coefficients in the Grothendieck ring of $\operatorname{SL_{2}}(\mathbb{Q})$ is

[TABLE]

and the ordinary Hodge polynomial is $(1+uv)^{2}T_{n-1}(u,v)$.

Section 3 represents the module H^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}},{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\displaystyle\bullet}}}\hfil}}{\hbox to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\textstyle\bullet}}}\hfil}}{\hbox to4.0833pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptstyle\bullet}}}\hfil}}{\hbox to3.53883pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower 0.0pt\hbox{\scriptscriptstyle\bullet}}}\hfil}}}}}(U\!B) that corresponds to the right factor of eq. (4).