数理解析研究所講究録
第 1991 巻 2016 年 130-139

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TAUTOLOGICAL ALGEBRAS OF MODULI SPACES-SURVEY AND PROSPECTSHIGEYUKI MORITA

1. INTRODUCTION
This paper is based on the author’s talk at the Numazu workshop which in turn was
based on ajoint work with Takuya Sakasai and Masaaki Suzuki. It reproduces the actual
talk rather closely. The contents are roughly as follows. In \S 2, we briefly recall known results about the tautological algebras of various moduli spaces. In particular, an importantconjecture, called the Faber conjecture, concerning the tautological algebra of the moduli
space of curves is mentioned. In \S 3, we summarize our former topological approach to thetautological algebra. In \S 4, a complete description is given how the space of symplectic
invariant tensors degenerates with respect to the genus. Then in \S 5, we sketch our new
results which are obtained by combining a theorem of Manivel on a plethysm of certain
GL-representations and the result mentioned in \S 4. Finally in \S 6, we present several open
problems.

2. TAUTOLOGICAL ALGEBRAS OF MODULI SPACES$G_{k}(\mathbb{C}^{n})$,$A_{g},$ $M_{g}$In this section, we recall known results about the tautological algebras of various moduli
spaces. First we consider the elementary case of the Grassmann manifold$G_{k}(\mathbb{C}^{n})=${$V\subset \mathbb{C}^{n}$;$k$-dimensional linear subspace}.

Letdenote the tautological bundle overvector bundle and we have the following exact sequence$\xiarrow G_{k}(\mathbb{C}^{n})$$G_{k}(\mathbb{C}^{n})$which is a -dimensional$k$$0arrow\xiarrow G_{k}(\mathbb{C}^{n})\cross \mathbb{C}^{n}arrow Qarrow 0$

where denotes the quotient bundle. We denote by
the Chern classes of$Q$$c_{1}(\xi)$,...,$c_{k}(\xi)\in H^{*}(G_{k}(\mathbb{C}^{n});\mathbb{Z}\rangle$$\xi.$

Theorem 2.1 (well-known). We have the following presentation$H^{*}(G_{k}(\mathbb{C}^{n});\mathbb{Q})\cong \mathbb{Q}[c_{1}(\xi), \cdots, c_{k}(\xi)]$/relations

relations: $c_{i}(Q)=[ \frac{1}{1+c_{1}(\xi)+\cdots+c_{k}(\xi)}]_{2i}=0$ for all $i>n-k$
and it satisfies Poincare duality of $\dim=2k(n-k)$ .

denote the Siegel upper half space on which the Siegel modular group
acts properly discontinuously. The quotient space

Next, let$Sp(2g,$$\mathbb{Z}\rangle$$\mathfrak{h}_{g}$$A_{9}=\mathfrak{h}_{g}/Sp(2g, \mathbb{Z})$The author was partially supported by KAKENHI $(No.15H03618)$ , iapan Society for the Promotionof Science, Japan, Received December 31, 2015.

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is the moduli space of principally polarized abelian varieties. The Siegel modular groupplays the role of the orbifold fundamental group of th\’is moduli space, thus
$Sp(2g, \mathbb{Z})\cong\pi_{1}^{orb}A_{g}.$as a discrete subgroup and theis contained in
On the other hand,group $U(g)$ . Hence we haveunitary
group
the
is
lattercompactofsubgroupthemaximalthe Chern classes$Sp(2g, \mathbb{R})$$Sp(2g, \mathbb{Z})$.

$c_{\tau}\epsilon H^{*}(A_{g};\mathbb{Q})\cong H^{*}(Sp(2g, \mathbb{Z});\mathbb{Q})$Then the tautological algebra in cohomology of$\mathcal{R}^{*}(A_{g})=$subalgebra of$A_{g}$

is defined as$H^{*}(A_{g};\mathbb{Q})$generated by$c_{i}’ s.$

Theorem 2.2 (van der Geer [5], true at the Chow algebra level). The following presentation holds
relationswhere the relations are described as: (i) $p_{i}=0$ (Pontnjagin classes) (ii) $c_{g}=0$ . Also it
satasfies Poincar\’e duatity of $\dim=g(g-1)$ .
$\mathcal{R}^{*}(A_{g}\rangle\cong \mathbb{Q}[c_{1}, \ldots,c_{g}]/$.
It is also known that, additively
denote the MMM
Finally we consider the moduli space of curves. Let
denotes the mapping class group of a closed
tautological class ([27][20][19]) whereis defined asoriented surface of genus . Then the tautological algebra of$\mathcal{R}^{*}(A_{g})\cong H^{*}(S^{2}\cross S^{4}\cross\cdots\cross S^{2g-2};\mathbb{Q})$$e_{i}\in H^{2i}(\mathcal{M}_{g};\mathbb{Q})$$\mathcal{M}_{g}$$\mathcal{M}_{g}$$g$$\mathcal{R}^{*}(\mathcal{M}_{g})=$subalgebra of$H^{*}(\mathcal{M}_{g};\mathbb{Q})$generated by$e_{t}’ s.$

Let $A^{i}(M_{9})$ be the Chow algebra of the moduli space of curves of genus 9 defined by
Mumford [27] and let$\kappa_{i}\in A’(M_{g})$be the Mumford kappa class. Then the tautological algebra of$\mathcal{R}^{*}(M_{g})=$subalgebra of$\mathcal{A}^{*}(M_{g})$$M_{g}$

generated by

There exists a canonical surjectiongroup is also called the tautological algebra in cohomology ofis defined as)$\kappa_{i}s$.

$\mathcal{R}^{*}(M_{g})arrow \mathcal{R}^{*}(\mathcal{M}_{g})(\kappa_{i}\mapsto(-1)^{i+1}e_{i})$Conjecture 2.3 (Faber [3]).
$\mathcal{R}^{*}(M_{g})\cong H^{*}$$\langle$2)$\mathcal{R}^{*}(M_{g})$and the latter$M_{g}.$(1) Gorenstein conjecture, including Poincare duality( tsmooth projective variety” of $\dim=g-2;\mathbb{Q}$ )?

is generated by the first $[g/3]MMM$ -classes with no relations in degrees$\leq[g/3].$(3) Explicit formula for the intersection numbers, namely proportionality in degree(proved by Looijenga [15] and Faber [3]).
$g-2:\mathcal{R}^{g-2}(M_{9})\cong \mathbb{Q}$etc. and
There are generalizations of this conjecture to the cases ofGetzler,
Faber,
Zagier,PandhariLooijenga,
many results due to many people, including
pande, Vakil, Graber, Lee, Randal-Williams, Pixton, Liu, Xu, Yin, have been obtained(we refer to Faber’s survey paper [4] for details as well as references). Returning to theabove original conjecture, we have the following.
$\overline{M}_{g,n\rangle}M_{g,n}^{ct},$$M_{g,n}^{rt}$132(1) is still open, Faber (and Faber and Pandharipande) verified the claim for $g\leq 23$
using the Faber-Zagier relations. On the other hand, Pandharipande and Pixton [28]showed that the Faber-Zagier relations are actual ones.
(2) was proved by M. [23] at the cohomological level, and later by Ionel [10] at the
Chow algebra level. No relation part is due to Harer [9] (and improvements by Ivanov
and Boldsen).
(3) There are three proofs, first by Givental [6], and the others by Liu-Xu [14] andBuryak-Shadrin [2].

More recently, Faber and Pandharipande found that some new situation happens for(again see [4] for details).

$g\geq 24$

3. TOPOLOGICAL STUDY OF THE TAUTOLOGICAL ALGEBRA OF$M_{g}$In this section, we recall a topological approach of investigating the structure of thetautological algebra. Let
be a closed oriented surface of genus $g(\geq 1)$ as before.
We denote
simply bywhich is the fundamental representation of Sp. Let$\Sigma_{g}$$H_{1}(\Sigma_{g};\mathbb{Q})$$=$$H_{\mathbb{Q}}$$Sp(2g, \mathbb{Q})$$\mu:H_{\mathbb{Q}}\otimes H_{\mathbb{Q}}arrow \mathbb{Q}$be the intersection pairing. If we fix a symplectic basis ofexists an isomorphismAut$(H_{\mathbb{Q}}, \mu)\cong Sp(2g, \mathbb{Q})$The Torelli group is the subgroup of$\mathcal{M}_{g}$$H_{\mathbb{Q}}$, then as is well known there.

defined by$\mathcal{I}_{g}=Ker(\mathcal{M}_{g}4 Aut (H_{\mathbb{Q}}, \mu)\cong Sp(2g,\mathbb{Q})\rangle$

where$\rho_{0}$denotes the natural homomorphism induced by the action of$\mathcal{M}_{g}$on$H_{\mathbb{Q}}.$

Theorem 3.1 (Johnson [11]).
$H_{1}(\mathcal{I}_{g\rangle}\cdot \mathbb{Q})\cong\wedge^{3}H_{\mathbb{Q}}/H_{\mathbb{Q}} (g\geq 3)$.

Let us use the following notation:$U_{\mathbb{Q}}$$:=\wedge^{3}H_{\mathbb{Q}}/H_{\mathbb{Q}}=$

irrep.

$[1^{3}]_{Sp}.$In [21], a linear representation$\rho_{1}:\mathcal{M}_{g}arrow H_{1}(\mathcal{I}_{g};\mathbb{Q})xSp(2g, \mathbb{Q})$of$\mathcal{M}_{g}$was constructed and it induces the following homomorphism$\Phi:H^{*}(U_{\mathbb{Q}}=\wedge^{3}H_{\mathbb{Q}}/H_{\mathbb{Q}})^{Sp}arrow H^{*}(\mathcal{M}_{g};\mathbb{Q})$.

Theorem 3.2 (Kawazumi-M. [12]).
${\rm Im}\Phi=\mathcal{R}^{*}(\mathcal{M}_{g})=\mathbb{Q}[MMM$classesJ/relations.

Here recall that Madsen and Weiss [17] determined Harer’s stable cohomology group([9]) of the mapping class group to be$H^{*}(\mathcal{M}_{\infty};\mathbb{Q})=\mathbb{Q}$[ $MMM$-classes].

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on the third nilpotent quotient ofThen by analyzing the natural action of
the author constructed in [22] the following commutative diagram$\mathcal{M}_{g}$$7r_{1}\Sigma_{g},$$7r_{x^{\Sigma_{g}}}arrow [1^{2}]_{Sp\cross}^{\sim}H_{\mathbb{Q}}$$\downarrow$$\downarrow$$\mathcal{M}_{g,*}\frac{\tilde{\rho}_{2}}{\prime}(([1^{2}]\mathscr{X}\oplus[2^{2}]_{Sp})\sim\cross\Lambda^{3}H_{\mathbb{Q}})xSp(2g,\mathbb{Q})$$\downarrow p \downarrow$$\mathcal{M}_{g}arrow^{\rho_{2}} ([2^{2}]_{S_{I)}}\tilde{\cross}U_{\mathbb{Q}})xSp(2g, \mathbb{Q})$Here$\mathcal{M}_{g,*}=\pi_{0}Diif_{+}(\Sigma_{9}, *)$point$*\in\Sigma_{g}$and$[2^{2}]_{S_{I}},$.

relative to the basedenotes the mapping class group of
[7].
Hain
is the summand identified by$\Sigma_{g}$$cH^{2}(U_{\mathbb{Q}})$

Theorem 3.3 (Kawazumi-M. [13]). In a certain stable range, the $h_{omomo7}phism\rho_{2}^{*}$ on
cohomology induces an isomorphism$(H^{*}(U_{\mathbb{Q}})/([2^{2}]_{Sp}))^{s_{1)}}\cong \mathbb{Q}[MMM$

Similarly, in a certain stable range,$\tilde{p}_{2}^{*}$-ciassesJ.

induces an isomorphism[$(H^{*}(\wedge^{3}H_{Q})/([1^{2}]_{Sp}^{torelli}\oplus[2^{2}]_{Sp}))^{Sp}\cong \mathbb{Q}$ $e,$

$MMM$ -classes].

4. DEGENERATION OF SYMPLECTIC INVARIANT TENSORSLet us consider the Sp-invariant subspace$(H_{Q}^{\otimes 2k})^{Sp}$of the tensor productthe following mapping$H_{\mathbb{Q}}^{\otimes 2k}$

. We analyze the structure of this space completely. Consider$\mu^{\otimes 2k}:H_{\mathbb{Q}}^{\otimes 2k}\otimes H_{\mathbb{Q}}^{\otimes 2k}arrow \mathbb{Q}$defined by$(u_{1}\otimes\cdots\otimes u_{2k})\otimes(v_{1}\otimes\cdots\otimes v_{2k})\mapsto II_{i=1}^{2k}\mu(u_{i}, v_{i}) (u_{l}\prime, v_{i}\epsilon H_{\mathbb{Q}})$

Clearly$\mu^{\otimes 2k}$.

is a symmetric bilinear form.

is positive definite for any
onTheorem4.1 (M. [25]). The symmetricspace. Furthermore, there exists an orthogonal direct
so that it defines a metric on$pair’ing\mu^{\otimes 2k}$$(H_{\mathbb{Q}}^{\otimes 2k})^{S_{i)}}$$th?\dot{s}$sum decomposition$(H_{\mathbb{Q}}^{\otimes 2k})^{Sp} \cong\bigoplus_{|\lambda|=k,h(\lambda)\leq g}U_{\lambda}$

for a Young diagram
number of rows. Also$\uparrow/4here$$\lambda,$$|\lambda|$

denotes the number of boxes andas an$U_{\lambda}\cong(\lambda^{\delta})_{6_{2k}}$$\mathfrak{S}_{2k}$-moduleand there exists a bijective correspondence
bijective$\{\lambda;|\lambda|=k\}$$\{\mu_{\lambda};|\lambda|=k\}$eigenvalues.

$h(\lambda)$

denotes the

$g$

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degenerates according to the genus
Table 1 below indicates how the space$g\geq3k$
to $g=3k-1,$ $3k-2$ , .. . , 1.
changes from the stable range$(H_{\mathbb{Q}}^{\otimes 2k})^{Sp}$TABLE 1. 0rthogonal decomposition of$g$$(H_{Q}^{\otimes 6k})^{Sp}$

Remark 4.2. Related eigenvalues already appeared in Hanlon-Wales [8] in the context
of Brauer’s centralizer algebras.
Now we consider the following mappingssurj.
$(H_{\mathbb{Q}}^{\otimes 6k})^{Sp}arrow(\wedge^{2k}U_{\mathbb{Q}})^{Sp}\epsilonurj.arrow \mathcal{R}^{2k}(\mathcal{M}_{g})$together with the following degenerations of symplec invariant tensors$[3k]’\mapsto 0$$(g\leq 3k-1)$(enough to prove Faber conjecture (2))$[3k-1, 1]’\mapsto 0 (g\leq 3k-2)$$[3k-2, 2]’[3k-2, 1^{2}]’\mapsto 0 (g\leq 3k-3)$$[3k-3,3]’[3k-3, 21]’[3k-3, 1^{3}]’\mapsto 0 (g\leq 3k-4)$$[6k-8, 8]’[6k-8,62]’\cdots[6k-8, 2^{4}]’\mapsto 0 (g\leq 3k-5)$ .

In this way, we obtain many (hopefully all? the) relations and we proposed the following.
Conjecture 4.3 (M. [24]).
$\mathcal{R}^{*}(\mathcal{M}_{g})\cong(\wedge^{*}U_{\mathbb{Q}}/([2^{2}]_{Sp}))^{Sp},$$\mathcal{R}^{*}(\mathcal{M}_{g,*})\underline{\simeq}(\wedge^{*}(\wedge^{3}H_{\mathbb{Q}})/([1^{2}]_{Sp}^{t\circ oelti}\oplus[2^{2}]_{Sp}))^{Sp}$5. PLETHYSM OF GL REPRESENTATIONS AND TAUTOLOGICAL ALGEBRAIn this section, we consider certain plethysm of GL-representations. Recall that plethysm
is a composition of two Schur functors. Determination of a given plethysm is a very important but extremely diﬃcult problem and a complete answer is known for only thefollowing four cases.
Theorem 5.1 (Formula of Littlewood). There exists a complete description of the following plethysms$S^{*}(S^{2}H_{\mathbb{Q}}) , \wedge^{*}(S^{2}H_{Q})$,$S^{*}(\wedge^{2}H_{Q}) , \wedge^{*}(\wedge^{2}H_{Q})$.

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The following result of Manivel plays a key role in our work. Here we describe his resultin a simplified form and we refer to his original paper for details.

Theorem 5.2 (Manivel [18]). The plethysm
Furthermore the super stable decomposition of$S^{k}(S^{l}H_{\mathbb{Q}})t$$S^{\infty}(S^{3}H_{\mathbb{Q}})i\mathcal{S}$$S^{*}(S^{2}H_{\mathbb{Q}}\oplus S^{3}H_{\mathbb{Q}})$

uper stabilizes” as $karrow\infty.$
given by.

We apply the well-known involution on the space of symmetric polynomials (see [16])to the following particular case
$H_{k}H_{3}dua\Leftrightarrow^{)}\mathcal{B}_{k}E_{3}$denotes the k-th elementary symmetric polynomial andwherecomplete symmetric polynomial. We obtain the following result.
$E_{k}$

Theorem 5.3 (Sakasai Suzuki-M. [26]).
and letthird exterior power of$Let\wedge^{k}(A^{3}H_{\mathbb{Q}})$$H_{k}$

denotes the k-th

be the k-th exterior power of the$H_{Q}$$\wedge^{k}(\wedge^{3}H_{\mathbb{Q}})=\bigoplus_{\lambda,|\lambda|=3k}m_{\lambda}\lambda_{GL}$be the stable irreducible decomposition as a $GL$ -module. Then, for any , the mapping$k$$A^{k}(\wedge^{3}H_{\mathbb{Q}})arrow\wedge^{k+1}(\wedge^{3}H_{\mathbb{Q}})$is injective and bijective for the part
induced by the operation$2k\leq h(\lambda)\leq 3k$ . In other words, we have the inequality$\lambda\mapsto\lambda^{+}=[\lambda 1^{3}]$$\lambda_{GL}^{+}$

with$m_{\lambda}\{\begin{array}{l}\leq m_{\lambda+}=m_{\lambda+} (2k\leq h(\lambda\rangle\leq 3k) .\end{array}$

Theorem 5.4 (Sakasai Suzuki-M. [26]). We have $deter\gamma nined$ the super stable irreducible
up to codimension 30.
decomposition
up to
Table 2 below indicates the super stable irreducible decomposition ofcodimension 10.
Corollary 5.5 (Sakasai-Suzuki-M. [26]). We have determined the super stable $Sp$ -invariant
up to codimension 30.
part$of\wedge^{\infty}[1^{3}]_{GL}$$\Lambda^{\infty}[1^{3}]_{GL}$$\langle\wedge^{\infty}[1^{3}]_{GL}\rangle^{Sp}$

Table 3 below indicates the super stable Sp-inval.iant partsion 10.

$(\Lambda^{\infty}[1^{3}]_{GL})^{Sp}$up to codimen-

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TABLE 2. Super stable irreducible decomposition

TABLE 3. Super stable irred. summands$of\wedge^{\infty}[1^{3}]_{GL}$$of\wedge^{\infty}[1^{3}]_{GL}$with double floors

Let us consider the following series of mappings (see [4] for the definition of the Gorenstein quotients).
$\mathcal{R}^{*}(M_{9})arrow \mathcal{R}^{*}(\mathcal{M}_{g})arrow G^{*}(M_{g})$$\mathcal{R}^{*}(M_{g}^{1})arrow \mathcal{R}^{*}(\mathcal{M}_{g_{)}*})arrow G^{*}(M_{g}^{1})$Here$M_{g}^{1}$(Gorenstein quotient),(Gorenstein quotient).

denotes the moduli space of curves of genus$g$with one marked point.

Expectation 5.6 (Faber-Zagier [3][4]; Faber Bergvall [1], Y\’in [29]). The number$p(k)-\dim G^{2k}(M_{g})=$ numberof relations of codimension$k$depends only on $P=3k-1-g$ in the range $2k\leq g-2$ $(i.e. k\geq\ell+3)$ . Similarly the
number$1+p(1)+\cdots+p(k)-\dim G^{2k}(M_{g}^{1})$depends only on $\ell=3k-1-g$ in the range $2k\leq g-2$ $(i.e. k\geq\ell+3)$ ,in case $2k=g-1$ , something happens

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Expectation 5.7 (continued, Faber-Zagier [3][4]; Faber-Bergvall [1], Yin [29]). If the$f_{07}mer$ partof the previous Expectation holds, then the above number can be described as?

number of partitions of with parts:, . ..
1, 2, 3, 4, 6, 7, , 10, 12, 13,$n\neq 2$ is excluded
mod3.
if
Similarly, if the latter part of the previous Expectation holds, then the above number canbe described as$a(\ell)=$$\ell$$g$$15,\backslash 16$$n\equiv 2$$b(P)?= \sum_{i=0,i\neq 3m+2}^{\ell}a(P-i)=a(P)+a(P-1)+a(P-3)+(t(l-4)+\cdot \cdot\cdot$

We have the following theorems which may serve as a supporting evidences for theabove expectations.
Theorem 5.8 (Sakasai-Suzuki-M.

$[26]\rangle$.

(1) The number$\tilde{a}(l) :=p(k)-\dim(\wedge^{2k}U_{\mathbb{Q}}/([2^{2}]_{Sp}))^{s_{I)}}$

depends only on $\ell=3k-1-g$ in the range $2k\leq g-2$ $(i.e. k\geq\ell+3)$ .
(2) The number$\tilde{b}(\ell):=1+p(1)+\cdots+p(k)-\dim(\wedge^{2k}(\wedge^{3}H_{Q})/([1^{2}]_{Sp}^{torelh}\oplus[2^{2}]_{S_{I)}}))^{Sp}$depends only on $\ell=3k-1-g$ in the same range.
Furthermore, we have the following more precise result.

orthogonal complement of$(\Lambda^{2k}(A^{3}H_{\mathbb{Q}}))^{s_{I)}}$$\Rightarrow$in$(\Lambda^{2k}(\wedge^{3}H_{\mathbb{Q}}^{\infty}))^{Sp}mod ([1^{2}]_{Sp}^{tore11i}\oplus[2^{2}]_{Sp})^{Sp}$tautological relations ininorthogonal complement oftautological relations in$(\wedge^{2k}U_{\mathbb{Q}})^{Sp}$$\mathcal{R}^{2k}(\mathcal{M}_{g,*})$$(\wedge^{2k}U_{\mathbb{Q}}^{\infty})^{Sp}mod ([2^{2}]_{Sp})^{Sp}$$\Rightarrow$$\mathcal{R}^{2k}(\mathcal{M}_{g})$.

.

Theorem 5.9 $(Saﬃsai-S\mathfrak{u}zuki-M.)$ If we fix $P=3k-1-g$ , then all the above orthogonal complements are canonically isomorphic to each other in the range $2k\leq g-$$2$$(i.e. k\geq P+3)$.
6. PROBLEMS

Problem 6.1. Construct the “fundamental cycles”$\mu_{g,*}\in(\Lambda^{2g-2}(\wedge^{3}H_{\mathbb{Q}}^{\langle g\rangle}))^{Sp}$$\mu_{g}\in(\wedge^{2g-4}U_{\mathbb{Q}}^{\langle g\rangle})^{Sp}$and give a topological proof of the intersection number formula.

Problem 6.2. Give a topological proof of the Faber-Zagier relations.
Problem 6.3. Study the relation between our tautological relations with those of FaberZagier as well as those of $Yin.$

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Problem 6.4 (suggested by Faber [4]). Which part (and/or in which degrees) of thefollowing $homomo\gamma$phisms is isomorphic or non-isomorphi :$c^{}$$\mathcal{R}^{*}(M_{g})arrow \mathcal{R}^{*}(\mathcal{M}_{g})arrow G^{*}(M_{g})$$\mathcal{R}^{*}(M_{g}^{1})arrow \mathcal{R}^{*}(\mathcal{M}_{g,*})arrow G^{*}(M_{g}^{1})$Problem 6.5 (suggested by Faber [4]).

(Gorenstein quotient),(Gorenstein quotient).

(1) Are Faber-Zagier relations linearly inde-

pendent2(2) Are Faber-Zagier relations complete up to the half $dimension^{9}$(3) Are there more relations above the half dimension$($$)^{g}$REFERENCES[1] Q. Bergvall, Relations in the tautologicd ring of the universal curve, Master thesis, KTH Stockholm,
2011.
[2] A. Buryak, S. Shadrin,
new proof of Faber’s intersection number conjecture, Adv. Math. 228(2011), 22-42.
[3] C. Faber, conjectural description of of the tautological ring of the moduli space of curwes, in “Moduli
of curves and abelian varieties”, 109-129, Aspects Math., E33, Vieweg, Braunschweig, 1999.
[4] C. Faber, Tautological algebras of the moduli space of curves, in “Moduli Spaces of Riemann Surfaces”, IAS/Park City Mathematical Series, Vol. 20, 197-219, Amer. Math. Soc., 2013.
[5] G. van der Geer, Cycles on the moduli space of abelian varieties, in “Moduli oi curves and abelianvarieties 65-89, Aspects Math., E33, Vieweg, Braunschweig, 1999.
[6] A. Givental, Gromov- Wiuen invariants and quantization of quadratic Hamiltonians, Mosc. Math.
J. 1 (2001), 551-568.
[7] R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597-651.
[8J P. Hanlon, D. Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989)$\mathcal{A}$$\mathcal{A}$409-445.

[9] J. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math.
(2 121 (1985), 215-249.
[10] E. Ionel, Relations in the tautological ring of, Duke Math. J. 129 (2005), 157-186.
[11] D. Johnson,
abelian quotient of the mapping class group , Math. Ann. 249 (1980) 225-242.
[12] N. Kawazumi, S. Morita, The pr\’imary approximateon to the cohomology of the moduli space of curves
and stable characteristic classes, Math. Res. Lett. 3 (1996) 629-642.
[13 N. Kawazumi, S. Morita, The primary approximation to the cohomology of the moduli space of curves
and cocycles for the Mumford-Morita-Miller classes, preprint, Univ. of Tokyo, 2001.
[14] K. Liu, H. Xu, Computing top intersections in the tautological ring of, Math. Zeit. 270 (2012),$\rangle$$\mathcal{M}_{g}$$\mathcal{A}n$$\mathcal{I}_{9}$$|$$\mathcal{M}_{g}$819-837.
[15] E. Looijenga, On the tautological ring of, Invent. Math. 121 (1995), 411-419.
[16] I. Macdonald, “Symmetric Functions and Hall Polynomials”, Oxford Mathematical Monographs,$\mathcal{M}_{g}$2nd edition, Oxford University Press, 1995.
[17] I. Madsen, M. Weiss, The stable moduli space of Riemann surfaces: Mumford’ s conjecture, Ann. of
Math. (2) 165 (2007), 843-94.
[18] L. Manivel, Gaussian maps and plethysm, In: Algebraic Geometry (Catania, 1993/Barcelona, 1994),
Lecture Notes in Pure and Appl. Math., vol. 200, 91-117, Dekker (1998).
[19] E.T. Miller, The homology of the mapping class group, J. Diﬀerential Geometry 24 (1986) 1-14.
[20] S. Morita, Charactenstic dasses of surface bundles, Invent. Math. 90 (1987) 551-577.
[21] S. Morita, A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles, from: “Proceedings of the Taniguchi Symposium on Topology andTeichm\"uller Spaces”, held in Finland, July 1995, World Scientific (1996) 159-186.
[22J S. Morita, Structure of the mapping class group and symplectic representation theory, l’Enseignement
Math. Monographs 38 (2001), 577-596.

139[23] S. Morita, Generators for the tautological algebra of the moduli space of curves, Topology 42 2003),$\langle$

787-819.
[24] S. Morita, Cohomological structure of the mapping dass group and beyond, in: “Problems on Mapping
Class Groups and related Topics edited by Benson Farb, Proc. Sympos. Pure Math. 74 (2006),
American Mathematical Society, 329-354.
[25] S. Morita, Canonical metric on the space of syrnplectic invariant tensors and its applications,pre.)rint, arXiv: 1404.3354[math.GT].
[26] S. Morita, T. Sakasai, M. Suzuki, A topological study of the tautological algebra of the moduli spaceof curves, in preparation.
[27] D. Mumford, Towards an enumerative geometry of the moduli space of curves, in; Arithmetic andGeometry, Progress in Math. 36 (1983), 271-328.
[28] R. Pandharipande, A. Pixton, Relations in the tautological ring of the moduli space of curves,preprint, arXiv: 1301.4561 [math.AG].
[29] Q. Yin, Tautological cycles on curves and Jacobians, thesis, 2013, Nijmegen-Paris.

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY 0 TOKYO, 3-8-1 KOMABA,
MEGURO-KU, TOKYO, 153-8914, JAPAN
$E$-mail address: morita@ns.u-tokyo.ac.jp