Tautological Algebra of Moduli Space of Curves and
Representation Theory
.
Shigeyuki MORITA
.
based on jw/w Takuya SAKASAI and Masaaki SUZUKI
October 27, 2014
Shigeyuki MORITA
Tautological Algebra and Representation Theory
Contents
Contents
.
.
1
Tautological algebra of moduli space of curves
2
Topological approach to the tautological algebra
3
Degeneration of symplectic invariant tensors
4
Plethysm of GL representations and tautological algebra
5
Prospects
.
.
Shigeyuki MORITA
Tautological Algebra and Representation Theory
Tautological algebra of moduli space of curves (1)
Mg = π0 Diff + Σg : Mapping class group of Σg
Tg : Teichmüller space, Mg acts properly discontinuously
Mg = Tg /Mg : Riemann moduli space
Mg ∼
= π1orb Mg : orbifold fundamental group
characteristic classes of surface bundles =
cohomology of moduli space
H ∗ (BDiff + Σg ; Q)
Earle−Eells
∼
=
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H ∗ (Mg ; Q) ∼
= H ∗ (Mg ; Q)
Tautological Algebra and Representation Theory
Tautological algebra of moduli space of curves (2)
H 2i (Mg ; Q) 3 ei : MMM tautological class
R∗ (Mg ) = subalgebra of H ∗ (Mg ; Q) generated by ei ’s
tautological algebra in cohomology of Mg
Ai (Mg ) 3 κi : Mumford kappa class
R∗ (Mg ) = subalgebra of A∗ (Mg ) generated by κi ’s
tautological algebra of Mg
canonical surjection R∗ (Mg ) → R∗ (Mg )
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Tautological Algebra and Representation Theory
Tautological algebra of moduli space of curves (3)
Conjecture (Faber 1993)
.
1
(most difficult part)
R∗ (Mg ) ∼
= H ∗ (smooth proj. variety of dim = g − 2; Q)?
Gorenstein conjecture, including Poincaré duality
2
R∗ (Mg ) is generated by the first [g/3] MMM-classes
with no relations in degrees ≤ [g/3]
3
.
explicit formula for the intersection numbers, namely
.
proportionality in degree g − 2:
Rg−2 (Mg ) ∼
= Q (proved by Looijenga and Faber)
rt
generalizations to the cases of Mg,n , Mct
g,n , Mg,n etc.
.
Shigeyuki MORITA
Tautological Algebra and Representation Theory
Tautological algebra of moduli space of curves (4)
many results due to many people:
Faber (verified g ≤ 23), Looijenga, Getzler, Pandharipande
Zagier, Vakil, Graber, Lee,...
2010: Pandharipande-Pixton proved Faber-Zagier relations
(2): Morita,1998 (cohomology), Ionel, 2003 (Chow algebra)
no relation due to Harer (stability theorem), Lee
(3): three proofs Givental, 2001, Liu-Xu, Buryak-Shadrin
Faber-Pandharipande: some new situation happens for g ≥ 24
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Tautological Algebra and Representation Theory
Topological approach to the tautological algebra (1)
HQ = H1 (Σg ; Q) Σg : closed oriented surface, genus g (≥ 1)
µ : HQ ⊗ HQ → Q: intersection pairing
HQ : fundamental representation of Sp = Sp(2g, Q)
Torelli group:
Ig = Ker (Mg → Aut (HQ , µ) ∼
= Sp(2g, Q))
Theorem (Johnson)
H1 (Ig ; Q) ∼
= ∧3 HQ /HQ
(g ≥ 3)
UQ := ∧3 HQ /HQ = irrep. [13 ]Sp
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Tautological Algebra and Representation Theory
Topological approach to the tautological algebra (2)
representation of Mg :
ρ1 : Mg → H1 (Ig ; Q) o Sp(2g, Q)
(M.)
⇒
Φ : H ∗ (UQ = ∧3 HQ /HQ )Sp → H ∗ (Mg ; Q)
Theorem (Kawazumi-M.)
Im Φ = R∗ (Mg ) = Q[MMM-classes]/relations
tautological algebra in cohomology
Madsen-Weiss: H ∗ (M∞ ; Q) = Q[MMM-classes]
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Tautological Algebra and Representation Theory
Topological approach to the tautological algebra (3)
Furthermore, by analyzing the natural action of Mg on the third
nilpotent quotient of π1 Σg , I have constructed the following
commutative diagram
π1 Σg −−−−→


y
˜ Q
[12 ]Sp ×H


y
ρ̃2
˜ ∧3 HQ ) o Sp(2g, Q)
Mg,∗ −−−−→ (([12 ]torelli
⊕ [22 ]Sp )×
Sp


p

y
y
ρ2
Mg −−−−→
˜ Q ) o Sp(2g, Q).
([22 ]Sp ×U
Mg,∗ = π0 Diff + (Σg , ∗), [22 ]Sp ⊂ H 2 (UQ ) (Hain)
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Tautological Algebra and Representation Theory
Topological approach to the tautological algebra (4)
Theorem (Kawazumi-M.)
ρ∗2 on H ∗ induces an isomorphism
(H ∗ (UQ )/([22 ]Sp ))Sp ∼
= Q[MMM-classes]
in a certain stable range. Also ρ̃∗2 induces an isomorphism
⊕ [22 ]Sp ))Sp ∼
(H ∗ (∧3 HQ )/([12 ]torelli
= Q[e, MMM-classes]
Sp
in a certain stable range.
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Tautological Algebra and Representation Theory
Degeneration of symplectic invariant tensors (1)
(HQ⊗2k )Sp : Sp-invariant subspace of the tensor product HQ⊗2k
We analyze the structure of this space completely. Consider
µ⊗2k : HQ⊗2k ⊗ HQ⊗2k → Q defined by
(u1 ⊗ · · · ⊗ u2k ) ⊗ (v1 ⊗ · · · ⊗ v2k ) 7→ Π2k
i=1 µ(ui , vi ) (ui , vi ∈ HQ ).
Clearly µ⊗2k is a symmetric bilinear form.
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Tautological Algebra and Representation Theory
Degeneration of symplectic invariant tensors (2)
Theorem (M.)
µ⊗2k on (HQ⊗2k )Sp is positive definite for any g
⇒ it defines a metric on this space
Furthermore, ∃ an orthogonal direct sum decomposition
⊕
(HQ⊗2k )Sp ∼
Uλ
=
|λ|=k, h(λ)≤g
λ: Young diagram
|λ|: number of boxes, h(λ): number of rows
Uλ ∼
= (λδ )S2k as an S2k -module
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Tautological Algebra and Representation Theory
Degeneration of symplectic invariant tensors (3)
{λ; |λ| = k}
bijective
⇔
{µλ ; |λ| = k} eigenvalues
Table: Orthogonal decomposition of (HQ⊗8 )Sp
λ
[14 ]
[212 ]
[22 ]
[31]
[4]
total
µλ (eigen value of Uλ )
(2g − 6)(2g − 4)(2g − 2)2g
(2g − 4)(2g − 2)2g(2g + 1)
(2g − 2)(2g − 1)2g(2g + 1)
(2g − 2)2g(2g + 1)(2g + 2)
2g(2g + 1)(2g + 2)(2g + 3)
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dim Uλ
1
20
14
56
14
105
g
g
g
g
g
g
for Uλ 6= {0}
= 4, 5, · · ·
= 3, 4, 5, · · ·
= 2, 3, 4, 5, · · ·
= 2, 3, 4, 5, · · ·
= 1, 2, 3, 4, 5, · · ·
Tautological Algebra and Representation Theory
Degeneration of symplectic invariant tensors (4)
(HQ⊗6k )Sp −→ (∧2k UQ )Sp −→ R2k (Mg )
onto
onto
[6k]0 7−→ 0 (g ≤ 3k − 1) (enough to prove Faber conj. (2))
[6k − 2, 2]0 7−→ 0 (g ≤ 3k − 2)
[6k − 4, 4]0 [6k − 4, 22 ]0 7−→ 0 (g ≤ 3k − 3)
[6k − 6, 6]0 [6k − 6, 42]0 [6k − 6, 23 ]0 7−→ 0
0
0
(g ≤ 3k − 4)
4 0
[6k − 8, 8] [6k − 8, 62] · · · [6k − 8, 2 ] 7−→ 0 (g ≤ 3k − 5)
In this way, we obtain many (hopefully all? the) relations in
R∗ (Mg ) as well as in R∗ (Mg,∗ )
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Tautological Algebra and Representation Theory
Degeneration of symplectic invariant tensors (5)
Conjecture (M.)
.
1
R∗ (Mg ) ∼
= (∧∗ UQ /([22 ]Sp ))Sp
(
)⊥
∼
= ([22 ]Sp )Sp in (∧∗ UQ )Sp
2
.
(
)Sp
2
R∗ (Mg,∗ ) ∼
⊕
[2
]
)
= ∧∗ (∧3 HQ )/([12 ]torelli
Sp
Sp
(
)⊥ (
)Sp
2 torelli
2
Sp
∼
([1
]
⊕
[2
]
)
in ∧∗ (∧3 HQ )
=
Sp
Sp
.
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (1)
Plethysm: composition of two Schur functors
determination of plethysm: very important but extremely difficult
Theorem (Formula of Littlewood)
Complete description of the following plethysms
S ∗ (S 2 HQ ), ∧∗ (S 2 HQ ), S ∗ (∧2 HQ ), ∧∗ (∧2 HQ )
.
Theorem (Manivel)
Plethysm S k (S l HQ ) “super stabilizes” as k → ∞, in particular
super stable decomposition of S ∞ (S 3 HQ ) is given by
S ∗ (S 2 HQ ⊕ S 3 HQ )
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.
Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (2)
Theorem (Sakasai-Suzuki-M.)
Let
∧k (∧3 HQ ) =
⊕
mλ λGL
λ,|λ|=3k
be the stable irreducible decomposition as a GL-module. Then,
for any k, the mapping
∧k (∧3 HQ ) −→ ∧k+1 (∧3 HQ )
induced by the operation λ 7→ λ+ = [λ13 ] is injective and
surjective for the part λ+
GL with 2k + 1 ≤ h(λ) ≤ 3k.
In other words, we have the inequality
{
≤ m λ+
.
mλ
= mλ+ (2k + 1 ≤ h(λ) ≤ 3k)
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (3)
Theorem (Sakasai-Suzuki-M.)
.
We have determined the super stable irreducible
decomposition of ∧∞ [13 ]GL up to codimension 30
.
.
Table: Super stable irreducible decomposition of ∧∞ [13 ]GL
cod.
0
1
2
3
4
5
6
7
irreducible decomposition
[1∗ ]
[21∗ ]
[22 1∗ ]
[23 1∗ ]
[24 1∗ ][32 1∗ ]
[25 1∗ ][323 1∗ ][32 21∗ ]
2[26 1∗ ]2[32 22 1∗ ][42 1∗ ]
[27 1∗ ][325 1∗ ]2[32 23 1∗ ][33 21∗ ][4322 1∗ ][42 21∗ ]
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (4)
Corollary (Sakasai-Suzuki-M.)
.
We have determined super stable Sp-invariant part
( ∞ 3 )Sp
∧ [1 ]GL
up to codimension 30
.
.
Table: Super stable irred. summands of ∧∞ [13 ]GL with double floors
cod.
0
2
4
6
8
10
irreducible decomposition
[1∗ ]
[22 1∗ ]
[24 1∗ ][32 1∗ ]
2[26 1∗ ]2[32 22 1∗ ][42 1∗ ]
2[28 1∗ ]3[32 24 1∗ ]2[34 1∗ ]2[42 22 1∗ ][52 1∗ ]
2[210 1∗ ]4[32 26 1∗ ]4[34 22 1∗ ]4[42 24 1∗ ]3[42 32 1∗ ]2[52 22 1∗ ][62 1∗ ]
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (5)
R∗ (Mg ) → R∗ (Mg ) → G∗ (Mg ) (Gorenstein quotient)
R∗ (Mg,1 ) → R∗ (Mg,∗ ) → G∗ (Mg,1 ) (Gorenstein quotient)
Expectation (Faber, Zagier, Bergvall, Yin)
The number
p(k) − dim G2k (Mg )
depends only on ` = 3k − 1 − g in the range
2k ≤ g − 2 (i.e. k ≥ ` + 3). Similarly the number
1 + p(1) + · · · + p(k) − dim G2k (Mg,1 )
depends only on ` = 3k − 1 − g in the range
2k ≤ g − 1 (i.e. k ≥ ` + 2).
.
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (6)
We have the following two theorems which may serve as
supporting evidences for the above expectation
Theorem (Sakasai-Suzuki-M.)
The number
(
)Sp
(
)
hgi Sp
− dim ∧2k UQ
dim ∧2k UQ
depends only on ` = 3k − 1 − g in the range
hgi
2k ≤ g − 2 (i.e. k ≥ ` + 3), where UQ denotes UQ for the
specific genus g while we assume that the genus of UQ is in the
stable range, namely it is sufficiently large.
.
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (7)
Theorem (Sakasai-Suzuki-M.)
The number
(
)Sp
(
)
hgi Sp
dim ∧2k (∧3 HQ )
− dim ∧2k (∧3 HQ )
depends only on ` = 3k − 1 − g in the range
hgi
2k ≤ g − 1 (i.e. k ≥ ` + 2), where HQ denotes HQ for the
specific genus g while we assume that the genus of HQ. is in the
stable range, namely it is sufficiently large.
Similar statements should also hold
modulo the corresponding ideals
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Tautological Algebra and Representation Theory
Plethysm of GL representations and tautological algebra (8)
Furthermore, we have the following more precise result.
(
)
(
)Sp
hgi Sp
orthogonal complement of ∧2k (∧3 HQ )
in ∧2k (∧3 HQ )
⇒ tautological relations in R2k (Mg,∗ )
)
(
)Sp
(
hgi Sp
orthogonal complement of ∧2k UQ
in ∧2k UQ
⇒ tautological relations in R2k (Mg )
Theorem (Sakasai-Suzuki-M.)
If we fix ` = 3k − 1 − g, then all the above orthogonal
complements are canonically isomorphic to each other in the
range 2k ≤ g − 1 (i.e. k ≥ ` + 2) or 2k ≤ g − 2 (i.e. k ≥ ` + 3).
Similar statements should hold modulo corresponding ideals
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Tautological Algebra and Representation Theory
Prospects (1)
(I) Construction of the “fundamental cycles” :
(
)
hgi Sp
µg,∗ ∈ ∧2g−2 (∧3 HQ )
(
)
hgi Sp
µg ∈ ∧2g−4 UQ
and topological proof of the intersection number formula
(II) Investigation of the relation between our tautological
relations with those of Faber-Zagier as well as those of Yin
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Tautological Algebra and Representation Theory
Prospects (2)
(III) Which part is isomorphic or non-isomorphic ?:
R∗ (Mg ) → R∗ (Mg ) → G∗ (Mg ) (Gorenstein quotient)
R∗ (Mg,1 ) → R∗ (Mg,∗ ) → G∗ (Mg,1 ) (Gorenstein quotient)
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Tautological Algebra and Representation Theory