Exceptional Lie groups and some related geometry

99909_6talk.pdf : Exceptional Lie groups and some related geometryOctober 8, 2021 1 / 26

Exceptional Lie groups and some related geometry

October 8, 2021Outline of the talk

Compact simple Lie groups { Maximal tori, Weyl groups and root systemsCrash course in exceptional groups | Applications |Exceptional groups occur in many contexts of mathematics and

physics, and the main interest in this talk isSome instances of geometry arising from arrangements dened by

Weyl groups of exceptional groups

Exceptional Lie groups and some related geometryOctober 8, 2021 1 / 26

Examples

LetTGbe a maximal torus in a compact simple Lie group.Example LetVbe a real vector space of dimensionnwith an inner product; then the group of linear automorphisms ofVpreservingthe inner product is a compact group denotedO(n)(theorthogonalgroup).

s(x;y) =txSymeanss(x;y) =s(gx;gy) =t(Mx)S(My))M= (tM)1.LetVbe a complex vector space of dimensionnwith an inner

product; then the group of (complex) linear automorphisms ofV which preserve the inner product is a compact Lie group denoted U(n)(theunitarygroup). ThenM= (M)1.In each case, the determinant gives rise to an exact sequence

1!SG!Gdet!K!1

whereKis the eldRorCandGis the group above. Then:SGis a

simple compact Lie groupSO(n)resp.SU(n).Exceptional Lie groups and some related geometryOctober 8, 2021 2 / 26

Examples

LetTGbe a maximal torus in a compact simple Lie group.Example LetVbe a real vector space of dimensionnwith an inner product; then the group of linear automorphisms ofVpreservingthe inner product is a compact group denotedO(n)(theorthogonalgroup).

s(x;y) =txSymeanss(x;y) =s(gx;gy) =t(Mx)S(My))M= (tM)1.LetVbe a complex vector space of dimensionnwith an inner

product; then the group of (complex) linear automorphisms ofV which preserve the inner product is a compact Lie group denoted U(n)(theunitarygroup). ThenM= (M)1.In each case, the determinant gives rise to an exact sequence

1!SG!Gdet!K!1

whereKis the eldRorCandGis the group above. Then:SGis a

simple compact Lie groupSO(n)resp.SU(n).Exceptional Lie groups and some related geometryOctober 8, 2021 2 / 26

Examples

LetTGbe a maximal torus in a compact simple Lie group.Example LetVbe a real vector space of dimensionnwith an inner product; then the group of linear automorphisms ofVpreservingthe inner product is a compact group denotedO(n)(theorthogonalgroup).

s(x;y) =txSymeanss(x;y) =s(gx;gy) =t(Mx)S(My))M= (tM)1.LetVbe a complex vector space of dimensionnwith an inner

product; then the group of (complex) linear automorphisms ofV which preserve the inner product is a compact Lie group denoted U(n)(theunitarygroup). ThenM= (M)1.In each case, the determinant gives rise to an exact sequence

1!SG!Gdet!K!1

whereKis the eldRorCandGis the group above. Then:SGis a

simple compact Lie groupSO(n)resp.SU(n).Exceptional Lie groups and some related geometryOctober 8, 2021 2 / 26

Examples

LetTGbe a maximal torus in a compact simple Lie group.Example LetVbe a real vector space of dimensionnwith an inner product; then the group of linear automorphisms ofVpreservingthe inner product is a compact group denotedO(n)(theorthogonalgroup).

s(x;y) =txSymeanss(x;y) =s(gx;gy) =t(Mx)S(My))M= (tM)1.LetVbe a complex vector space of dimensionnwith an inner

product; then the group of (complex) linear automorphisms ofV which preserve the inner product is a compact Lie group denoted U(n)(theunitarygroup). ThenM= (M)1.In each case, the determinant gives rise to an exact sequence

1!SG!Gdet!K!1

whereKis the eldRorCandGis the group above. Then:SGis a

simple compact Lie groupSO(n)resp.SU(n).Exceptional Lie groups and some related geometryOctober 8, 2021 2 / 26

Maximal tori

For the two groups considered, the maximal tori are easily described.Example ForU(n), it is the groupT=fdiag(t1;:::;tn)gof diagonal matrices with complex entriesti=eiqi. ForSU(n)one has in additionnÕ iti=1.LetB(q):=sin(q)cos(q) cos(q)sin(q) be the22real matrix representing a rotation ofqdegrees inR2. For the groupO(n)Tis the group of diagonalblockmatricesT=fdiag(B1(q1);:::;Bn2 (qn2 ))g whenn=2mis even andT=fB1(q1);:::;B[n2 ](q[n2 ]);1)gwhen n=2m+1is odd.Exceptional Lie groups and some related geometryOctober 8, 2021 3 / 26

Maximal tori

For the two groups considered, the maximal tori are easily described.Example ForU(n), it is the groupT=fdiag(t1;:::;tn)gof diagonal matrices with complex entriesti=eiqi. ForSU(n)one has in additionnÕ iti=1.LetB(q):=sin(q)cos(q) cos(q)sin(q) be the22real matrix representing a rotation ofqdegrees inR2. For the groupO(n)Tis the group of diagonalblockmatricesT=fdiag(B1(q1);:::;Bn2 (qn2 ))g whenn=2mis even andT=fB1(q1);:::;B[n2 ](q[n2 ]);1)gwhen n=2m+1is odd.Exceptional Lie groups and some related geometryOctober 8, 2021 3 / 26

The Weyl group

LetN(T)Gdenote the normalizer ofT; the groupW(G) =N(T)=Tis called theWeyl groupofG.Example ForU(n), in addition to multiplication on itself, permutations of the factorst1;:::;tnpreserve (normalize) the torusT. The relation deningSU(n)makestndependent on the others, so the group of permutations isSn1.ForSO(n), one has the permutations of theblocks; but in addition, in each blockqi7! qipreserves the torus. The Weyl group is W(SO(2m+1)) =SmoZm2, forSO(2m)the number of sign changes

needs to be even,W(SO(2m)) =SmoZm12.Exceptional Lie groups and some related geometryOctober 8, 2021 4 / 26

The Weyl group

LetN(T)Gdenote the normalizer ofT; the groupW(G) =N(T)=Tis called theWeyl groupofG.Example ForU(n), in addition to multiplication on itself, permutations of the factorst1;:::;tnpreserve (normalize) the torusT. The relation deningSU(n)makestndependent on the others, so the group of permutations isSn1.ForSO(n), one has the permutations of theblocks; but in addition, in each blockqi7! qipreserves the torus. The Weyl group is W(SO(2m+1)) =SmoZm2, forSO(2m)the number of sign changes

needs to be even,W(SO(2m)) =SmoZm12.Exceptional Lie groups and some related geometryOctober 8, 2021 4 / 26

Using quaternions

His theR-algebra generated by 1 and elementsi;j;ksuch that i

2=j2=k2=1;ij=ji;ik=ki;jk=kj:

Hforms adivision algebraoverR, meaning it is a multiplicative algebra overRwithout zero-divisors.Hsatises all the axioms of a eld except commutativity, hence it can be used as the scalar eld for a vector spaceV of dimensionnoverH(dimension 4 overR).Convention One needs to x the scalar multiplication; standard is:Vis aright

H-vector space, then matrices operate from the left.Using aH-hermitian form, what was done above for the inner products

can be extended to the case ofquaternionic matrices. This leads to compact simple Lie groups called thesymplectic groupsSp(n). The Weyl group isSnoZn2.Exceptional Lie groups and some related geometryOctober 8, 2021 5 / 26

Root systems and classication

A representation ofGis a group homomorphism ofGin the automorphism group of a vector space. Conjugation inG,x7!cs(x):=sxs1, denes a map

Inn :G!Aut(G);s7!cs:G!G:

This leads to theadjoint representationofG, dened as follows:

Ad :G!GL(Te(G));

s7!T(cs):Te(G)!Te(G):(1) Representations are studied by restricting toT: any representation ofTis multiplication by(ta11;:::;tann)with integerai. The torus itself is T=Rn=Zn, and the exponents correspond to elements in alattice. The roots ofGare the eigenvalues of the adjoint representation. In this way a root systemis dened, and leads to the classication A n;Bn;Cn;Dn;G2;F4;E6;E7;E8.Exceptional Lie groups and some related geometryOctober 8, 2021 6 / 26

Classical and exceptional groups

In the notation, the subscript denotes thedimension of the maximal torusClassical groups A nthe special unitary groupsSU(n+1).B nthe special orthogonal groupSO(2n+1).C nthe symplectic groupSp(n).D nthe special orthogonal groupSO(2n).Exceptional groups G

2;F4;E6;E7;E8

The classication is called the Cartan-Killing classication and goes back to independent determinations by those two mathematicians. Any idea what these exceptional groups represent? Exceptional Lie groups and some related geometryOctober 8, 2021 7 / 26

Classical and exceptional groups

In the notation, the subscript denotes thedimension of the maximal torusClassical groups A nthe special unitary groupsSU(n+1).B nthe special orthogonal groupSO(2n+1).C nthe symplectic groupSp(n).D nthe special orthogonal groupSO(2n).Exceptional groups G

2;F4;E6;E7;E8

The classication is called the Cartan-Killing classication and goes back to independent determinations by those two mathematicians. Any idea what these exceptional groups represent? Exceptional Lie groups and some related geometryOctober 8, 2021 7 / 26

Classical and exceptional groups

In the notation, the subscript denotes thedimension of the maximal torusClassical groups A nthe special unitary groupsSU(n+1).B nthe special orthogonal groupSO(2n+1).C nthe symplectic groupSp(n).D nthe special orthogonal groupSO(2n).Exceptional groups G

2;F4;E6;E7;E8

The classication is called the Cartan-Killing classication and goes back to independent determinations by those two mathematicians. Any idea what these exceptional groups represent? Exceptional Lie groups and some related geometryOctober 8, 2021 7 / 26

Classical and exceptional groups

In the notation, the subscript denotes thedimension of the maximal torusClassical groups A nthe special unitary groupsSU(n+1).B nthe special orthogonal groupSO(2n+1).C nthe symplectic groupSp(n).D nthe special orthogonal groupSO(2n).Exceptional groups G

2;F4;E6;E7;E8

The classication is called the Cartan-Killing classication and goes back to independent determinations by those two mathematicians. Any idea what these exceptional groups represent? Exceptional Lie groups and some related geometryOctober 8, 2021 7 / 26

Classical and exceptional groups

In the notation, the subscript denotes thedimension of the maximal torusClassical groups A nthe special unitary groupsSU(n+1).B nthe special orthogonal groupSO(2n+1).C nthe symplectic groupSp(n).D nthe special orthogonal groupSO(2n).Exceptional groups G

2;F4;E6;E7;E8

The classication is called the Cartan-Killing classication and goes back to independent determinations by those two mathematicians. Any idea what these exceptional groups represent? Exceptional Lie groups and some related geometryOctober 8, 2021 7 / 26

Octonions

Note thatC=R+iR;i2=1;H=C+jC;j2=1. Such algebras are called composisition algebras. The construction can be extended one more time,O=H+e3Hto give a division algebra. This algebra is given by the relations:The division algebra of octonions (Cayley-Graves algebra) e

2j=e0=1j=1;:::;7

e jek=ekejj6=k;j;k=1;:::;7 e i
ei+1=ei+3;ei+1
ei+3=ei;ei+3
ei=ei+1;indices taken mod 7 (2)

in whiche0;ei;ei+1;ei+3form a subalgebra isomorphic toH.Exceptional Lie groups and some related geometryOctober 8, 2021 8 / 26

Multiplication inO?e

1?e 2?e 3?e 4? ?e 6?e 7e

5Exceptional Lie groups and some related geometryOctober 8, 2021 9 / 26

Multiplication inO?e

1?e 2?e 3?e 4? ?e 6?e 7e

5Exceptional Lie groups and some related geometryOctober 8, 2021 10 / 26

Multiplication inO?e

1?e 2?e 3?e 4? ?e 6?e 7e

5Each colored triangle is a subalgebra

=H. For example, this shows e

3
e5=e2amde5
e1=e6.Exceptional Lie groups and some related geometryOctober 8, 2021 11 / 26

Octonions and exceptional groups

The octonionsOare not associative. Consequently, matrix algebras of arbitrary size cannot be dened.A33algebraisdened, which is evencommutative, but not

associative ... It forms what is called aJordan algebra.Jordan algebras were studied at the beginning of the twentieth

century in a quest to nd interesting algebras for quantum physics. Again there is a classication, with aunique exceptionalone, a

27-dimensional algebra (the33algebra above).There is an exotic way to dene, starting with a Jordan algebra and a

R-division algebra, a Lie algebra, the Tits-Vinberg construction.Exceptional Lie groups and some related geometryOctober 8, 2021 12 / 26

Octonions and exceptional groups

The octonionsOare not associative. Consequently, matrix algebras of arbitrary size cannot be dened.A33algebraisdened, which is evencommutative, but not

associative ... It forms what is called aJordan algebra.Jordan algebras were studied at the beginning of the twentieth

century in a quest to nd interesting algebras for quantum physics. Again there is a classication, with aunique exceptionalone, a

27-dimensional algebra (the33algebra above).There is an exotic way to dene, starting with a Jordan algebra and a

R-division algebra, a Lie algebra, the Tits-Vinberg construction.Exceptional Lie groups and some related geometryOctober 8, 2021 12 / 26

Octonions and exceptional groups

The octonionsOare not associative. Consequently, matrix algebras of arbitrary size cannot be dened.A33algebraisdened, which is evencommutative, but not

associative ... It forms what is called aJordan algebra.Jordan algebras were studied at the beginning of the twentieth

century in a quest to nd interesting algebras for quantum physics. Again there is a classication, with aunique exceptionalone, a

27-dimensional algebra (the33algebra above).There is an exotic way to dene, starting with a Jordan algebra and a

R-division algebra, a Lie algebra, the Tits-Vinberg construction.Exceptional Lie groups and some related geometryOctober 8, 2021 12 / 26

Octonions and exceptional groups

The octonionsOare not associative. Consequently, matrix algebras of arbitrary size cannot be dened.A33algebraisdened, which is evencommutative, but not

associative ... It forms what is called aJordan algebra.Jordan algebras were studied at the beginning of the twentieth

century in a quest to nd interesting algebras for quantum physics. Again there is a classication, with aunique exceptionalone, a

27-dimensional algebra (the33algebra above).There is an exotic way to dene, starting with a Jordan algebra and a

R-division algebra, a Lie algebra, the Tits-Vinberg construction.Exceptional Lie groups and some related geometryOctober 8, 2021 12 / 26

The Tits-Vinberg construction

LetAbe a composition algebra (one ofR;C;H;O) andJa Jordan algebra. Dene the algebra

T(A;J) =Der(A)Der(J)+A0

J0: Provided with a rather complicated bracket, this can be made to a Lie algebra. The algebraH3(A)of \hermitian"33-matrices for a composition algebraAis a Jordan algebra; taking forA,Bthe various composition algebras withT=T(A;H3(B))leads to themagic square:RCHO RA 1A 2C 3F 4CA 2A 2A2A 5E 6HC 3A 5D 6E 7OF 4E 6E 7E

8Exceptional Lie groups and some related geometryOctober 8, 2021 13 / 26

The Tits-Vinberg construction

LetAbe a composition algebra (one ofR;C;H;O) andJa Jordan algebra. Dene the algebra

T(A;J) =Der(A)Der(J)+A0

J0: Provided with a rather complicated bracket, this can be made to a Lie algebra. The algebraH3(A)of \hermitian"33-matrices for a composition algebraAis a Jordan algebra; taking forA,Bthe various composition algebras withT=T(A;H3(B))leads to themagic square:RCHO RA 1A 2C 3F 4CA 2A 2A2A 5E 6HC 3A 5D 6E 7OF 4E 6E 7E

8Exceptional Lie groups and some related geometryOctober 8, 2021 13 / 26

Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26 Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26 Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26 Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26 Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26 Appearances of exceptional groups in various contexts The octonions are closely related toCliord algebrasandspin representations(work of G. Dixon on the \Algebraic design of

physics") and even Bott periodicity (as pointed out by J. Baez).Dynkin diagrams of equally laced groups arise as resolution graphs of

rational singularities and universal deformations as singular orbits in the adjoint representations.The exceptional groups give rise to a series ofgeneralized projective planes. The book \Locally mixed symmetric spaces", just appeared at Springer, details this as an example ofsymmetric spaces.E

8E8is the gauge group of theheterotic string.The exceptional groups, in fact the entire magic square, arises in

context of compactication ofsupergravity in dimension 11.The Weyl groups and root systems dene arrangements which arise in

various fascinating situations in algebraic geometry. Exceptional Lie groups and some related geometryOctober 8, 2021 14 / 26

Main topic of this talk

Arrangements dened by Weyl groups

The Weyl groupW(G)can be dened as a re

ection group, generated in GL n(R)(nthe rank ofG) by there ections on the roots. Considering the correspondingprojectivearrangement one dimension lower denes a hyperplane arrangement inPn(R)(or in complex projective space). A particular case in considered in \The geometry of some special arithmetic quotients". More details will be given in the upcoming monograph by the author.Weyl group actions and arrangements The groupG2, being only of rank 2, denes only a set of points onP1(C), but the others give rise to interesting arrangements.F

4denes an arrangement of planes inP3(C): the 24 planes form an

arrangement which has two subsets of 12 roots, each corresponding to a subroot system of typeD4, called atriple of desmic tetrahedra, one arising from the long roots, one from the short ones. Exceptional Lie groups and some related geometryOctober 8, 2021 15 / 26

Main topic of this talk

Arrangements dened by Weyl groups

The Weyl groupW(G)can be dened as a re

ection group, generated in GL n(R)(nthe rank ofG) by there ections on the roots. Considering the correspondingprojectivearrangement one dimension lower denes a hyperplane arrangement inPn(R)(or in complex projective space). A particular case in considered in \The geometry of some special arithmetic quotients". More details will be given in the upcoming monograph by the author.Weyl group actions and arrangements The groupG2, being only of rank 2, denes only a set of points onP1(C), but the others give rise to interesting arrangements.F

4denes an arrangement of planes inP3(C): the 24 planes form an

arrangement which has two subsets of 12 roots, each corresponding to a subroot system of typeD4, called atriple of desmic tetrahedra, one arising from the long roots, one from the short ones. Exceptional Lie groups and some related geometryOctober 8, 2021 15 / 26

Arrangements dened by Weyl groups

Weyl group actions and arrangements

E

6denes an arrangement of 36 hyperplanes (P4) inP5(C). The

groupW(E6)is isomorphic to the group of incidence-preserving permutations of the 27 lines on a smooth cubic surface, and all the geometry has wonderful interpretations of geometric congurations. This geometry has been studied in detail in \The geometry of some special arithmetic quotients".E

7denes an arrangement of 63 hyperplanes (P5) inP6(C). The

groupW(E7)is similarly related to the 28 bitangents of a quartic curve inP2(C), and a similar correspondence to geometric properties is mirrored in the arrangement.E

8denes an arrangement of 120 hyperplanes (P6) inP7(C). This

arrangement is very \universal": all the above can be derived as subarrangements of this arrangement. The lattice inR8dened by the roots can be identied with a very special order inO, that is a sublattice which is also a subring. Many other connections exist. Exceptional Lie groups and some related geometryOctober 8, 2021 16 / 26

Arrangements dened by Weyl groups

Weyl group actions and arrangements

E

6denes an arrangement of 36 hyperplanes (P4) inP5(C). The

groupW(E6)is isomorphic to the group of incidence-preserving permutations of the 27 lines on a smooth cubic surface, and all the geometry has wonderful interpretations of geometric congurations. This geometry has been studied in detail in \The geometry of some special arithmetic quotients".E

7denes an arrangement of 63 hyperplanes (P5) inP6(C). The

groupW(E7)is similarly related to the 28 bitangents of a quartic curve inP2(C), and a similar correspondence to geometric properties is mirrored in the arrangement.E

8denes an arrangement of 120 hyperplanes (P6) inP7(C). This

arrangement is very \universal": all the above can be derived as subarrangements of this arrangement. The lattice inR8dened by the roots can be identied with a very special order inO, that is a sublattice which is also a subring. Many other connections exist. Exceptional Lie groups and some related geometryOctober 8, 2021 16 / 26

Arrangements dened by Weyl groups

Weyl group actions and arrangements

E

6denes an arrangement of 36 hyperplanes (P4) inP5(C). The

groupW(E6)is isomorphic to the group of incidence-preserving permutations of the 27 lines on a smooth cubic surface, and all the geometry has wonderful interpretations of geometric congurations. This geometry has been studied in detail in \The geometry of some special arithmetic quotients".E

7denes an arrangement of 63 hyperplanes (P5) inP6(C). The

groupW(E7)is similarly related to the 28 bitangents of a quartic curve inP2(C), and a similar correspondence to geometric properties is mirrored in the arrangement.E

8denes an arrangement of 120 hyperplanes (P6) inP7(C). This

arrangement is very \universal": all the above can be derived as subarrangements of this arrangement. The lattice inR8dened by the roots can be identied with a very special order inO, that is a sublattice which is also a subring. Many other connections exist. Exceptional Lie groups and some related geometryOctober 8, 2021 16 / 26

The arrangement dened by the Weyl groupW(E6)

The Weyl groupW(E6)is isomorphic to the group of permutations of the

27 lines on a cubic surface. This isomorphism has a beautiful geometric

expression in the properties of the arrangment dened.Properties of the 27 lines on a smooth cubic surface

The are 45 special hyperplane sections, each of which intersects the cubic surface in 3 of the 27 lines; this plane is tangent at the 3

intersection points of the lines, hence the nametritangent.There are 36double sixeswhich are sets of 12 of the lines, arranged

in a26matrix, such that two lines meet if and only if they are dierent rows and columns.There are 120trihedral pairs, sets of nine of the lines, cut out by 6 of

the tritangents, which determine the equation of the cubic surface.There are 40triads of trihedral pairs, such that the triple contains all

27 lines.

Exceptional Lie groups and some related geometryOctober 8, 2021 17 / 26

The arrangement dened by the Weyl groupW(E6)

The Weyl groupW(E6)is isomorphic to the group of permutations of the

27 lines on a cubic surface. This isomorphism has a beautiful geometric

expression in the properties of the arrangment dened.Properties of the 27 lines on a smooth cubic surface

The are 45 special hyperplane sections, each of which intersects the cubic surface in 3 of the 27 lines; this plane is tangent at the 3

intersection points of the lines, hence the nametritangent.There are 36double sixeswhich are sets of 12 of the lines, arranged

in a26matrix, such that two lines meet if and only if they are dierent rows and columns.There are 120trihedral pairs, sets of nine of the lines, cut out by 6 of

the tritangents, which determine the equation of the cubic surface.There are 40triads of trihedral pairs, such that the triple contains all

27 lines.

Exceptional Lie groups and some related geometryOctober 8, 2021 17 / 26

The arrangement dened by the Weyl groupW(E6)

The Weyl groupW(E6)is isomorphic to the group of permutations of the

27 lines on a cubic surface. This isomorphism has a beautiful geometric

expression in the properties of the arrangment dened.Properties of the 27 lines on a smooth cubic surface

The are 45 special hyperplane sections, each of which intersects the cubic surface in 3 of the 27 lines; this plane is tangent at the 3

intersection points of the lines, hence the nametritangent.There are 36double sixeswhich are sets of 12 of the lines, arranged

in a26matrix, such that two lines meet if and only if they are dierent rows and columns.There are 120trihedral pairs, sets of nine of the lines, cut out by 6 of

the tritangents, which determine the equation of the cubic surface.There are 40triads of trihedral pairs, such that the triple contains all

27 lines.

Exceptional Lie groups and some related geometryOctober 8, 2021 17 / 26

The arrangement dened by the Weyl groupW(E6)

The Weyl groupW(E6)is isomorphic to the group of permutations of the

27 lines on a cubic surface. This isomorphism has a beautiful geometric

expression in the properties of the arrangment dened.Properties of the 27 lines on a smooth cubic surface

The are 45 special hyperplane sections, each of which intersects the cubic surface in 3 of the 27 lines; this plane is tangent at the 3

intersection points of the lines, hence the nametritangent.There are 36double sixeswhich are sets of 12 of the lines, arranged

in a26matrix, such that two lines meet if and only if they are dierent rows and columns.There are 120trihedral pairs, sets of nine of the lines, cut out by 6 of

the tritangents, which determine the equation of the cubic surface.There are 40triads of trihedral pairs, such that the triple contains all

27 lines.

Exceptional Lie groups and some related geometryOctober 8, 2021 17 / 26

Subloci of the arrangement dened byW(E6)

As a Weyl group,W(E6)is generated byre

ections on the roots. Since the rank is 6, the projective arrangement is dened inP5(C).36 hyperplanes SinceE6has 36 positive roots, this denes 36 hyperplanes (each a P

4(C)), one corresponding to each double-six.Corresponding to the 120 triples of azygetic double-sixes, there are

120P3(C)'s which are the intersection of 3 of the 36.Corresponding to the 270 pairs of syzegetic pairs of double sixes,

there are 270P3(C)'s which are the intersection of two of the 36.Corresponding to the 120 trihedral pairs, there are 120P1(C)'s (lines)

which are the intersection of 6 of the 36.Where are the 27 lines? Exceptional Lie groups and some related geometryOctober 8, 2021 18 / 26

The 27 weights

The groupE6has two complex-conjugate 27-dimensional representations, in the exceptional Jordan algebra (the highest weight is the fundamental weight corresponding to the rst (resp. last) roota1(resp.a6)). This denes 27 weights, which are accordingly an orbit under the Weyl group. It is these weights which correspond to the 27 lines. (Note: everything taking

place in the universal cover of the maximal torus identied withH1(T;Z)).The arrangement dened by the 27 hyperplanes

This denes 27 hyperplanes (P4). Some of the interesting loci:Corresponding to the 45 tritangent planes, there are 45P3(C)'s

which are the intersection of 3 of the 27.Corresponding to the 216 pairs of skew lines there are 216P3(C)'s

which are the intersection of two of the 27.Recalling that each of the 120 trihedral pairs denes a set of 9 lines,

there are 120 lines which are the intersection of 9 of the 27.etc... Exceptional Lie groups and some related geometryOctober 8, 2021 19 / 26

The 27 weights

The groupE6has two complex-conjugate 27-dimensional representations, in the exceptional Jordan algebra (the highest weight is the fundamental weight corresponding to the rst (resp. last) roota1(resp.a6)). This denes 27 weights, which are accordingly an orbit under the Weyl group. It is these weights which correspond to the 27 lines. (Note: everything taking

place in the universal cover of the maximal torus identied withH1(T;Z)).The arrangement dened by the 27 hyperplanes

This denes 27 hyperplanes (P4). Some of the interesting loci:Corresponding to the 45 tritangent planes, there are 45P3(C)'s

which are the intersection of 3 of the 27.Corresponding to the 216 pairs of skew lines there are 216P3(C)'s

which are the intersection of two of the 27.Recalling that each of the 120 trihedral pairs denes a set of 9 lines,

there are 120 lines which are the intersection of 9 of the 27.etc... Exceptional Lie groups and some related geometryOctober 8, 2021 19 / 26

Invariant polynomials ofW(E6)

The invariants of the classical groups described above are polynomials invariant under permutations of the coordinates, or under permutations of the squares of the coordinates. Here it is a more challenging matter. The degrees of the invariants are: 2, 5, 6, 8, 9, 12. Of these, only the quintic is unique. As a general denition, for invariants ofevendegree one can take the sum of the powers of the roots. Forodddegrees, this sum is always 0

because with any root also its negative is a root.Thekthpowers of the 27 fundamental weightsUsing the 27 weights (with equationsai=0;bi=0;cij=0), one gets for

any degreekan invariant of the corresponding degree. I k:=å i;j aki+bki+ckij : Exceptional Lie groups and some related geometryOctober 8, 2021 20 / 26

The invariant quintic

A whole chapter in \The geometry of some special arithmetic quotients" is devoted to the quinticI5=0, a 4-dimensional variety inP5(C). Some of the beautiful properties are just listed here.Loci on the quintic Thesingular locusof the quintic consists of 120 lines which intersect in 36 points.The 45P3(C)'s mentioned above are all contained in the quintic; in fact the intersection ofI5with any of the 27 hyperplanes, a quintic, splits into 5 of these 45.The 36 intersections ofI5with the root hyperplanes are copies of a

quintic threefold known as theNieto quintic.Both the Nieto quintic and the invariant quintic 4-fold are

compactications ofball quotients. This was conjectured in the lecture notes mentioned above and has been veried in the mean

time. In fact, also the 45P3(C)'s above are ball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 21 / 26

The invariant quintic

A whole chapter in \The geometry of some special arithmetic quotients" is devoted to the quinticI5=0, a 4-dimensional variety inP5(C). Some of the beautiful properties are just listed here.Loci on the quintic Thesingular locusof the quintic consists of 120 lines which intersect in 36 points.The 45P3(C)'s mentioned above are all contained in the quintic; in fact the intersection ofI5with any of the 27 hyperplanes, a quintic, splits into 5 of these 45.The 36 intersections ofI5with the root hyperplanes are copies of a

quintic threefold known as theNieto quintic.Both the Nieto quintic and the invariant quintic 4-fold are

compactications ofball quotients. This was conjectured in the lecture notes mentioned above and has been veried in the mean

time. In fact, also the 45P3(C)'s above are ball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 21 / 26

The invariant quintic

A whole chapter in \The geometry of some special arithmetic quotients" is devoted to the quinticI5=0, a 4-dimensional variety inP5(C). Some of the beautiful properties are just listed here.Loci on the quintic Thesingular locusof the quintic consists of 120 lines which intersect in 36 points.The 45P3(C)'s mentioned above are all contained in the quintic; in fact the intersection ofI5with any of the 27 hyperplanes, a quintic, splits into 5 of these 45.The 36 intersections ofI5with the root hyperplanes are copies of a

quintic threefold known as theNieto quintic.Both the Nieto quintic and the invariant quintic 4-fold are

compactications ofball quotients. This was conjectured in the lecture notes mentioned above and has been veried in the mean

time. In fact, also the 45P3(C)'s above are ball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 21 / 26

The invariant quintic

A whole chapter in \The geometry of some special arithmetic quotients" is devoted to the quinticI5=0, a 4-dimensional variety inP5(C). Some of the beautiful properties are just listed here.Loci on the quintic Thesingular locusof the quintic consists of 120 lines which intersect in 36 points.The 45P3(C)'s mentioned above are all contained in the quintic; in fact the intersection ofI5with any of the 27 hyperplanes, a quintic, splits into 5 of these 45.The 36 intersections ofI5with the root hyperplanes are copies of a

quintic threefold known as theNieto quintic.Both the Nieto quintic and the invariant quintic 4-fold are

compactications ofball quotients. This was conjectured in the lecture notes mentioned above and has been veried in the mean

time. In fact, also the 45P3(C)'s above are ball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 21 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26

This years discovery: the invariant nonticI9

The only invariants of odd degree areI5andI9, but while the quintic is unique,I9is only unique up to addition of termsI22I5. Nevertheless it seemed natural to consider the invariantI9, which had not been done back in the 1990's whenI5was studied. An initial study already showed the following.The geometry ofI9The intersectionI5\I9consists of the union of the 45P3(C)'s. (This

implies thatanyinvariant of degree 9 intersectsI5in the same union.)The singular locusS9I9is a curve of degree 1200 (!). What is this

curve?The Jacobian ideal is generated by 6 octics, and it is not dicult to see

that these octics vanish at the 120 lines (the singular locus ofI5).The 120 lines are containedsimply, leaving a curve of degree 1080 to

explain.The 120 lines are arranged inP5(C)such that 4 of them lie in certain planes. Exceptional Lie groups and some related geometryOctober 8, 2021 22 / 26 More geometry ofI9Restricted to these planes, 5 of the octics vanish and the remaining one splits o these 4 lines, leaving a quartic curve in each of the planes. These planes are the so-callc-planes, and the 4 lines intersect at 6 of

the 36 singular points of intersection.There are 270c-planes, each the intersection of 6 of the 36 root

hyperplanes. Hence270
4=1080is the degree of the union of these

smooth quartics in thec-planes.The16intersection points on each quartic with the 4 lines are disjoint

from the 6 nodes in that plane.The intersection ofI9with thesymmetrichyperplanesP4(C)is a nontic threefoldK9. It intersects the Nieto quintic (the intersection ofI5with the same hyperplane) in the union of 15 + 15 planes, the c-planes mentioned above, and the so-called Segre planes, each containing 3 of the nodes and 4 of the 10 isolated double points of N

5, thec-planes being double in the intersection.Exceptional Lie groups and some related geometryOctober 8, 2021 23 / 26

More geometry ofI9Restricted to these planes, 5 of the octics vanish and the remaining one splits o these 4 lines, leaving a quartic curve in each of the planes. These planes are the so-callc-planes, and the 4 lines intersect at 6 of

the 36 singular points of intersection.There are 270c-planes, each the intersection of 6 of the 36 root

hyperplanes. Hence270
4=1080is the degree of the union of these

smooth quartics in thec-planes.The16intersection points on each quartic with the 4 lines are disjoint

from the 6 nodes in that plane.The intersection ofI9with thesymmetrichyperplanesP4(C)is a nontic threefoldK9. It intersects the Nieto quintic (the intersection ofI5with the same hyperplane) in the union of 15 + 15 planes, the c-planes mentioned above, and the so-called Segre planes, each containing 3 of the nodes and 4 of the 10 isolated double points of N

5, thec-planes being double in the intersection.Exceptional Lie groups and some related geometryOctober 8, 2021 23 / 26

More geometry ofI9Restricted to these planes, 5 of the octics vanish and the remaining one splits o these 4 lines, leaving a quartic curve in each of the planes. These planes are the so-callc-planes, and the 4 lines intersect at 6 of

the 36 singular points of intersection.There are 270c-planes, each the intersection of 6 of the 36 root

hyperplanes. Hence270
4=1080is the degree of the union of these

smooth quartics in thec-planes.The16intersection points on each quartic with the 4 lines are disjoint

from the 6 nodes in that plane.The intersection ofI9with thesymmetrichyperplanesP4(C)is a nontic threefoldK9. It intersects the Nieto quintic (the intersection ofI5with the same hyperplane) in the union of 15 + 15 planes, the c-planes mentioned above, and the so-called Segre planes, each containing 3 of the nodes and 4 of the 10 isolated double points of N

5, thec-planes being double in the intersection.Exceptional Lie groups and some related geometryOctober 8, 2021 23 / 26

More geometry ofI9Restricted to these planes, 5 of the octics vanish and the remaining one splits o these 4 lines, leaving a quartic curve in each of the planes. These planes are the so-callc-planes, and the 4 lines intersect at 6 of

the 36 singular points of intersection.There are 270c-planes, each the intersection of 6 of the 36 root

hyperplanes. Hence270
4=1080is the degree of the union of these

smooth quartics in thec-planes.The16intersection points on each quartic with the 4 lines are disjoint

from the 6 nodes in that plane.The intersection ofI9with thesymmetrichyperplanesP4(C)is a nontic threefoldK9. It intersects the Nieto quintic (the intersection ofI5with the same hyperplane) in the union of 15 + 15 planes, the c-planes mentioned above, and the so-called Segre planes, each containing 3 of the nodes and 4 of the 10 isolated double points of N

5, thec-planes being double in the intersection.Exceptional Lie groups and some related geometryOctober 8, 2021 23 / 26

Part of a bigger story?

The proof thatN5is (birational to) a ball quotient requires manipulations of Chern numbers onN5and a branched coverW!N5ofN5. The branch locus of that cover is contained in the union of thec-planes and the Segre planes, i.e., in the intersectionN5\K9. This means that the computation done already forN5applies immediately toK9! Apart from

the singular quartics ofK9in each of thec-planes, ...A conjecture onK9There is a coverY!P4(C)which restricted toN5is the cover

mentioned above. Then the coverZ!K9, branched at the union of the c-planes and the Segre planes, is (birational to) a compactication of a ball quotient.A more daring conjecture LetVP4(C)be a symmetric variety which contains the Segre planes and/or thec-planes. Then the restriction ofYtoVis (birational to) a

ball quotient.Yis itself a 4-dimensional ball quotient.Exceptional Lie groups and some related geometryOctober 8, 2021 24 / 26

Part of a bigger story?

The proof thatN5is (birational to) a ball quotient requires manipulations of Chern numbers onN5and a branched coverW!N5ofN5. The branch locus of that cover is contained in the union of thec-planes and the Segre planes, i.e., in the intersectionN5\K9. This means that the computation done already forN5applies immediately toK9! Apart from

the singular quartics ofK9in each of thec-planes, ...A conjecture onK9There is a coverY!P4(C)which restricted toN5is the cover

mentioned above. Then the coverZ!K9, branched at the union of the c-planes and the Segre planes, is (birational to) a compactication of a ball quotient.A more daring conjecture LetVP4(C)be a symmetric variety which contains the Segre planes and/or thec-planes. Then the restriction ofYtoVis (birational to) a

ball quotient.Yis itself a 4-dimensional ball quotient.Exceptional Lie groups and some related geometryOctober 8, 2021 24 / 26

Part of a bigger story?

The proof thatN5is (birational to) a ball quotient requires manipulations of Chern numbers onN5and a branched coverW!N5ofN5. The branch locus of that cover is contained in the union of thec-planes and the Segre planes, i.e., in the intersectionN5\K9. This means that the computation done already forN5applies immediately toK9! Apart from

the singular quartics ofK9in each of thec-planes, ...A conjecture onK9There is a coverY!P4(C)which restricted toN5is the cover

mentioned above. Then the coverZ!K9, branched at the union of the c-planes and the Segre planes, is (birational to) a compactication of a ball quotient.A more daring conjecture LetVP4(C)be a symmetric variety which contains the Segre planes and/or thec-planes. Then the restriction ofYtoVis (birational to) a

ball quotient.Yis itself a 4-dimensional ball quotient.Exceptional Lie groups and some related geometryOctober 8, 2021 24 / 26

The big conjecture and beyond

A conjecture on the nonticI9...The nonticI9is birational to the compactication of a ball quotient. It

contains the following subball quotients.The 36 nonticsK9above.The 45P3's which are the intersectionI5\I9.The 36 exceptional divisors of the 36 nodes, each a copy of theSegre

cubic.The compactication locus is the image of the 120 singular lines in the birational model.... and beyond Assuming the above conjectures are correct, we would have the following situation: inP5(C)there are two hypersurfacesI5andI9which are (birational to) compactications of ball quotients; these meet in 45 linear

spacesP3(C)which are 3-dimensional subball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 25 / 26

The big conjecture and beyond

A conjecture on the nonticI9...The nonticI9is birational to the compactication of a ball quotient. It

contains the following subball quotients.The 36 nonticsK9above.The 45P3's which are the intersectionI5\I9.The 36 exceptional divisors of the 36 nodes, each a copy of theSegre

cubic.The compactication locus is the image of the 120 singular lines in the birational model.... and beyond Assuming the above conjectures are correct, we would have the following situation: inP5(C)there are two hypersurfacesI5andI9which are (birational to) compactications of ball quotients; these meet in 45 linear

spacesP3(C)which are 3-dimensional subball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 25 / 26

The big conjecture and beyond

A conjecture on the nonticI9...The nonticI9is birational to the compactication of a ball quotient. It

contains the following subball quotients.The 36 nonticsK9above.The 45P3's which are the intersectionI5\I9.The 36 exceptional divisors of the 36 nodes, each a copy of theSegre

cubic.The compactication locus is the image of the 120 singular lines in the birational model.... and beyond Assuming the above conjectures are correct, we would have the following situation: inP5(C)there are two hypersurfacesI5andI9which are (birational to) compactications of ball quotients; these meet in 45 linear

spacesP3(C)which are 3-dimensional subball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 25 / 26

The big conjecture and beyond

A conjecture on the nonticI9...The nonticI9is birational to the compactication of a ball quotient. It

contains the following subball quotients.The 36 nonticsK9above.The 45P3's which are the intersectionI5\I9.The 36 exceptional divisors of the 36 nodes, each a copy of theSegre

cubic.The compactication locus is the image of the 120 singular lines in the birational model.... and beyond Assuming the above conjectures are correct, we would have the following situation: inP5(C)there are two hypersurfacesI5andI9which are (birational to) compactications of ball quotients; these meet in 45 linear

spacesP3(C)which are 3-dimensional subball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 25 / 26

The big conjecture and beyond

A conjecture on the nonticI9...The nonticI9is birational to the compactication of a ball quotient. It

contains the following subball quotients.The 36 nonticsK9above.The 45P3's which are the intersectionI5\I9.The 36 exceptional divisors of the 36 nodes, each a copy of theSegre

cubic.The compactication locus is the image of the 120 singular lines in the birational model.... and beyond Assuming the above conjectures are correct, we would have the following situation: inP5(C)there are two hypersurfacesI5andI9which are (birational to) compactications of ball quotients; these meet in 45 linear

spacesP3(C)which are 3-dimensional subball quotients.Exceptional Lie groups and some related geometryOctober 8, 2021 25 / 26

Bigger and bigger...

On the ambient spaceP(L(TE6))It is natural then to wonder whetherP5(C)is a ball quotient such thatI5 andI9are codimension 1 subballs?... and one step further Move up toP6(C), which is the ambient space for the W(E7)-arrangement. Each of the 28 hyperplanes arising from the 28 weights contains exactly the conguration just considered. IsP6(C)a ball quotient with 28 subball quotientsP5(C)as above?The biggest challange to proving the above Currently our understanding of branched covers and birational geometry in dimension>3is not quite complete enough to describe explicitly what happens and what the birational models should precisely look like. Some progress here will be necessary.It's time for some pictures. Exceptional Lie groups and some related geometryOctober 8, 2021 26 / 26

Bigger and bigger...

On the ambient spaceP(L(TE6))It is natural then to wonder whetherP5(C)is a ball quotient such thatI5 andI9are codimension 1 subballs?... and one step further Move up toP6(C), which is the ambient space for the W(E7)-arrangement. Each of the 28 hyperplanes arising from the 28 weights contains exactly the conguration just considered. IsP6(C)a ball quotient with 28 subball quotientsP5(C)as above?The biggest challange to proving the above Currently our understanding of branched covers and birational geometry in dimension>3is not quite complete enough to describe explicitly what happens and what the birational models should precisely look like. Some progress here will be necessary.It's time for some pictures. Exceptional Lie groups and some related geometryOctober 8, 2021 26 / 26

Bigger and bigger...

On the ambient spaceP(L(TE6))It is natural then to wonder whetherP5(C)is a ball quotient such thatI5 andI9are codimension 1 subballs?... and one step further Move up toP6(C), which is the ambient space for the W(E7)-arrangement. Each of the 28 hyperplanes arising from the 28 weights contains exactly the conguration just considered. IsP6(C)a ball quotient with 28 subball quotientsP5(C)as above?The biggest challange to proving the above Currently our understanding of branched covers and birational geometry in dimension>3is not quite complete enough to describe explicitly what happens and what the birational models should precisely look like. Some progress here will be necessary.It's time for some pictures. Exceptional Lie groups and some related geometryOctober 8, 2021 26 / 26