The Broue-Malle-Rouquier conjecture for the exceptionalgroups of rank 2

Ph.D thesis defense

supervised by Ivan Marin

Presented by Eirini Chavli

12 May 2016

Introduction

Real reflection group

Generalized braid group Iwahori Hecke algebra

Introduction

Introduction

Complex reflection group

Complex braid group Generic Hecke algebra

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections

i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.

G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,

where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex reflection groups

Definition

A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).

Examples:

Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸

mij−factors

= sjsi sj . . .︸ ︷︷ ︸mij−factors

〉

G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.

Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.

By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.

Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .

1→ π1(X , x)→ π1(X/W , x)→W → 1

Definition (Broue-Malle-Rouquier 1998)

The complex braid group B associated to W is the fundamental groupπ1(X/W , x).

Examples

For the symmetric group Sn, B is the usual braid group on n strands.

For any finite Coxeter group B is a generalized braid group.

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).

We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

x

H

•

•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

x

H

•

•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

x

H

•

•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

x

H

•

•x0

•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

x

H

•H⊥

•x0

•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

s · x

x

H

•

•H⊥

••

x0

s · x0•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

s · x

x

H

•

•H⊥

••

x0

s · x0•0

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

s · x

x

H

•

•H⊥

••

x0

s · x0•0

The complex braid group B is generated by distinguished braidedreflections, whose image inside W are the distinguished pseudo-reflectionsthat generate W .

Complex braid groups

Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:

s · x

x

H

•

•H⊥

••

x0

s · x0•0

The complex braid group B is generated by distinguished braidedreflections, whose image inside W are the distinguished pseudo-reflectionsthat generate W .

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).

o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

Generic Hecke algebras

Let (us,j)1≤j≤o(s) be a set of indeterminates, where

s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.

Let R := Z[u±1s,1 , ..., u±1s,o(s)].

Definition

The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.

Examples

G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉

HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏

j=1

(σi − uj) = 0〉

G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉

HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏

j=1

(σ1 − uj) =4∏

j=1

(σ2 − vj) = 0〉

The BMR freeness conjecture

Let W be a complex reflection group of rank n.

Conjecture (Broue-Malle-Rouquier 1998)

HW is a free R-module of rank |W |.

Proposition (Broue-Malle-Rouquier 1998)

If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.

Every complex reflection group is a direct product of so-called irreducibleones.

Conjecture (equivalent form)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

The BMR freeness conjecture

Let W be a complex reflection group of rank n.

Conjecture (Broue-Malle-Rouquier 1998)

HW is a free R-module of rank |W |.

Proposition (Broue-Malle-Rouquier 1998)

If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.

Every complex reflection group is a direct product of so-called irreducibleones.

Conjecture (equivalent form)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

The BMR freeness conjecture

Let W be a complex reflection group of rank n.

Conjecture (Broue-Malle-Rouquier 1998)

HW is a free R-module of rank |W |.

Proposition (Broue-Malle-Rouquier 1998)

If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.

Every complex reflection group is a direct product of so-called irreducibleones.

Conjecture (equivalent form)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

The BMR freeness conjecture

Let W be a complex reflection group of rank n.

Conjecture (Broue-Malle-Rouquier 1998)

HW is a free R-module of rank |W |.

Proposition (Broue-Malle-Rouquier 1998)

If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.

Every complex reflection group is a direct product of so-called irreducibleones.

Conjecture (equivalent form)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

The BMR freeness conjecture

Let W be a complex reflection group of rank n.

Conjecture (Broue-Malle-Rouquier 1998)

HW is a free R-module of rank |W |.

Proposition (Broue-Malle-Rouquier 1998)

If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.

Every complex reflection group is a direct product of so-called irreducibleones.

Conjecture (equivalent form)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to

The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).

The finite set of 34 exceptions.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to

The infinite family G (de, e, n) of groups of n × n monomialmatrices

such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).

The finite set of 34 exceptions.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to

The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C)

andtheir product to µd(C).

The finite set of 34 exceptions.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to

The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).

The finite set of 34 exceptions.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to

The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).

The finite set of 34 exceptions.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

We have some recent results from Marin and Pfeiffer (2015).

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

The finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

We have some recent results from Marin and Pfeiffer (2015).

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

A finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

We have some recent results from Marin and Pfeiffer (2015).

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

A finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

We have some recent results from Marin and Pfeiffer (2015).

Marin has proved the case of G26 (2012).

Classification of irreducible complex reflection groups

The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to

The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)

A finite set of 34 exceptions.

We only have to deal with the 34 exceptional groups, nicknamed as

G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.

Among them there are 6 finite Coxeter groups.

We have some recent results from Marin and Pfeiffer (2015).

Marin has proved the case of G26 (2012).

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.

It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.

X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.

X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.

X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.

X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.

A first approach: The finite quotients of Bn

Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations

si si+1si = si+1si si+1, for i = 1, ..., n − 1

si sj = sjsi , for |i − j | > 1

Theorem (Coxeter 1957)

The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1

k + 1n >

12 .

n = 2. Then, W is a cyclic group.X

k = 2. Then, W = Sn.X

n = 5 and k = 3. Then W = G32.X (Marin 2011)

n = 4 and k = 3. Then W = G25.X (Broue- Malle 1994, Marin 2011)

n = 3 and k = 3, 4, 5. Then W = G41, G8, G16. X (C. 2013)

1The case of G4 has been proven by Broue - Malle 1994, Berceanu - Funar 1994,Marin 2011.

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.

Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).

Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.

We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir

inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1

The cases of G4, G8, G16

Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn

(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1︸ ︷︷ ︸U

Since 1 ∈ U, it is enough to prove that σiU ⊂ U i.e. it is enough to provethat σ2U ⊂ U.

The cases of G4, G8, G16

Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn

(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1︸ ︷︷ ︸U

Since 1 ∈ U, it is enough to prove that σiU ⊂ U

i.e. it is enough to provethat σ2U ⊂ U.

The cases of G4, G8, G16

Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.

Gn =⊔w∈J

W1 · w ·W1.

For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn

(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.

HGn

?=

∑w∈J

u1 · Tfw · u1︸ ︷︷ ︸U

Since 1 ∈ U, it is enough to prove that σiU ⊂ U i.e. it is enough to provethat σ2U ⊂ U.

An application: A classification of small dimensionalirreducible representations of B3

There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:

If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .

Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .

Example: The irreducible representations of dimension 3

A =

λ3 0 0λ1λ3 + λ22 λ2 0

λ2 1 λ1

, B =

λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3

,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:

If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .

Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .

Example: The irreducible representations of dimension 3

A =

λ3 0 0λ1λ3 + λ22 λ2 0

λ2 1 λ1

, B =

λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3

,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:

If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .

Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .

Example: The irreducible representations of dimension 3

A =

λ3 0 0λ1λ3 + λ22 λ2 0

λ2 1 λ1

, B =

λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3

,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:

If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .

Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .

Example: The irreducible representations of dimension 3

A =

λ3 0 0λ1λ3 + λ22 λ2 0

λ2 1 λ1

, B =

λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3

,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Questions

What is the reason we do not expect to have similar results for anydimension beyond 5?

Why are the matrices in this neat form?

What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?

An application: A classification of small dimensionalirreducible representations of B3

Questions

What is the reason we do not expect to have similar results for anydimension beyond 5?

Why are the matrices in this neat form?

What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?

An application: A classification of small dimensionalirreducible representations of B3

Questions

What is the reason we do not expect to have similar results for anydimension beyond 5?

Why are the matrices in this neat form?

What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .

By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.

ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2),

(ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.

Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u

±k ].

Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .

We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra,

i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e.

t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Assumption

Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:

t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .

The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.

Theorem (Malle 1999)

There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.

Theorem (Geck 1990)

t =∑

χ∈Irr(KHk )

1

sχχ,

where sχ is the Schur element of χ with respect to t.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

An application: A classification of small dimensionalirreducible representations of B3

Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.

Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where

V is a finite-dimensional KHk -module.

We can define the decomposition map dθ associated to θ.

R+0 (KHk)

dθ−→ R+0 (CHk)

Let S be an irreducible k-dimensional CHk module.

Theorem (C. 2014)

There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].

θ(sχM) 6= 0.

The cases of G4, G8, G16

Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .

We know that Z (B3) = 〈z〉.

HG4 = u1 + u1z−1 + u1u2u1

HG8 = u1 + u1z + u1z−1 + u1z

−2 + u1u2u1 + u1σ−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1

HG16 = u1 + u1z + u1z−1 + u1z

2 + u1z−2 + u1z

3 + u1z−3+

+u1z4 + u1z

−4 + u1z−5 + u1u2u1 + u1σ

−12 σ2

1σ−12 u1+

+u1σ2σ−21 σ2u1 + u1σ

22σ

21σ

22u1 + u1σ

−22 σ−21 σ−22 u1+

+u1σ2σ−21 σ2

2u1 + u1σ−12 σ2

1σ−22 u1 + u1σ

−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1 + u1σ

−22 σ−21 σ2

2u1 + u1σ22σ

21σ−22 u1+

+u1σ22σ−21 σ2

2u1 + u1σ−22 σ2

1σ−22 u1 + u1σ

−22 σ1σ

−12 u1+

+u1σ−12 σ1σ

−22 u1 + u1σ

−22 σ2

1σ−12 σ1σ

−12 u1+

+u1σ22σ−21 σ2σ

−11 σ2u1 + u1σ2σ

−21 σ2

2σ−21 σ2

2u1++u1σ

−12 σ2

1σ−22 σ2

1σ−22 u1

The cases of G4, G8, G16

Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.

HG4 = u1 + u1z−1 + u1u2u1

HG8 = u1 + u1z + u1z−1 + u1z

−2 + u1u2u1 + u1σ−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1

HG16 = u1 + u1z + u1z−1 + u1z

2 + u1z−2 + u1z

3 + u1z−3+

+u1z4 + u1z

−4 + u1z−5 + u1u2u1 + u1σ

−12 σ2

1σ−12 u1+

+u1σ2σ−21 σ2u1 + u1σ

22σ

21σ

22u1 + u1σ

−22 σ−21 σ−22 u1+

+u1σ2σ−21 σ2

2u1 + u1σ−12 σ2

1σ−22 u1 + u1σ

−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1 + u1σ

−22 σ−21 σ2

2u1 + u1σ22σ

21σ−22 u1+

+u1σ22σ−21 σ2

2u1 + u1σ−22 σ2

1σ−22 u1 + u1σ

−22 σ1σ

−12 u1+

+u1σ−12 σ1σ

−22 u1 + u1σ

−22 σ2

1σ−12 σ1σ

−12 u1+

+u1σ22σ−21 σ2σ

−11 σ2u1 + u1σ2σ

−21 σ2

2σ−21 σ2

2u1++u1σ

−12 σ2

1σ−22 σ2

1σ−22 u1

The cases of G4, G8, G16

Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.

HG4 = u1 + u1z−1 + u1u2u1

HG8 = u1 + u1z + u1z−1 + u1z

−2 + u1u2u1 + u1σ−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1

HG16 = u1 + u1z + u1z−1 + u1z

2 + u1z−2 + u1z

3 + u1z−3+

+u1z4 + u1z

−4 + u1z−5 + u1u2u1 + u1σ

−12 σ2

1σ−12 u1+

+u1σ2σ−21 σ2u1 + u1σ

22σ

21σ

22u1 + u1σ

−22 σ−21 σ−22 u1+

+u1σ2σ−21 σ2

2u1 + u1σ−12 σ2

1σ−22 u1 + u1σ

−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1 + u1σ

−22 σ−21 σ2

2u1 + u1σ22σ

21σ−22 u1+

+u1σ22σ−21 σ2

2u1 + u1σ−22 σ2

1σ−22 u1 + u1σ

−22 σ1σ

−12 u1+

+u1σ−12 σ1σ

−22 u1 + u1σ

−22 σ2

1σ−12 σ1σ

−12 u1+

+u1σ22σ−21 σ2σ

−11 σ2u1 + u1σ2σ

−21 σ2

2σ−21 σ2

2u1++u1σ

−12 σ2

1σ−22 σ2

1σ−22 u1

The cases of G4, G8, G16

Let ui be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.

HG4 = u1 + u1z−1 + u1u2u1

HG8 = u1 + u1z + u1z−1 + u1z

−2 + u1u2u1 + u1σ−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1

HG16 = u1 + u1z + u1z−1 + u1z

2 + u1z−2 + u1z

3 + u1z−3+

+u1z4 + u1z

−4 + u1z−5 + u1u2u1 + u1σ

−12 σ2

1σ−12 u1+

+u1σ2σ−21 σ2u1 + u1σ

22σ

21σ

22u1 + u1σ

−22 σ−21 σ−22 u1+

+u1σ2σ−21 σ2

2u1 + u1σ−12 σ2

1σ−22 u1 + u1σ

−12 σ1σ

−12 u1+

+u1σ2σ−11 σ2u1 + u1σ

−22 σ−21 σ2

2u1 + u1σ22σ

21σ−22 u1+

+u1σ22σ−21 σ2

2u1 + u1σ−22 σ2

1σ−22 u1 + u1σ

−22 σ1σ

−12 u1+

+u1σ−12 σ1σ

−22 u1 + u1σ

−22 σ2

1σ−12 σ1σ

−12 u1+

+u1σ22σ−21 σ2σ

−11 σ2u1 + u1σ2σ

−21 σ2

2σ−21 σ2

2u1++u1σ

−12 σ2

1σ−22 σ2

1σ−22 u1

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.

We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C))

' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).

Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.

The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...,G15.

Icosahedral family G16, ...,G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...G11, ...,G15.

Icosahedral family G16, ...,G19, ...G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

The exceptional groups of rank 2

Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:

Tetrahedral family G4, ...,G7.

Octahedral family G8, ...G11, ...,G15.

Icosahedral family G16, ...,G19, ...G22.

In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .

Proposition (C. 2013)

If HW is torsion free and the BMR conjecture is true for HW

, then theconjecture holds for HW , as well.

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1,

i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).

Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].

A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies

mij∏k=1

(YiYj − tij ,k) = 0 〉

A+(C ) = 〈Aij := YiYj , i 6= j〉

Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:

< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1

(A21 − t21,1) . . . (A21 − t21,4) = 0 >

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3.

Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3.

Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

A more global approach: Etingof-Rains 2004

For every reduced word wx that represents x inside W , we denote by Twx

the corresponding element in A+(C ).

Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.

Proposition (Etingof-Rains 2004)

A+(C ) is generated as R-module by the elements Twx , x ∈W .

Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .

Proposition (Etingof-Rains 2004)

There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .

wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3

From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)

Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :

wx =

{wx for x = 1

wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.

wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :

`(wx) = `(wx), where `(w) denotes the length of the word w .

Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .

wx = wx if and only if wx = wx . In particular, w1 = w1.

wx = y2y1y1y2y2y1y1y3

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

RzkTwx ,

where Twx is the corresponding element in A+(C ).

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

RzkTwx ,

where Twx is the corresponding element in A+(C ).

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

RzkTwx ,

where Twx is the corresponding element in A+(C ).

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

Rzk Twx ,

where Twx is the corresponding element in A+(C )⊗θ R+.

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

RzkΦ(Twx ),

where Twx is the corresponding element in A+(C )⊗θ R+.

Theorem (C. 2015)

HW = U, and, hence, the BMR freeness conjecture is true for the groupsbelonging to the first two families.

The conjecture for the first two families

Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.

For every x ∈ W we found a specific word wx and we set

U :=∑x∈W

|Z(W )|−1∑k=0

RzkΦ(Twx ),

where Twx is the corresponding element in A+(C )⊗θ R+.

Theorem (C. 2015)

HW = U, and, hence, the BMR freeness conjecture is true for the groupsbelonging to the first two families.

Remaining cases

Conjecture (equivalent version)

HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.

G4, ...,G7︸ ︷︷ ︸T

, G8, . . . ,G15︸ ︷︷ ︸O

, G16,G17,G18,G19,G20,G21,G22︸ ︷︷ ︸I

, G23,G24, ...G36,G37.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2),

where u2 is the subalgebra of HG6 generated

by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉

HG9 =7∑

k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra

of HG9 generated by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉

HG9 =7∑

k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra

of HG9 generated by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉

HG9 =7∑

k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra

of HG9 generated by σ2.

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉

HG9 =7∑

k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra

of HG9 generated by σ2.

• G17 = 〈s1, s2 | s21 = s52 = 1, (s1s2)3 = (s2s1)3〉

HG17

?=

19∑k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1 + zk

5∑i=1

u2wi ), where u2 is

the subalgebra of HG17 generated by σ2 and wi the image of an element ofB inside HG17 .

Open questions

• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉

HG6 =3∑

k=0

(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated

by σ2.

• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉

HG9 =7∑

k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra

of HG9 generated by σ2.

• G17 = 〈s1, s2 | s21 = s52 = 1, (s1s2)3 = (s2s1)3〉

HG17

?=

19∑k=0

(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1 + zk

5∑i=1

u2wi ), where u2 is

the subalgebra of HG17 generated by σ2 and wi the image of an element ofB inside HG17 .

Open questions

• G7 = 〈s1, s2, s3 | s21 = s32 = s33 = 1, s1s2s3 = s2s3s1 = s3s1s2〉

HG7 =11∑k=0

(zku3u2 + zkσ2σ−13 u2), where u2 and u3 are the subalgebras of

HG7 generated by σ2 and σ3, respectively.

• G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉

HG11 =23∑k=0

(zku3u2 + zku3σ2σ−13 u2), where u2 and u3 are the subalgebras

of HG7 generated by σ2 and σ3, respectively.

• G19 = 〈s1, s2, s3 | s21 = s32 = s53 = 1, s1s2s3 = s2s3s1 = s3s1s2〉

HG19

?=

59∑k=0

(zku3u2 + zku3σ2σ−13 u2 + zk

2∑i=1

u3wiu2), where u2 and u3 are

the subalgebras of HG19 generated by σ2 and σ3, respectively and wi theimage of an element of B inside HG19 .