Action of GL2

Action of GL ₂ ( Z ) on the Drinfeld double of a finite group

Michel Brou´ e (Universit´ e Paris-Diderot Paris 7)

“Finite Chevalley groups, reflection groups and braid groups”

Centre Bernoulli, September 2016 in honour of

(and Fran¸ cois Digne)

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

The algebra D _k G

Let G be a finite group and let k be a field (at the beginning a commutative ring, at the end a large enough characteristic zero field).

Let F (G , k) be the k -algebra of functions on G with values in k. Thus F (G , k) = M

s∈G

kδ

where (δ

)

is a family of orthogonal idempotents . G acts on F (G , k ) (g δ

g

= δ

), and we set

D

G := F (G , k) o G . Thus

D

G = M

s,g∈G

kδ

g with δ

g δ

h =

( δ

gh if s = gtg

0 if not.

and the unity element of D

G is 1

= P

s∈G

δ

.

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

The center ZD _k G

• Let Com(G × G ) denote the set of commuting pairs in G , and let C(G × G) denote the set of orbits of Com(G × G ) under conjugation by G .

• For Γ ∈ C(G × G ), we set S

:= P

(s,g)∈Γ

δ

g .

Then (S

)

is a basis of ZD

G.

• The first projection G × G → G , (s, g) 7→ s induces a bijection C(G × G ) −→ {(s

, C ) |

}/G-conjugation , and the k-linear map

M

s∈[Cl(G)]

ZkC

(s ) → ZD

G , z

7→ Tr

G(s)

(δ

z

) is an isomorphism.

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

The abelian category _D

_G mod

Objects: the G -graded kG-modules X = L

s∈G s

X .

Morphisms: the kG -morphisms X → Y such that

X →

Y .

The following functors define “inverse” equivalences of abelian categories:



 



 



DkG

mod → M

s∈[Cl(G)]

kC_G(s)

mod , M

s∈G

s

X 7→ M

s∈[Cl(G)]

s

X ,

M

s∈[Cl(G)]

kC_G(s)

mod →

mod , M

s∈[Cl(G)]

S

7→ M

s∈[Cl(G)]

Ind

S

.

⇒ D

G is a symmetric algebra.

Actually,

(

0 if

1 if

= 1

is a symmetrizing form.

⇒ The map z

7→ Tr

G(s)

(δ

z

) induces an algebra isomorphism M

s∈[Cl(G)]

Tr

(δ

·) : M

s∈[Cl(G)]

ZkC

(s) → ZD

G .

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

D

G mod as a ribbon category

Tensor product:

X ⊗ Y = M

s∈G

s

(X ⊗ Y ) where

(X ⊗ Y ) := M

t,u|tu=s

t

X ⊗

Y .

Dual: X

= M

s∈G

s

(X

) where

(X

) := (

X )

,

with obvious evaluation

and coevaluation

.

Braiding: c

: X ⊗ Y −→

Y ⊗ X , (

t

X ⊗

Y →

Y ⊗

X x ⊗ y 7→ ty ⊗ x

Twist: θ

: X −→

X , (

s

X →

X x 7→ sx

We have θ

= θ

· θ

· c

· c

.

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

Graded characters and Grothendieck ring

From now on, K is a characteristic zero field, which contains the |G |-th roots of unity.

Graded character:

For X a D

G -module, we set grchar

:= P

s∈G

tr(· |

X )s , that is grchar

(t) = P

s∈C_G(t)

tr(t |

X )s ∈ ZKC

(t) and

grchar

(t ) = grchar

(t)grchar

(t ) . If Y = X (t, T ), define

σ

:







Gr(D

G ) → K ,

X 7→ σ(X ) = ω

(grchar

(t)) Then σ

is a ring morphism.

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

D

G mod as a modular category

The S-matrix:

S

:= 1

|G | tr(c

· c

| X ⊗ Y )

.

t

X ⊗

Y contributes to the trace of c

· c

only if tu = ut , in which case c

· c

:

(

t

X ⊗

Y →

X ⊗

Y x ⊗ y 7→ ux ⊗ ty .

DKG

mod is a modular category since S is nondegenerate.

For X , Y ∈ Irr(D

G ),

σ

(X ) = |G | χ

(1) S

.

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

Now just for fun: name, where, when ?

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

Two bases of CF(D _K G )

CF(D

G ) has two bases, both partitioned according to s ∈ [Cl(G)] :

• (γ

)

,

where Γ = Γ(s, C) for s ∈ [Cl(G )] and C ∈ Cl(C

(s)), and γ

:= γ

denotes the characteristic function of Γ,

• (χ

)

,

where X = Ind

S for s ∈ [Cl(G )] and S ∈ Irr

(C

(s)), and we set χ

:= χ

.

From one basis to the other:











χ_s,S

=

g∈[Cl(C_G(s)]

χ_S

(g)γ

g),

γ_Γ(s,g)

=

)|

|G|

X

S∈Irr_K(C_G(s))

χ_S

(g

)χ

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

Action of GL ₂ ( Z ) on CF(D _K G )

• Let S be the endomorphism of CF(D

G ) such that S : χ

7→ X

Y∈Irr(D_KG)

S

· χ

for X ∈ Irr(D

G ) .

• Let Ω be the endomorphism of CF(D

G ) such that Ω : χ

7→ θ

· χ

for X ∈ Irr(D

G) .

• Let ∆

(n ∈ ( Z /|G | Z )

) be the endomorphism of CF(D

G ) such that

∆

: χ

7→ χ

for X ∈ Irr(D

G ) .

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

Action of GL ₂ ( Z ) on CF(D _K G ), continued

• GL

( Z ) acts on the set Com(G × G ) of commuting pairs of elements of G :

a b c d

· (s, g ) := (s

g

, s

g

) ,

⇒ hence on the set C(G × G ) of its orbits under G ,

⇒ hence on the basis (γ

)

of CF(D

G).

Theorem



 

 

 

 

 

 

 

 

 

 

S : γ

7→ γ

hence acts like 0 1

−1 0

!

Ω : γ

7→ γ

hence acts like 1 0

−1 1

!

∆

: γ

7→ γ

hence acts like 1 0 0 n

!

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group

Action of GL ₂ ( Z ) on CF(D _K G ), continued

Hence

the group hS, Ωi acts like SL

( Z /|G | Z ), hS, Ω, ∆

i acts like GL

( Z /|G | Z ).

• S

corresponds to permutation matrices

χ

7→ χ

and γ

7→ χ

,

• For Sh := SΩS

, we have

Ω · Sh · Ω = Sh · Ω · Sh .

• Verlinde formula: If X ⊗ Y ' L

Z

z

, then N

= X

X

S

S

S

S

∈ N .

Michel Brou´e Action ofGL₂(Z)on the Drinfeld double of a finite group

Again !

Michel Brou´e Action ofGL( )on the Drinfeld double of a finite group