Braid group actions on D

Braid group actions on D

(O)

Fabian Lenzen

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Irving’s shuffling functor) Endofunctors ofO₀ with properties:

exact sequence idO0 ⇒Θs ⇒Shs ⇒0,

ShsM(w ·0) =M(ws ·0),

Θ_sM(w·0) = Θ_sM(ws·0),

Θ²_s = Θ_s⊕Θ_s.

Theorem (Rouquier)

Br_n acts on D^b(O₀)via LSh_s.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(E_i,E_i)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(Ei,Ei)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(Ei,Ei)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(Ei,Ei)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(Ei,Ei)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Setup

principal block O₀ of BGG category for sl_n.

Vermas M(w ·0), simplesL(w ·0), projectives P(w·0).

Definition (Seidel and Thomas’ spherical twist)

{E_i} ⊆D^b(O₀) is an An-collectionofd -spherical objects if

Hom^∗(Ei,Ei)∼=C[ε]/(ε²) withεof degreed,

Hom^∗(Ei,F)×Hom^d−∗(F,Ei)−→^◦ Hom^d(Ei,Ei)/(idε) non-degenerate

dim Hom^∗(Ei,Ej) =n_{1 if|i−j|= 1,}

0 o/w.

Definetwist functor T_E:F 7→cone(hom^•(E,F)⊗F −→^ev F)

Theorem (Seidel, Thomas) Br_n acts on D^b(O₀)via T_Ei.

Braid group actions onD^b(O)

Question

Are thereEi’s such thatTEi ∼=LShsi?

Example (spherical objects in D^b(O₀) for sl₂) P(s1) has C[ε]/(ε²) as endomorphisms:

L(s1) L(e) L(s1)

L(e) L(s1)

L(e) L(s1)

L(s1)

P(s₁) P(e) P(s₁)

Hence{P(s₁)} is an A₁-collection ford = 0.

Braid group actions onD^b(O)

Question

Are thereEi’s such thatTEi ∼=LShsi?

Example (spherical objects in D^b(O₀) for sl₂) P(s₁) has C[ε]/(ε²) as endomorphisms:

L(s1) L(e) L(s1)

L(e) L(s1)

L(e) L(s1)

L(s1)

P(s₁) P(e) P(s₁)

Hence{P(s₁)} is an A₁-collection ford = 0.

Braid group actions onD^b(O)

Results

Start withO₀ forsl₂:

Theorem

{P(s)}is 0-spherical and T_P(s)∼=LSh_s[−1]

{L(e)} is 2-spherical and T_L(e)∼=LShs

as auto-equivalences of D^b(O₀).

Braid group actions onD^b(O)

Results

Start withO₀ forsl₂: Theorem

{P(s)}is 0-spherical and T_P(s)∼=LSh_s[−1]

{L(e)} is 2-spherical and T_L(e) ∼=LShs

as auto-equivalences of D^b(O₀).

Braid group actions onD^b(O)

Results

Start withO₀ forsl₂: Theorem

{P(s)}is 0-spherical and T_P(s)∼=LSh_s[−1]

{L(e)} is 2-spherical and T_L(e) ∼=LShs

as auto-equivalences of D^b(O₀).

Braid group actions onD^b(O)

Results

Start withO₀ forsl₂: Caveat

Remain not spherical under sl₂,→sl_n:

L(s) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

L(t) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

L(s) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

P(s) P(t) P(s)

D^b(O₀(sl₂))( @ (D^b(O₀(sl₃)) in which P(s) is spherical.

Braid group actions onD^b(O)

Results

Start withO₀ forsl₂: Caveat

Remain not spherical under sl₂,→sl_n:

L(s) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

L(t) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

L(s) L(st) L(ts)

L(w0) L(e)

L(s) L(t) L(st) L(ts)

L(w0)

P(s) P(t) P(s)

D^b(O₀(sl2))( @ (D^b(O₀(sl3)) in which P(s) is spherical.

Braid group actions onD^b(O)

Results

Pass to parabolic categoryO^p:

Theorem

For maximal parabolic subalgebrap=

∗ ∗ ··· ∗ ... ... ... ...

∗ ∗ ··· ∗ 0∗ ··· ∗

⊂sl_n,

{P^p(s1), . . . ,P^p(s1· · ·sn−1)}is an An−2-collection.

T_P^p_(s1···si) ∼=LSh_si[−1].

• −1 ←−−→− • −←−−→− · · · −←−−→−ⁿ⁻¹• .











•1 = 0,

• → • → •= 0,

• ← • ← •= 0,

•i = •ⁱ











Braid group actions onD^b(O)

Results

Pass to parabolic categoryO^p: Theorem

For maximal parabolic subalgebrap=

∗ ∗ ··· ∗ ... ... ... ...

∗ ∗ ··· ∗ 0∗ ··· ∗

⊂sl_n,

{P^p(s1), . . . ,P^p(s1· · ·sn−1)}is an An−2-collection.

T_Pp(s1···s_i) ∼=LSh_si[−1].

• −1 ←−−→− • −←−−→− · · · −←−−→−ⁿ⁻¹• .











•1 = 0,

• → • → •= 0,

• ← • ← •= 0,

•i = •ⁱ











Braid group actions onD^b(O)

Results

Pass to parabolic categoryO^p: Theorem

For maximal parabolic subalgebrap=

∗ ∗ ··· ∗ ... ... ... ...

∗ ∗ ··· ∗ 0∗ ··· ∗

⊂sl_n,

{P^p(s1), . . . ,P^p(s1· · ·sn−1)}is an An−2-collection.

T_Pp(s1···s_i) ∼=LSh_si[−1].

• −1 ←−−→− • −←−−→− · · · −←−−→−ⁿ⁻¹• .











•1 = 0,

• → • → •= 0,

• ← • ← •= 0,

•i = •ⁱ











Braid group actions onD^b(O)

Results

Pass to parabolic categoryO^p: Caveat

Condition is necessary:

there is no A_n-collection of projective objects for other maximal parabolic subalgebras.

Example: p=

_{∗ ∗ ∗ ∗}

∗ ∗ ∗ ∗ 0 0∗ ∗ 0 0∗ ∗

L(ts)L(tsut)L(tu) L(t)

L(tsu) L(e)

L(t) L(tsut)

L(t) L(tsut)

L(ts) L(tu)

L(tsu) L(tsu)

L(ts) L(tsu)

L(tu)L(tsut) L(tsu)

L(ts)L(tsut)L(tu) L(t)

L(tsu) L(e)

L(t) L(tsut)

P^p(t) P^p(tsu) P^p(t)

Braid group actions onD^b(O)

Results

Pass to parabolic categoryO^p: Caveat

Condition is necessary:

there is no A_n-collection of projective objects for other maximal parabolic subalgebras.

Example: p=

_{∗ ∗ ∗ ∗}

∗ ∗ ∗ ∗ 0 0∗ ∗ 0 0∗ ∗

L(ts)L(tsut)L(tu) L(t)

L(tsu) L(e)

L(t) L(tsut)

L(t) L(tsut)

L(ts) L(tu)

L(tsu) L(tsu)

L(ts) L(tsu)

L(tu)L(tsut) L(tsu)

L(ts)L(tsut)L(tu) L(t)

L(tsu) L(e)

L(t) L(tsut)

P^p(t) P^p(tsu) P^p(t)