Quivers and relations via the Bruhat order of the symmetric group - - PowerPoint PPT Presentation

Quivers and relations via the Bruhat order of the symmetric group

Daiva Puˇ cinskait˙ e

Florida Atlantic University

Conference on Geometric Methods in Representation Theory

Iowa November 22-24, 2014

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0}

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j + 1) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk ⊳ τ σ ⊳ ν ⇔ ν = (i, j) · σ and l(ν) = l(σ) + 1

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j + 1) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk ⊳ τ σ ⊳ ν ⇔ ν = (i, j) · σ and l(ν) = l(σ) + 1

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j + 1) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk ⊳ τ σ ⊳ ν ⇔ ν = (i, j) · σ and l(ν) = l(σ) + 1

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Bruhat order on Sym(n):

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk−1 ⊳ νk = τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk−1 ⊳ νk = τ σ ⊳ ν

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk−1 ⊳ νk = τ σ ⊳ ν ⇔ ν = (i, j) · σ and l(ν) = l(σ) + 1

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Lie-algebra sln(C)

sln := sln(C) = {C = (cij)ij ∈ Matn(C) | tr(C) = 0} sln =

 

i<j

C·Eij  

• (sln)+

⊕

n−1

• i=1

C·(Eii − Ei+1,i+1)

• hn

⊕

 

i>j

C·Eij  

• (sln)−

W (n) =

• sij := sαij | i < j
• ←

→ Sym(n) = {(i, j) | i < j}

sij → (i, j) α12 α13 . . . α1n α23 . . . α2n · · · . . . αn−1n (1, 2) (1, 3) . . . (1, n) (2, 3) . . . (2, n) · · · . . . (n-1, n)

Bruhat order on Sym(n): σ < τ if σ ⊳ ν1 ⊳ · · · ⊳ νk−1 ⊳ νk = τ σ ⊳ ν ⇔ ν = (i, j) · σ and l(ν) = l(σ) + 1

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(2), )

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(2), )

(1, 2)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(2), )

(1, 2)

(1234) (4321) (123) (321) (12) (21)

Sym(n + 1) = ˙

• 1≤k≤n

Sym(n)·(k, n + 1)

• (...,∗,n+1,∗,...)

, σ ⊳ τ σ·(k, n + 1)⊳τ ·(k, n + 1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(3), )

(1, 2) (1, 3) (2, 3)

(1234) (4321) (123) (321) (12) (21)

Sym(n + 1) = ˙

• 1≤k≤n

Sym(n)·(k, n + 1)

• (...,∗,n+1,∗,...)

, σ ⊳ τ σ·(k, n + 1)⊳τ ·(k, n + 1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(3), )

(1, 2) (1, 3) (2, 3)

(1234) (2134) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n + 1) = ˙

• 1≤k≤n

Sym(n)·(k, n + 1)

• (...,∗,n+1,∗,...)

, σ ⊳ τ σ·(k, n + 1)⊳τ ·(k, n + 1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n + 1) = ˙

• 1≤k≤n

Sym(n)·(k, n + 1)

• (...,∗,n+1,∗,...)

, σ ⊳ τ σ·(k, n + 1)⊳τ ·(k, n + 1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n)

Φn

− → Sym(n) σ → ωn·σ·ωn , σ ⊳ τ ⇔ Φn(σ) ⊳ Φn(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n)

Φn

− → Sym(n) σ → ωn·σ·ωn , σ ⊳ τ ⇔ Φn(σ) ⊳ Φn(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(3), )

Φ3 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Φn

− → Sym(n) σ → ωn·σ·ωn , σ ⊳ τ ⇔ Φn(σ) ⊳ Φn(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Φn

− → Sym(n) σ → ωn·σ·ωn , σ ⊳ τ ⇔ Φn(σ) ⊳ Φn(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Φn

− → Sym(n) σ → ωn·σ·ωn , σ ⊳ τ ⇔ Φn(σ) ⊳ Φn(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Un

− → Sym(n) σ → ωn·σ , σ ⊳ τ ⇔ Un(σ) ⊲ Un(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Un

− → Sym(n) σ → ωn·σ , σ ⊳ τ ⇔ Un(σ) ⊲ Un(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

U4

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4 U3 U2

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Un

− → Sym(n) σ → ωn·σ , σ ⊳ τ ⇔ Un(σ) ⊲ Un(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

Φ4 (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

U4

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321)

Φ4 U3 U2

(123) (213) (132) (312) (231) (321)

Φ3

(12) (21)

Sym(n)

Un

− → Sym(n) σ → ωn·σ , σ ⊳ τ ⇔ Un(σ) ⊲ Un(τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321) (12) (21)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (213) (132) (312) (231) (321)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (123) (2134) (1324) (3124) (2314) (3214)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213) (1342) (3142) (1432) (4132) (3412) (4312)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213) (1342) (3142) (1432) (4132) (3412) (4312) (2341) (3241) (2431) (4231) (3421) (4321)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213) (1342) (3142) (1432) (4132) (3412) (4312) (2341) (3241) (2431) (4231) (3421) (4321)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213) (1342) (3142) (1432) (4132) (3412) (4312) (2341) (3241) (2431) (4231) (3421) (4321)

Sym(n+1) = ˙

• 1≤k≤n+1

(k, . . . , n+1) · Sym(n)

• (∗,··· ,∗,k)

, σ ⊳ τ (k, . . . , n+1)·σ⊳(k, . . . , n+1)·τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214)

Sym(n+1) = ˙

• 1≤k≤n

Sym(n) · (k, . . . , n+1)

• (...∗,n+1,∗...)

, σ ⊳ τ σ·(k, . . . , n+1)⊳τ ·(k, . . . , n+1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214)

Sym(n+1) = ˙

• 1≤k≤n+1

Sym(n) · (k, . . . , n+1)

• (...∗,n+1,∗...)

, σ ⊳ τ σ·(k, . . . , n+1)⊳τ ·(k, . . . , n+1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1342) (3142) (2341) (3241)

Sym(n+1) = ˙

• 1≤k≤n+1

Sym(n) · (k, . . . , n+1)

• (...∗,n+1,∗...)

, σ ⊳ τ σ·(k, . . . , n+1)⊳τ ·(k, . . . , n+1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1342) (3142) (2341) (3241) (1423) (2413) (1432) (3412) (2431) (3421) (4123) (4213) (4132) (4312) (4231) (4321)

Sym(n+1) = ˙

• 1≤k≤n+1

Sym(n) · (k, . . . , n+1)

• (...∗,n+1,∗...)

, σ ⊳ τ σ·(k, . . . , n+1)⊳τ ·(k, . . . , n+1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Bruhat order on Sym(n)

• Example. Hasse-diagram on (Sym(4), )

(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) (1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1342) (3142) (2341) (3241) (1423) (2413) (1432) (3412) (2431) (3421) (4123) (4213) (4132) (4312) (4231) (4321)

Sym(n+1) = ˙

• 1≤k≤n+1

Sym(n) · (k, . . . , n+1)

• (...∗,n+1,∗...)

, σ ⊳ τ σ·(k, . . . , n+1)⊳τ ·(k, . . . , n+1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• Daiva Puˇ

cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) =

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order),

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable,

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ))

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ)) = µ(Un(σ), Un(τ))

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ)) = µ(Un(σ), Un(τ)) Q1(n)

• Q1(n + 1)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ)) = µ(Un(σ), Un(τ)) Q1(n)

• Q1(n + 1)

µ(σ, τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ)) = µ(Un(σ), Un(τ)) Q1(n)

• Q1(n + 1)

µ(σ, τ) = µ((k, . . . , n + 1)·σ, (k, . . . , n + 1)·τ)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

The quiver Q(n) of A(n)

Q(n) = (Q0(n), Q1(n)) Q0(n) ← → Sym(n) Q1(n) coefficients µ(σ, τ) of Kazhdan-Lusztig polynomials µ(σ, τ) =

• β ∈ Q1(n) | σ

β

→ τ

• =

µ(τ, σ) = 1

if σ ⊳ τ (Bruhat order), if σ and τ are incomparable, ? in general.

= µ(Φn(σ), Φn(τ)) = µ(Un(σ), Un(τ)) Q1(n)

• Q1(n + 1)

µ(σ, τ) = µ((k, . . . , n + 1)·σ, (k, . . . , n + 1)·τ) = µ(σ·(k, . . . , n + 1), τ ·(k, . . . , n + 1))

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Example.

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(n) of A(n) for n = 2, 3, 4

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(n) of A(n) for n = 2, 3, 4

Q(2)

(21) (12)

Q(3)

(321) (312) (231) (132) (213) (123)

Q(4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(n) of A(n) for n = 2, 3, 4

Q(2)

(21) (12)

Q(3)

(321) (312) (231) (132) (213) (123)

Q(4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(1234) (2134) (1324) (1243) (2314) (3124) (2143) (1342) (1423) (3214) (2341) (2413) (3142) (4123) (1432) (3241) (2431) (3412) (4213) (4132) (3421) (4231) (4312) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345)(13245)(12435) (23145)(31245)(21435)(13425)(14235) (32145)(23415) (24135)(31425) (41235)(14325) (32415)(24315)(34125)(42135)(41325) (34215)(42315)(43125) (43215) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(4, 5) · σ (12345) (21345)(13245)(12435) (23145)(31245)(21435)(13425)(14235) (32145)(23415) (24135)(31425) (41235)(14325) (32415)(24315)(34125)(42135)(41325) (34215)(42315)(43125) (43215) (12354) (21354)(13254)(12534) (23154)(31254)(21534)(13524)(15234) (32154)(23514) (25134)(31524) (51234)(15324) (32514)(25314)(35124)(52134)(51324) (35214)(52314)(53124) (53214) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(4, 5) · σ (3, 4) · σ (12345) (21345)(13245)(12435) (23145)(31245)(21435)(13425)(14235) (32145)(23415) (24135)(31425) (41235)(14325) (32415)(24315)(34125)(42135)(41325) (34215)(42315)(43125) (43215) (12354) (21354)(13254)(12534) (23154)(31254)(21534)(13524)(15234) (32154)(23514) (25134)(31524) (51234)(15324) (32514)(25314)(35124)(52134)(51324) (35214)(52314)(53124) (53214) ( (21453)( (24153)(41253)( (42153)(24513) (251 (42513)(25413)( (45213)( ( Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

• Example. The quiver Q(5) of A(5) from Q(4)

(12345) (21345) (13245) (12435) (12354) (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54312) (54231) (53421) (45321) (54321) Φ5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1· · ·νm τ 2.2. σ > τ τ ν1· · ·νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)}

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1· · ·νm τ 2.2. σ > τ τ ν1· · ·νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 2.2. σ > τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 2.2. σ > τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 2.2. σ > τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 2.2. σ > τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 2.2. σ < τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 2.1. σ > τ σ ν1 · · · νm τ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 · · · νm σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 νi · · · νi+1 · · · νm τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 τ σ ν1 τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

A(n) is quadratic ⇒ I(n) = {ρ | ρ =

i ci(σ → νi → τ)}

Let σ, τ ∈ Sym(n) and {ν1, . . . , νm} = {ν | σ → ν → τ ∈ Q(n)} = ∅ νi is a neighbour of σ and τ for any i

• 1. |l(σ) − l(τ)| = 0
• 2. |l(σ) − l(τ)| = 2

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm σ < τ τ ν1 ν2 σ 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ νi not a neighbour of σ or τ for some i

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

• 1. |l(σ) − l(τ)| = 0

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

• 1. |l(σ) − l(τ)| = 0

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ Lemma

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

• 1. |l(σ) − l(τ)| = 0

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ Lemma The paths {σ→υ→τ | υ ⊳ σ, τ} are linearly independent in A(n).

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

• 1. |l(σ) − l(τ)| = 0

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ Lemma The paths {σ→υ→τ | υ ⊳ σ, τ} are linearly independent in A(n). Let ν ∈ Sym(n) with σ, τ ⊳ ν

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

• 1. |l(σ) − l(τ)| = 0

1.1. σ = τ σ ν1 νi · · · νi+1 · · · νm 1.2. σ = τ σ ν1 τ σ ν1 τ σ ν1 ν2 τ σ ν1 ν2 ν3 τ σ ν1 ν2 ν3 τ Lemma The paths {σ→υ→τ | υ ⊳ σ, τ} are linearly independent in A(n). Let ν ∈ Sym(n) with σ, τ ⊳ ν, then (σ → ν → τ) −

σ,τ⊲υ cυ(σ → υ → τ) ∈ I(n)

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Lemma

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Lemma Let σ, νi, τ ∈ Sym(n) and

• ci(σ → νi → τ) ∈ I(n).

Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Lemma Let σ, νi, τ ∈ Sym(n) and

• ci(σ → νi → τ) ∈ I(n). Then
• ci((σ, n + 1) → (νi, n + 1) → (τ, n + 1)) ∈ I(n + 1)
• ci(σ·(n, n + 1) → νi ·(n, n + 1) → τ ·(n, n + 1)) ∈ I(n + 1)

(1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1342) (3142) (2341) (3241) (1423) (2413) (1432) (3412) (2431) (3421) (4123) (4213) (4132) (4312) (4231) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Lemma Let σ, νi, τ ∈ Sym(n) and

• ci(σ → νi → τ) ∈ I(n). Then
• ci((σ, n + 1) → (νi, n + 1) → (τ, n + 1)) ∈ I(n + 1)
• ci(σ·(n, n + 1) → νi ·(n, n + 1) → τ ·(n, n + 1)) ∈ I(n + 1)

σ → (n, n + 1)·σ → σ ∈ I(n + 1)

(1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1342) (3142) (2341) (3241) (1423) (2413) (1432) (3412) (2431) (3421) (4123) (4213) (4132) (4312) (4231) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

Relations of A(n)

Lemma Let σ, νi, τ ∈ Sym(n) and

• ci(σ → νi → τ) ∈ I(n). Then
• ci((σ, n + 1) → (νi, n + 1) → (τ, n + 1)) ∈ I(n + 1)
• ci(σ·(n, n + 1) → νi ·(n, n + 1) → τ ·(n, n + 1)) ∈ I(n + 1)

σ → (n, n + 1)·σ → σ ∈ I(n + 1)

(1234) (2134) (1324) (3124) (2314) (3214) (1243) (2143) (1423) (4123) (2413) (4213) (1342) (3142) (1432) (4132) (3412) (4312) (2341) (3241) (2431) (4231) (3421) (4321) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group