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Quivers and relations via the Bruhat order of the symmetric group Daiva Pu cinskait e Florida Atlantic University Conference on Geometric Methods in Representation Theory Iowa November 22-24, 2014 Daiva Pu cinskait e Quivers and

  1. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(3) , � ) (1 , 2) (1 , 3) (2 , 3) (4321) (321) (231) (312) (213) (132) (21) (2134) (123) (12) (1234) ˙ σ ⊳ τ � Sym( n + 1) = Sym( n ) · ( k , n + 1) , σ · ( k , n + 1) ⊳τ · ( k , n + 1) � �� � 1 ≤ k ≤ n ( ..., ∗ , n +1 , ∗ ,... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  2. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (321) (231) (312) (213) (132) (21) (123) (12) (1234) ˙ σ ⊳ τ � Sym( n + 1) = Sym( n ) · ( k , n + 1) , σ · ( k , n + 1) ⊳τ · ( k , n + 1) � �� � 1 ≤ k ≤ n ( ..., ∗ , n +1 , ∗ ,... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  3. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  4. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ Φ n ( σ ) ⊳ Φ n ( τ ) , σ �→ ω n · σ · ω n Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  5. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ Φ n ( σ ) ⊳ Φ n ( τ ) , σ �→ ω n · σ · ω n Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  6. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(3) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 3 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ 3 Φ n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ Φ n ( σ ) ⊳ Φ n ( τ ) , σ �→ ω n · σ · ω n Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  7. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ 3 Φ 4 Φ n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ Φ n ( σ ) ⊳ Φ n ( τ ) , σ �→ ω n · σ · ω n Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  8. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ 3 Φ 4 Φ n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ Φ n ( σ ) ⊳ Φ n ( τ ) , σ �→ ω n · σ · ω n Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  9. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ 3 Φ 4 U n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ U n ( σ ) ⊲ U n ( τ ) , σ �→ ω n · σ Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  10. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Φ 3 Φ 4 U n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ U n ( σ ) ⊲ U n ( τ ) , σ �→ ω n · σ Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  11. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) U 4 (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) U 3 (213) (132) (21) (2134) (1324) (1243) U 2 (123) (12) (1234) Φ 3 Φ 4 U n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ U n ( σ ) ⊲ U n ( τ ) , σ �→ ω n · σ Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  12. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) Φ 4 (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) U 4 (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) U 3 (213) (132) (21) (2134) (1324) (1243) U 2 (123) (12) (1234) Φ 3 Φ 4 U n Sym( n ) − → Sym( n ) σ ⊳ τ ⇔ U n ( σ ) ⊲ U n ( τ ) , σ �→ ω n · σ Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  13. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  14. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  15. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  16. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (21) (2134) (1324) (1243) (123) (12) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  17. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (321) (3214) (2341) (2413) (3142) (4123) (1432) (231) (312) (2314) (3124) (2143) (1342) (1423) (213) (132) (2134) (1324) (1243) (123) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  18. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3214) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (2134) (1324) (1243) (123) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  19. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (4213) (3214) (2413) (4123) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1423) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  20. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (4312) (3241) (2431) (3412) (4213) (4132) (4213) (3412) (4132) (3214) (2413) (4123) (3142) (1432) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1423) (1342) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  21. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (4321) (3 , 4) (3421) (4231) (4312) (4312) (3421) (4231) (3241) (2431) (3412) (4213) (4132) (4213) (3412) (4132) (3241) (2431) (3214) (2413) (4123) (3142) (1432) (2341) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1423) (1342) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  22. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (4321) (3 , 4) (3421) (4231) (4312) (4312) (3421) (4231) (3241) (2431) (3412) (4213) (4132) (4213) (3412) (4132) (3241) (2431) (3214) (2413) (4123) (3142) (1432) (2341) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1423) (1342) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  23. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (4321) (3 , 4) (3421) (4231) (4312) (4312) (3421) (4231) (3241) (2431) (3412) (4213) (4132) (4213) (3412) (4132) (3241) (2431) (3214) (2413) (4123) (3142) (1432) (2341) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1423) (1342) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ ( k , . . . , n +1) · Sym( n ) Sym( n +1) = , ( k , . . . , n +1) · σ⊳ ( k , . . . , n +1) · τ � �� � 1 ≤ k ≤ n +1 ( ∗ , ··· , ∗ , k ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  24. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3214) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ Sym( n ) · ( k , . . . , n +1) Sym( n +1) = , σ · ( k , . . . , n +1) ⊳τ · ( k , . . . , n +1) � �� � 1 ≤ k ≤ n ( ... ∗ , n +1 , ∗ ... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  25. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3214) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ Sym( n ) · ( k , . . . , n +1) Sym( n +1) = , σ · ( k , . . . , n +1) ⊳τ · ( k , . . . , n +1) � �� � 1 ≤ k ≤ n +1 ( ... ∗ , n +1 , ∗ ... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  26. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (3 , 4) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3241) (3214) (2341) (3142) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1342) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ Sym( n ) · ( k , . . . , n +1) Sym( n +1) = , σ · ( k , . . . , n +1) ⊳τ · ( k , . . . , n +1) � �� � 1 ≤ k ≤ n +1 ( ... ∗ , n +1 , ∗ ... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  27. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (4321) (3 , 4) (3421) (4231) (4312) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3241) (2431) (3412) (4213) (4132) (3214) (2341) (3142) (2413) (1432) (4123) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1342) (1423) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ Sym( n ) · ( k , . . . , n +1) Sym( n +1) = , σ · ( k , . . . , n +1) ⊳τ · ( k , . . . , n +1) � �� � 1 ≤ k ≤ n +1 ( ... ∗ , n +1 , ∗ ... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  28. Bruhat order on Sym( n ) Example. Hasse-diagram on (Sym(4) , � ) (1 , 2) (1 , 3) (1 , 4) (2 , 3) (2 , 4) (4321) (4321) (3 , 4) (3421) (4231) (4312) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3241) (2431) (3412) (4213) (4132) (3214) (2341) (3142) (2413) (1432) (4123) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1342) (1423) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (2134) (1324) (1243) (1234) (1234) � ˙ σ ⊳ τ Sym( n ) · ( k , . . . , n +1) Sym( n +1) = , σ · ( k , . . . , n +1) ⊳τ · ( k , . . . , n +1) � �� � 1 ≤ k ≤ n +1 ( ... ∗ , n +1 , ∗ ... ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  29. The quiver Q ( n ) of A ( n ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  30. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  31. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  32. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  33. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) = Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  34. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → 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

  35. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  36. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  37. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  38. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) = µ ( U n ( σ ) , U n ( τ )) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  39. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) = µ ( U n ( σ ) , U n ( τ )) Q 1 ( n ) Q 1 ( n + 1) � Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  40. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) = µ ( U n ( σ ) , U n ( τ )) Q 1 ( n ) Q 1 ( n + 1) � µ ( σ, τ ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  41. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) = µ ( U n ( σ ) , U n ( τ )) Q 1 ( n ) Q 1 ( 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

  42. The quiver Q ( n ) of A ( n ) Q ( n ) = ( Q 0 ( n ) , Q 1 ( n )) Q 0 ( n ) ← → Sym( n ) Q 1 ( n ) � coefficients µ ( σ, τ ) of Kazhdan-Lusztig polynomials � �� � β � � µ ( σ, τ ) = β ∈ Q 1 ( n ) | σ → τ � � = µ ( τ, σ ) � 1 if σ ⊳ τ (Bruhat order), = 0 if σ and τ are incomparable , ? in general. = µ (Φ n ( σ ) , Φ n ( τ )) = µ ( U n ( σ ) , U n ( τ )) Q 1 ( n ) Q 1 ( 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

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

  44. 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

  45. Example. The quiver Q ( n ) of A ( n ) for n = 2 , 3 , 4 Q (2) Q (4) (21) (4321) (12) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) Q (3) (321) (3214) (2341) (2413) (3142) (4123) (1432) (312) (231) (2314) (3124) (2143) (1342) (1423) (132) (213) (2134) (1324) (1243) (123) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  46. Example. The quiver Q ( n ) of A ( n ) for n = 2 , 3 , 4 Q (2) Q (4) (21) (4321) (12) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) Q (3) (321) (3214) (2341) (2413) (3142) (4123) (1432) (312) (231) (2314) (3124) (2143) (1342) (1423) (132) (213) (2134) (1324) (1243) (123) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  47. Example. The quiver Q (5) of A (5) from Q (4) (4321) (3421) (4231) (4312) (3241) (2431) (3412) (4213) (4132) (3214) (2341) (2413) (3142) (4123) (1432) (2314) (3124) (2143) (1342) (1423) (2134) (1324) (1243) (1234) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  48. Example. The quiver Q (5) of A (5) from Q (4) (43215) (34215)(42315)(43125) (32415)(24315)(34125)(42135)(41325) (32145)(23415) (24135)(31425) (41235)(14325) (23145)(31245)(21435)(13425)(14235) (21345)(13245)(12435) (12345) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

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

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

  51. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  52. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  53. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  54. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  55. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  56. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  57. Example. The quiver Q (5) of A (5) from Q (4) (54321) (54312) (54231) (53421) (45321) (54213) (54132) (53412) (53241) (45312) (52431) (45231) (43521) (35 (54123)(53214)(53142)(35241)(51432)(45213)(45132)(53241)(43512)(35412)(43251)(52413)(42531)(2 (35214)(42513)(52143)(53124)(51423)(43152)(43215)(45123)(51342)(25341)(52314)(42351)(34512)(15432)(41532)(34251 (43125)(42315)(51324)(34215)(52134)(24513)(32514)(41523)(31542)(32451)(25314)(41352)(51243)(42153)(34152)(25143)(35124 (41253)(31524)(34125)(42135)(32415)(25134)(51234)(32154)(24315)(14352)(41325)(15324)(21543)(23451)(23514)(15243 (32145)(41325)(24135)(14253)(23415)(31254)(23154)(14325)(21534)(21453)(15234)(31425)(13524)(1 (31245) (23145) (21435) (14235) (21354) (13425) (13254) (12534) (12 (21345) (13245) (12435) (12354) (12345) Φ 5 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  58. Relations of A ( n ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  59. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  60. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  61. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  62. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  63. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  64. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i 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

  65. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  66. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  67. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  68. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  69. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ > τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  70. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i τ · · · ν 1 · · · ν m 2.2. σ < τ 1.2. σ � = τ σ τ ν i +1 · · · ν m σ ν i 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

  71. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i σ 1.1. σ = τ σ 2.1. σ > τ ν 1 · · · ν m ν i +1 · · · ν m τ ν 1 ν i · · · 1.2. σ � = τ σ τ ν i +1 · · · ν m ν i 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

  72. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 · · · ν m ν i +1 · · · ν m σ ν 1 ν i · · · 1.2. σ � = τ σ τ ν i +1 · · · ν m ν i 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

  73. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 ν i · · · 1.2. σ � = τ σ τ ν i +1 · · · ν m ν i 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

  74. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 1.2. σ � = τ σ τ ν i 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

  75. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 1.2. σ � = τ σ τ σ τ ν 1 ν i 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

  76. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 ν 1 1.2. σ � = τ σ τ σ τ σ τ ν 1 ν 2 ν i 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

  77. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 ν 1 ν 1 ν 2 1.2. σ � = τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν i 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

  78. Relations of A ( n ) A ( n ) is quadratic ⇒ I ( n ) = �{ ρ | ρ = � i c i ( σ → ν i → τ ) }� Let σ, τ ∈ Sym( n ) and { ν 1 , . . . , ν m } = { ν | σ → ν → τ ∈ Q ( n ) } � = ∅ ν i is a neighbour of σ and τ for any i 1. | l ( σ ) − l ( τ ) | = 0 2. | l ( σ ) − l ( τ ) | = 2 ν 1 · · · ν i τ 1.1. σ = τ σ < τ σ ν 1 ν 2 ν i +1 · · · ν m σ ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 ν i 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

  79. Relations of A ( n ) 1. | l ( σ ) − l ( τ ) | = 0 ν 1 · · · ν i 1.1. σ = τ σ ν i +1 · · · ν m ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  80. Relations of A ( n ) 1. | l ( σ ) − l ( τ ) | = 0 ν 1 · · · ν i 1.1. σ = τ σ ν i +1 · · · ν m ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 Lemma Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

  81. Relations of A ( n ) 1. | l ( σ ) − l ( τ ) | = 0 ν 1 · · · ν i 1.1. σ = τ σ ν i +1 · · · ν m ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 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

  82. Relations of A ( n ) 1. | l ( σ ) − l ( τ ) | = 0 ν 1 · · · ν i 1.1. σ = τ σ ν i +1 · · · ν m ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 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

  83. Relations of A ( n ) 1. | l ( σ ) − l ( τ ) | = 0 ν 1 · · · ν i 1.1. σ = τ σ ν i +1 · · · ν m ν 1 ν 1 ν 1 ν 2 ν 3 1.2. σ � = τ σ τ σ τ σ τ σ τ σ τ ν 1 ν 2 ν 3 ν 1 ν 2 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 c υ ( σ → υ → τ ) ∈ I ( n ) Daiva Puˇ cinskait˙ e Quivers and relations via the Bruhat order of the symmetric group

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