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Combinatorial interpretations in affine Coxeter groups Christopher R. H. Hanusa Queens College, CUNY Joint work with Brant C. Jones, James Madison University Coxeter Groups Interpretations Application Future work What is a Coxeter group? A

  1. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups (Finite) n -permutations S n An n -permutation is a permutation of { 1 , 2 , . . . , n } , (e.g. 2 1 4 5 3 6). Every n -permutation is a product of adjacent transpositions . ◮ s i : ( i ) ↔ ( i + 1). (e.g. s 4 = 1 2 3 5 4 6). Example. Write 2 1 4 5 3 6 as s 3 s 4 s 1 . This is a Coxeter group: ◮ Generators: s 1 , . . . , s n − 1 ◮ s i s j = s j s i when | i − j | ≥ 2 (commutation relation) Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 4 / 37

  2. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups (Finite) n -permutations S n An n -permutation is a permutation of { 1 , 2 , . . . , n } , (e.g. 2 1 4 5 3 6). Every n -permutation is a product of adjacent transpositions . ◮ s i : ( i ) ↔ ( i + 1). (e.g. s 4 = 1 2 3 5 4 6). Example. Write 2 1 4 5 3 6 as s 3 s 4 s 1 . This is a Coxeter group: ◮ Generators: s 1 , . . . , s n − 1 ◮ s i s j = s j s i when | i − j | ≥ 2 (commutation relation) ◮ s i s j s i = s j s i s j when | i − j | = 1 (braid relation) Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 4 / 37

  3. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups (Finite) n -permutations S n An n -permutation is a permutation of { 1 , 2 , . . . , n } , (e.g. 2 1 4 5 3 6). Every n -permutation is a product of adjacent transpositions . ◮ s i : ( i ) ↔ ( i + 1). (e.g. s 4 = 1 2 3 5 4 6). Example. Write 2 1 4 5 3 6 as s 3 s 4 s 1 . This is a Coxeter group: ◮ Generators: s 1 , . . . , s n − 1 ◮ s i s j = s j s i when | i − j | ≥ 2 (commutation relation) ◮ s i s j s i = s j s i s j when | i − j | = 1 (braid relation) s 1 s 2 s 3 ... s n � 2 s n � 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 4 / 37

  4. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n ◮ Generators: s 0 , s 1 , . . . , s n − 1 s 0 ◮ Relations: s 1 s 2 s 3 ... s n � 2 s n � 1 ◮ s 0 has a braid relation with s 1 and s n − 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 5 / 37

  5. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n ◮ Generators: s 0 , s 1 , . . . , s n − 1 s 0 ◮ Relations: s 1 s 2 s 3 ... s n � 2 s n � 1 ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact 1-line notation? Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 5 / 37

  6. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n ◮ Generators: s 0 , s 1 , . . . , s n − 1 s 0 ◮ Relations: s 1 s 2 s 3 ... s n � 2 s n � 1 ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact 1-line notation? ◮ Perhaps interchanges 1 and n ? Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 5 / 37

  7. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n ◮ Generators: s 0 , s 1 , . . . , s n − 1 s 0 ◮ Relations: s 1 s 2 s 3 ... s n � 2 s n � 1 ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact 1-line notation? ◮ Perhaps interchanges 1 and n ? ◮ Not quite! (Would add a relation) Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 5 / 37

  8. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n ◮ Generators: s 0 , s 1 , . . . , s n − 1 s 0 ◮ Relations: s 1 s 2 s 3 ... s n � 2 s n � 1 ◮ s 0 has a braid relation with s 1 and s n − 1 s 0 ◮ How does this impact 1-line notation? s 1 s n � 1 ◮ Perhaps interchanges 1 and n ? ◮ Not quite! (Would add a relation) s 2 s n � 2 ◮ Better to view graph as: ◮ Every generator is the same. s 3 ... Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 5 / 37

  9. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n (G. Lusztig 1983, H. Eriksson, 1994) w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 6 / 37

  10. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n (G. Lusztig 1983, H. Eriksson, 1994) w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Generators transpose infinitely many pairs of entries: s i : (i) ↔ (i+1) . . . ( n + i ) ↔ ( n + i + 1) . . . ( − n + i ) ↔ ( − n + i + 1) . . . In � S 4 , · · · w (-4) w (-3) w (-2) w (-1) w (0) w (1) w (2) w (3) w (4) w (5) w (6) w (7) w (8) w (9) · · · s 1 · · · -4 -2 -3 -1 0 2 1 3 4 6 5 7 8 10 · · · s 0 · · · -3 -4 -2 -1 1 0 2 3 5 4 6 7 9 8 · · · Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 6 / 37

  11. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n (G. Lusztig 1983, H. Eriksson, 1994) w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Generators transpose infinitely many pairs of entries: s i : (i) ↔ (i+1) . . . ( n + i ) ↔ ( n + i + 1) . . . ( − n + i ) ↔ ( − n + i + 1) . . . In � S 4 , · · · w (-4) w (-3) w (-2) w (-1) w (0) w (1) w (2) w (3) w (4) w (5) w (6) w (7) w (8) w (9) · · · s 1 · · · -4 -2 -3 -1 0 2 1 3 4 6 5 7 8 10 · · · s 0 · · · -3 -4 -2 -1 1 0 2 3 5 4 6 7 9 8 · · · s 1 s 0 · · · -2 -4 -3 -1 2 0 1 3 6 4 5 7 10 8 · · · Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 6 / 37

  12. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n (G. Lusztig 1983, H. Eriksson, 1994) w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Generators transpose infinitely many pairs of entries: s i : (i) ↔ (i+1) . . . ( n + i ) ↔ ( n + i + 1) . . . ( − n + i ) ↔ ( − n + i + 1) . . . In � S 4 , · · · w (-4) w (-3) w (-2) w (-1) w (0) w (1) w (2) w (3) w (4) w (5) w (6) w (7) w (8) w (9) · · · s 1 · · · -4 -2 -3 -1 0 2 1 3 4 6 5 7 8 10 · · · s 0 · · · -3 -4 -2 -1 1 0 2 3 5 4 6 7 9 8 · · · s 1 s 0 · · · -2 -4 -3 -1 2 0 1 3 6 4 5 7 10 8 · · · Symmetry: Can think of as integers wrapped around a cylinder. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 6 / 37

  13. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n (G. Lusztig 1983, H. Eriksson, 1994) w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Generators transpose infinitely many pairs of entries: s i : (i) ↔ (i+1) . . . ( n + i ) ↔ ( n + i + 1) . . . ( − n + i ) ↔ ( − n + i + 1) . . . In � S 4 , · · · w (-4) w (-3) w (-2) w (-1) w (0) w (1) w (2) w (3) w (4) w (5) w (6) w (7) w (8) w (9) · · · s 1 · · · -4 -2 -3 -1 0 2 1 3 4 6 5 7 8 10 · · · s 0 · · · -3 -4 -2 -1 1 0 2 3 5 4 6 7 9 8 · · · s 1 s 0 · · · -2 -4 -3 -1 2 0 1 3 6 4 5 7 10 8 · · · Symmetry: Can think of as integers wrapped around a cylinder. w is defined by the window [ � � w (1) , � w (2) , . . . , � w ( n )]. s 1 s 0 = [0 , 1 , 3 , 6] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 6 / 37

  14. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 S 3 s 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  15. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 S 3 s 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  16. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 S 3 s 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  17. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 � S 3 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  18. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 � S 3 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  19. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 � S 3 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  20. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n s 2 � S 3 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  21. Coxeter Groups Interpretations Application Future work Examples of Coxeter groups Affine n -Permutations � S n — elements correspond to alcoves. s 2 � S 3 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 7 / 37

  22. Coxeter Groups Interpretations Application Future work Properties of Coxeter groups For a elements w in a Coxeter group W , ◮ w may have multiple expressions. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 8 / 37

  23. Coxeter Groups Interpretations Application Future work Properties of Coxeter groups For a elements w in a Coxeter group W , ◮ w may have multiple expressions. ◮ Transfer between them using relations. Example. In S 4 , w = s 1 s 2 s 3 s 1 = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 = s 2 s 1 s 2 s 3 s 1 s 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 8 / 37

  24. Coxeter Groups Interpretations Application Future work Properties of Coxeter groups For a elements w in a Coxeter group W , ◮ w may have multiple expressions. ◮ Transfer between them using relations. Example. In S 4 , w = s 1 s 2 s 3 s 1 = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 = s 2 s 1 s 2 s 3 s 1 s 1 ◮ w has a shortest expression (this length: Coxeter length ) Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 8 / 37

  25. Coxeter Groups Interpretations Application Future work Properties of Coxeter groups For a elements w in a Coxeter group W , ◮ w may have multiple expressions. ◮ Transfer between them using relations. Example. In S 4 , w = s 1 s 2 s 3 s 1 = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 = s 2 s 1 s 2 s 3 s 1 s 1 ◮ w has a shortest expression (this length: Coxeter length ) For a Coxeter group � W , ◮ An induced subgraph of � W ’s Coxeter graph is a subgroup W w ∈ � w = w 0 w , where ◮ Every element � W can be written � w 0 ∈ � W / W is a coset representative and w ∈ W . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 8 / 37

  26. Coxeter Groups Interpretations Application Future work S n as a subgroup of � S n Key concept: View S n as a subgroup of � S n . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 9 / 37

  27. Coxeter Groups Interpretations Application Future work S n as a subgroup of � S n Key concept: View S n as a subgroup of � S n . w = w 0 w , where w 0 ∈ � ◮ Write � S n / S n and w ∈ S n . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 9 / 37

  28. Coxeter Groups Interpretations Application Future work S n as a subgroup of � S n Key concept: View S n as a subgroup of � S n . w = w 0 w , where w 0 ∈ � ◮ Write � S n / S n and w ∈ S n . ◮ w 0 determines the entries; w determines their order. w = [ − 11 , 20 , − 3 , 4 , 11 , 0] ∈ � Example. For � S 6 , w 0 = [ − 11 , − 3 , 0 , 4 , 11 , 20] and w = [1 , 3 , 6 , 4 , 5 , 2] . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 9 / 37

  29. Coxeter Groups Interpretations Application Future work S n as a subgroup of � S n Key concept: View S n as a subgroup of � S n . w = w 0 w , where w 0 ∈ � ◮ Write � S n / S n and w ∈ S n . ◮ w 0 determines the entries; w determines their order. w = [ − 11 , 20 , − 3 , 4 , 11 , 0] ∈ � Example. For � S 6 , w 0 = [ − 11 , − 3 , 0 , 4 , 11 , 20] and w = [1 , 3 , 6 , 4 , 5 , 2] . Many interpretations of these minimal length coset representatives . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 9 / 37

  30. Coxeter Groups Interpretations Application Future work Combinatorial interpretations of � S n / S n � � 4, � 3,7,10 � window notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram elements of � � S n S n bounded core partition partition root lattice point � � 1,2,1, � 2 � Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 10 / 37

  31. Coxeter Groups Interpretations Application Future work An abacus model for � S n / S n (James and Kerber, 1981) Given w 0 = [ w 1 , . . . , w n ] ∈ � S n / S n , ◮ Place integers in n runners . � 15 � 14 � 13 � 12 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 11 / 37

  32. Coxeter Groups Interpretations Application Future work An abacus model for � S n / S n (James and Kerber, 1981) Given w 0 = [ w 1 , . . . , w n ] ∈ � S n / S n , ◮ Place integers in n runners . � 15 � 14 � 13 � 12 ◮ Circled: beads . Empty: gaps � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 11 / 37

  33. Coxeter Groups Interpretations Application Future work An abacus model for � S n / S n (James and Kerber, 1981) Given w 0 = [ w 1 , . . . , w n ] ∈ � S n / S n , ◮ Place integers in n runners . � 15 � 14 � 13 � 12 ◮ Circled: beads . Empty: gaps � 11 � 10 � 9 � 8 ◮ Bijection: Given w 0 , create � 7 � 6 � 5 � 4 an abacus where each runner � 3 � 2 � 1 0 has a lowest bead at w i . 1 2 3 4 Example: [ − 4 , − 3 , 7 , 10] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 11 / 37

  34. Coxeter Groups Interpretations Application Future work An abacus model for � S n / S n (James and Kerber, 1981) Given w 0 = [ w 1 , . . . , w n ] ∈ � S n / S n , ◮ Place integers in n runners . � 15 � 14 � 13 � 12 ◮ Circled: beads . Empty: gaps � 11 � 10 � 9 � 8 ◮ Bijection: Given w 0 , create � 7 � 6 � 5 � 4 an abacus where each runner � 3 � 2 � 1 0 has a lowest bead at w i . 1 2 3 4 Example: [ − 4 , − 3 , 7 , 10] 5 6 7 8 9 10 11 12 These abaci are flush and balanced. 13 14 15 16 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 11 / 37

  35. Coxeter Groups Interpretations Application Future work An abacus model for � S n / S n (James and Kerber, 1981) Given w 0 = [ w 1 , . . . , w n ] ∈ � S n / S n , ◮ Place integers in n runners . � 15 � 14 � 13 � 12 ◮ Circled: beads . Empty: gaps � 11 � 10 � 9 � 8 ◮ Bijection: Given w 0 , create � 7 � 6 � 5 � 4 an abacus where each runner � 3 � 2 � 1 0 has a lowest bead at w i . 1 2 3 4 Example: [ − 4 , − 3 , 7 , 10] 5 6 7 8 9 10 11 12 These abaci are flush and balanced. 13 14 15 16 The generators act nicely on the abacus. 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 11 / 37

  36. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  37. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Example: Consider [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . Start with id= [1 , 2 , 3 , 4] and apply the generators one by one: � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 [1 , 2 , 3 , 4] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  38. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Example: Consider [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . Start with id= [1 , 2 , 3 , 4] and apply the generators one by one: � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 � 3 � 2 � 1 0 s 0 → 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 [1 , 2 , 3 , 4] [0 , 2 , 3 , 5] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  39. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Example: Consider [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . Start with id= [1 , 2 , 3 , 4] and apply the generators one by one: � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 s 0 s 1 → → 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 [1 , 2 , 3 , 4] [0 , 2 , 3 , 5] [0 , 1 , 3 , 6] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  40. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Example: Consider [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . Start with id= [1 , 2 , 3 , 4] and apply the generators one by one: � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 s 0 s 1 s 3 → → → 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 13 14 15 16 [1 , 2 , 3 , 4] [0 , 2 , 3 , 5] [0 , 1 , 3 , 6] [ − 1 , 1 , 4 , 6] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  41. Coxeter Groups Interpretations Application Future work Action of generators on the abacus ◮ s i acts by interchanging runners i and i + 1. ◮ s 0 acts by interchanging runners 1 and n , with level shifts. Example: Consider [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . Start with id= [1 , 2 , 3 , 4] and apply the generators one by one: � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 � 3 � 2 � 1 0 s 0 s 1 s 3 s 0 → → → → 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 9 10 11 12 9 10 11 12 9 10 11 12 13 14 15 16 13 14 15 16 13 14 15 16 13 14 15 16 13 14 15 16 [1 , 2 , 3 , 4] [0 , 2 , 3 , 5] [0 , 1 , 3 , 6] [ − 1 , 1 , 4 , 6] [ − 1 , 0 , 5 , 6] Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 12 / 37

  42. Coxeter Groups Interpretations Application Future work Combinatorial interpretations of � S n / S n � � 4, � 3,7,10 � window notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram elements of � � S n S n bounded core partition partition root lattice point � � 1,2,1, � 2 � Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 13 / 37

  43. Coxeter Groups Interpretations Application Future work Integer partitions and n -core partitions For an integer partition λ = ( λ 1 , . . . , λ k ) drawn as a Ferrers diagram, The hook length of a box is # boxes below and to the right. 10 9 6 5 2 1 7 6 3 2 6 5 2 1 3 2 2 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 14 / 37

  44. Coxeter Groups Interpretations Application Future work Integer partitions and n -core partitions For an integer partition λ = ( λ 1 , . . . , λ k ) drawn as a Ferrers diagram, The hook length of a box is # boxes below and to the right. 10 9 6 5 2 1 7 6 3 2 6 5 2 1 3 2 2 1 An n-core is a partition with no boxes of hook length dividing n . Example. λ is a 4-core, 8-core, 11-core, 12-core, etc. λ is NOT a 1-, 2-, 3-, 5-, 6-, 7-, 9-, or 10-core. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 14 / 37

  45. Coxeter Groups Interpretations Application Future work Core partitions for � S n / S n Elements of � S n / S n are in bijection with n -cores. Bijection: { abaci } ← → { n -cores } Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 15 / 37

  46. Coxeter Groups Interpretations Application Future work Core partitions for � S n / S n Elements of � S n / S n are in bijection with n -cores. Bijection: { abaci } ← → { n -cores } Rule: Read the boundary steps of λ from the abacus: ◮ A bead ↔ vertical step ◮ A gap ↔ horizontal step � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 ← → 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 15 / 37

  47. Coxeter Groups Interpretations Application Future work Core partitions for � S n / S n Elements of � S n / S n are in bijection with n -cores. Bijection: { abaci } ← → { n -cores } Rule: Read the boundary steps of λ from the abacus: ◮ A bead ↔ vertical step ◮ A gap ↔ horizontal step � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 ← → 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Fact: Abacus flush with n -runners ↔ partition is n -core. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 15 / 37

  48. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  49. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 1 0 1 2 3 0 1 removes all removable 1-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  50. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 0 0 1 2 3 0 1 removes all removable 0-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  51. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 2 0 1 2 3 0 1 removes all removable 2-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  52. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 1 0 1 2 3 0 1 removes all removable 1-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  53. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 3 0 1 2 3 0 1 removes all removable 3-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  54. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 2 0 1 2 3 0 1 removes all removable 2-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  55. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 0 0 1 2 3 0 1 removes all removable 0-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  56. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 3 0 1 2 3 0 1 removes all removable 3-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  57. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 1 0 1 2 3 0 1 removes all removable 1-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  58. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : Applying generator s 0 0 1 2 3 0 1 removes all removable 0-boxes. 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  59. Coxeter Groups Interpretations Application Future work Action of generators on the core partition ◮ Label the boxes of λ with residues. ◮ s i acts by adding or removing boxes with residue i . Example: Let’s see the deconstruction of s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 : 0 1 2 3 0 1 3 0 1 2 3 0 2 3 0 1 2 3 1 2 3 0 1 2 0 1 2 3 0 1 3 0 1 2 3 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 16 / 37

  60. Coxeter Groups Interpretations Application Future work Combinatorial interpretations of � S n / S n � � 4, � 3,7,10 � window notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram elements of � � S n S n bounded core partition partition root lattice point � � 1,2,1, � 2 � Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 17 / 37

  61. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  62. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Elements of � S n / S n are in bijection with ( n − 1)-bounded partitions. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  63. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Elements of � S n / S n are in bijection with ( n − 1)-bounded partitions. Bijection: (Lapointe, Morse, 2005) { n -cores λ } ↔ { ( n − 1)-bounded partitions β } Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  64. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Elements of � S n / S n are in bijection with ( n − 1)-bounded partitions. Bijection: (Lapointe, Morse, 2005) { n -cores λ } ↔ { ( n − 1)-bounded partitions β } ◮ Remove all boxes of λ with hook length ≥ n Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  65. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Elements of � S n / S n are in bijection with ( n − 1)-bounded partitions. Bijection: (Lapointe, Morse, 2005) { n -cores λ } ↔ { ( n − 1)-bounded partitions β } ◮ Remove all boxes of λ with hook length ≥ n ◮ Left-justify remaining boxes. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  66. Coxeter Groups Interpretations Application Future work Bounded partitions for � S n / S n A partition β = ( β 1 , . . . , β k ) is b-bounded if β i ≤ b for all i . Elements of � S n / S n are in bijection with ( n − 1)-bounded partitions. Bijection: (Lapointe, Morse, 2005) { n -cores λ } ↔ { ( n − 1)-bounded partitions β } ◮ Remove all boxes of λ with hook length ≥ n ◮ Left-justify remaining boxes. 10 9 6 5 2 1 7 6 3 2 − → − → 6 5 2 1 3 2 2 1 λ = (6 , 4 , 4 , 2 , 2) β = (2 , 2 , 2 , 2 , 2) Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 18 / 37

  67. Coxeter Groups Interpretations Application Future work Canonical reduced expression for � S n / S n Given the bounded partition, read off the reduced expression: Method: (Berg, Jones, Vazirani, 2009) ◮ Fill β with residues i ◮ Tally s i reading right-to-left in rows from bottom-to-top Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 19 / 37

  68. Coxeter Groups Interpretations Application Future work Canonical reduced expression for � S n / S n Given the bounded partition, read off the reduced expression: Method: (Berg, Jones, Vazirani, 2009) ◮ Fill β with residues i ◮ Tally s i reading right-to-left in rows from bottom-to-top Example. [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 0 1 1 2 3 4 3 0 − → − → − → 5 6 7 8 2 3 9 10 11 12 1 2 13 14 15 16 0 1 17 18 19 20 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 19 / 37

  69. Coxeter Groups Interpretations Application Future work Canonical reduced expression for � S n / S n Given the bounded partition, read off the reduced expression: Method: (Berg, Jones, Vazirani, 2009) ◮ Fill β with residues i ◮ Tally s i reading right-to-left in rows from bottom-to-top Example. [ − 4 , − 3 , 7 , 10] = s 1 s 0 s 2 s 1 s 3 s 2 s 0 s 3 s 1 s 0 . � 7 � 6 � 5 � 4 � 3 � 2 � 1 0 0 1 1 2 3 4 3 0 − → − → − → 5 6 7 8 2 3 9 10 11 12 1 2 13 14 15 16 0 1 17 18 19 20 ◮ The Coxeter length of w is the number of boxes in β . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 19 / 37

  70. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  71. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  72. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Question: What is s 1 s 2 s 1 in 1-line notation? Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  73. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Question: What is s 1 s 2 s 1 in 1-line notation? Answer: 1 2 3 4 5 6 . . . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  74. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Question: What is s 1 s 2 s 1 in 1-line notation? Answer: 2 1 3 4 5 6 . . . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  75. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Question: What is s 1 s 2 s 1 in 1-line notation? Answer: 2 3 1 4 5 6 . . . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  76. Coxeter Groups Interpretations Application Future work Fully commutative elements Definition. An element in a Coxeter group is fully commutative if it has only one reduced expression (up to commutation relations). NO BRAIDS ALLOWED! Example. In S 4 , s 1 s 2 s 3 s 1 is not fully commutative because OK BAD = s 1 s 2 s 1 s 3 = s 2 s 1 s 2 s 3 s 1 s 2 s 3 s 1 Question: What is s 1 s 2 s 1 in 1-line notation? Answer: 3 2 1 4 5 6 . . . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 20 / 37

  77. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in S n ? Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 21 / 37

  78. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in S n ? Answer: Catalan many! S 1 : 1. id S 2 : 2. id, s 1 S 3 : 5. id, s 1 , s 2 , s 1 s 2 , s 2 s 1 S 4 : 14. id, s 1 , s 2 , s 3 , s 1 s 2 , s 2 s 1 , s 2 s 3 , s 3 s 2 , s 1 s 3 , s 1 s 2 s 3 , s 1 s 3 s 2 , s 2 s 1 s 3 , s 3 s 2 s 1 , s 2 s 1 s 3 s 2 Key idea: (Billey, Jockusch, Stanley, 1993) w is fully commutative ⇐ ⇒ w is 321-avoiding. (Knuth, 1973) These are counted by the Catalan numbers. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 21 / 37

  79. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in � S n ? Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 22 / 37

  80. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in � S n ? (Even in � Answer: Infinitely many! S 3 .) id , s 1 , s 1 s 2 , s 1 s 2 s 0 , s 1 s 2 s 0 s 1 , s 1 s 2 s 0 s 1 s 2 , . . . Multiplying the generators cyclically does not introduce braids. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 22 / 37

  81. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in � S n ? (Even in � Answer: Infinitely many! S 3 .) id , s 1 , s 1 s 2 , s 1 s 2 s 0 , s 1 s 2 s 0 s 1 , s 1 s 2 s 0 s 1 s 2 , . . . Multiplying the generators cyclically does not introduce braids. This is not the right question. Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 22 / 37

  82. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in � S n , with Coxeter length ℓ ? s 0 s 0 s 1 s 0 s 2 s 0 s 1 s 2 s 0 s 2 s 1 In � S 3 : id , , , , . . . s 1 s 1 s 0 s 1 s 2 s 1 s 0 s 2 s 1 s 2 s 0 s 2 s 2 s 0 s 2 s 1 s 2 s 0 s 1 s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 23 / 37

  83. Coxeter Groups Interpretations Application Future work Enumerating fully commutative elements Question: How many fully commutative elements are there in � S n , with Coxeter length ℓ ? s 0 s 0 s 1 s 0 s 2 s 0 s 1 s 2 s 0 s 2 s 1 In � S 3 : id , , , , . . . s 1 s 1 s 0 s 1 s 2 s 1 s 0 s 2 s 1 s 2 s 0 s 2 s 2 s 0 s 2 s 1 s 2 s 0 s 1 s 2 s 1 s 0 Question: Determine the coefficient of q ℓ in the generating function � q ℓ ( w ) . f n ( q ) = w ∈ f S FC e n f 3 ( q ) = 1 q 0 + 3 q 1 + 6 q 2 + 6 q 3 + . . . Combinatorial interpretations in affine Coxeter groups Binghamton University Combinatorics Seminar Christopher R. H. Hanusa Queens College, CUNY May 12, 2011 23 / 37

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