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Introduction Finite Groups Infinite Groups Research Before the talk: Open these websites: http://sourceforge.net/apps/trac/groupexplorer/wiki/ The First Five Symmetric Groups/ Bring Zome Tools: Permutohedron Affine A 3 Combinatorial

  1. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . t t ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  2. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . t t ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  3. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = tsttst t ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  4. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = tsst t ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  5. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = tt t ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  6. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = id t Shows ( st ) 3 = id is natural. ts s s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  7. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = id t Shows ( st ) 3 = id is natural. ts s ◮ Our group has six elements: { id , s , t , st , ts , sts } . s st Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  8. Introduction Finite Groups Infinite Groups Research Reflection Groups ◮ These regions can be thought of as group elements. Place id. ◮ The action of multiplying (on the left) by a generator s ( s 2 = id) corresponds to a reflection across a hyperplane H s . We see: t ◮ sts = tst ↔ ststst = id t Shows ( st ) 3 = id is natural. ts s ◮ Our group has six elements: { id , s , t , st , ts , sts } . s ◮ This is the group of st symmetries of a hexagon. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 4 / 24

  9. Introduction Finite Groups Infinite Groups Research Reflection Groups t t ts s s st 3 , relation is ( st ) 3 = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | = 6. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  10. Introduction Finite Groups Infinite Groups Research Reflection Groups t s 4 , relation is ( st ) 4 = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | = 8. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  11. Introduction Finite Groups Infinite Groups Research Reflection Groups t s 5 , relation is ( st ) 5 = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =10. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  12. Introduction Finite Groups Infinite Groups Research Reflection Groups t s 6 , relation is ( st ) 6 = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =12. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  13. Introduction Finite Groups Infinite Groups Research Reflection Groups t s n , relation is ( st ) n = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =2 n . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  14. Introduction Finite Groups Infinite Groups Research Reflection Groups t s n , relation is ( st ) n = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =2 n . ◮ All finite reflection groups in the plane are these dihedral groups . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  15. Introduction Finite Groups Infinite Groups Research Reflection Groups t s n , relation is ( st ) n = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =2 n . ◮ All finite reflection groups in the plane are these dihedral groups . ◮ Two directions: infinite and higher dimensional. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  16. Introduction Finite Groups Infinite Groups Research Reflection Groups t s n , relation is ( st ) n = id. ◮ When the angle between H s and H t is π ◮ The size of the group is | S | =2 n . ◮ All finite reflection groups in the plane are these dihedral groups . ◮ Two directions: infinite and higher dimensional. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 5 / 24

  17. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 52341 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  18. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 52341 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 31425 � 52341 � 35421 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  19. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 52341 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 31425 � 52341 � 35421 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  20. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 52341 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 52341 � 31425 � 51423 � 35421 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 (But not commutative) 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  21. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 12345 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 52341 � 31425 � 51423 � 35421 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 ◮ The identity permutation is id = 1 2 3 4 . . . n . 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  22. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 12345 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 31425 � 24135 � 12345 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 ◮ The identity permutation is id = 1 2 3 4 . . . n . ◮ Inverse permutation: Flip the 1 2 3 4 5 string diagram upside down! Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  23. Introduction Finite Groups Infinite Groups Research Permutations are a group An n -permutation is a permutation of { 1 , 2 , . . . , n } . ◮ Write in one-line notation or use a string diagram : 31425 12345 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 31425 � 24135 � 12345 n -Permutations form 1 2 3 4 5 the Symmetric group S n . ◮ We can multiply permutations. 1 2 3 4 5 ◮ The identity permutation is id = 1 2 3 4 . . . n . ◮ Inverse permutation: Flip the 1 2 3 4 5 string diagram upside down! Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 6 / 24

  24. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  25. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  26. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 31425 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  27. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . ◮ S = { s 1 , s 2 , . . . , s n − 1 } are generators of S n . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  28. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . ◮ S = { s 1 , s 2 , . . . , s n − 1 } are generators of S n . A reflection group also has relations : Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  29. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . ◮ S = { s 1 , s 2 , . . . , s n − 1 } are generators of S n . A reflection group also has relations : ◮ First, s 2 i = id. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  30. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . ◮ S = { s 1 , s 2 , . . . , s n − 1 } are generators of S n . 12345 12345 21 345 1 32 45 A reflection group also has relations : 2 31 45 31 245 32 145 3 21 45 ◮ First, s 2 i = id. ◮ Consecutive generators don’t commute: s i s i +1 s i = s i +1 s i s i +1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  31. Introduction Finite Groups Infinite Groups Research Permutations as a reflection group 12435 1 2 3 4 5 A special type of permutation is an adjacent transposition , switching two adjacent entries. 1 2 3 4 5 ◮ Write s i : ( i ) ↔ ( i + 1). (e.g. s 3 = 1 2 4 3 5). ⋆ Every n -permutation is a product of adjacent transpositions. ◮ (Construct any string diagram through individual twists.) ◮ Example. Write 3 1 4 2 5 as s 1 s 3 s 2 . ◮ S = { s 1 , s 2 , . . . , s n − 1 } are generators of S n . 12345 12345 21 345 1 32 45 A reflection group also has relations : 2 31 45 31 245 32 145 3 21 45 ◮ First, s 2 i = id. ◮ Consecutive generators don’t commute: s i s i +1 s i = s i +1 s i s i +1 ◮ Non-consecutive generators DO commute: s i s j = s j s i 21 3 45 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 7 / 24

  32. Introduction Finite Groups Infinite Groups Research Visualizing symmetric groups We have already seen S 3 , generated by { s 1 , s 2 } : 132 312 123 321 213 231 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 8 / 24

  33. Introduction Finite Groups Infinite Groups Research Visualizing symmetric groups We have already seen S 3 , generated by { s 1 , s 2 } : 132 312 123 321 213 231 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 8 / 24

  34. Introduction Finite Groups Infinite Groups Research Visualizing symmetric groups We have already seen S 3 , generated by { s 1 , s 2 } : 132 312 123 321 213 231 We can visualize S 4 as a permutohedron , generated by { s 1 , s 2 , s 3 } . sourceforge.net/apps/trac/groupexplorer/wiki/The First Five Symmetric Groups/ Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 8 / 24

  35. Introduction Finite Groups Infinite Groups Research Visualizing symmetric groups We have already seen S 3 , generated by { s 1 , s 2 } : 132 312 123 321 213 231 We can visualize S 4 as a permutohedron , generated by { s 1 , s 2 , s 3 } . sourceforge.net/apps/trac/groupexplorer/wiki/The First Five Symmetric Groups/ They also give a way to see S 5 . . . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 8 / 24

  36. Introduction Finite Groups Infinite Groups Research Higher-dimension symmetric groups How can we “see” a reflection group in higher dimensions? The relation ( s i s j ) m determines the angle between hyperplanes H i , H j : ◮ ( s i s j ) 2 = id ← → θ ( H i , H j ) = π/ 2 ◮ ( s i s j ) 3 = id ← → θ ( H i , H j ) = π/ 3 For S 6 , we expect an angle of 60 ◦ between the hyperplane pairs ( H 1 , H 2 ), ( H 2 , H 3 ), ( H 3 , H 4 ), and ( H 4 , H 5 ). Every other pair will be perpendicular. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 9 / 24

  37. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : ◮ Vertices: One for every generator i ◮ Edges: Between i and j when m i , j ≥ 3. Label edges with m i , j when ≥ 4. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  38. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : Dihedral groups ◮ Vertices: One for every generator i t ◮ Edges: Between i and j when m i , j ≥ 3. s Label edges with m i , j when ≥ 4. Generators: s and t . Relation: ( st ) m = id Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  39. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : Dihedral groups m ◮ Vertices: One for every generator i s t ◮ Edges: Between i and j when m i , j ≥ 3. Generators: s and t . Relation: ( st ) m = id Label edges with m i , j when ≥ 4. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  40. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : Dihedral groups m ◮ Vertices: One for every generator i s t ◮ Edges: Between i and j when m i , j ≥ 3. Generators: s and t . Relation: ( st ) m = id Label edges with m i , j when ≥ 4. Symmetric groups: 1 2 n .. .. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  41. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : Dihedral groups m ◮ Vertices: One for every generator i s t ◮ Edges: Between i and j when m i , j ≥ 3. Generators: s and t . Relation: ( st ) m = id Label edges with m i , j when ≥ 4. Symmetric groups: 1 2 n .. .. 4 1 1 2 n .. .. Infinite families: 3 n .. .. 2 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  42. Introduction Finite Groups Infinite Groups Research All finite reflection groups Or see with a Coxeter diagram : Dihedral groups m ◮ Vertices: One for every generator i s t ◮ Edges: Between i and j when m i , j ≥ 3. Generators: s and t . Relation: ( st ) m = id Label edges with m i , j when ≥ 4. Symmetric groups: 1 2 n .. .. 4 1 1 2 n .. .. Infinite families: 3 n .. .. 2 Exceptional types: 5 4 5 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 10 / 24

  43. Introduction Finite Groups Infinite Groups Research Wallpaper Groups The art of M. C. Escher plays upon symmetries in the plane. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 11 / 24

  44. Introduction Finite Groups Infinite Groups Research Wallpaper Groups The art of M. C. Escher plays upon symmetries in the plane. An isometry of the plane is a transformation that preserves distance. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 11 / 24

  45. Introduction Finite Groups Infinite Groups Research Wallpaper Groups The art of M. C. Escher plays upon symmetries in the plane. An isometry of the plane is a transformation that preserves distance. Think: translations, rotations, reflections, glide reflections. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 11 / 24

  46. Introduction Finite Groups Infinite Groups Research Wallpaper Groups The art of M. C. Escher plays upon symmetries in the plane. An isometry of the plane is a transformation that preserves distance. Think: translations, rotations, reflections, glide reflections. A wallpaper group is a group of isometries of the plane with two independent translations. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 11 / 24

  47. Introduction Finite Groups Infinite Groups Research Wallpaper Groups The art of M. C. Escher plays upon symmetries in the plane. An isometry of the plane is a transformation that preserves distance. A wallpaper group is a group of isometries of the plane with two independent translations. Some are also reflection groups: Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 11 / 24

  48. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . s 2 s 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  49. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . s 2 s 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  50. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . s 2 s 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  51. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . ◮ Add a new generator s 0 and a new affine hyperplane H 0 . s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  52. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . ◮ Add a new generator s 0 and a new affine hyperplane H 0 . s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  53. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . ◮ Add a new generator s 0 and a new affine hyperplane H 0 . s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  54. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . ◮ Add a new generator s 0 and a new affine hyperplane H 0 . s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  55. Introduction Finite Groups Infinite Groups Research Infinite Reflection Groups Constructing an infinite reflection group: the affine permutations � S n . ◮ Add a new generator s 0 and a new affine hyperplane H 0 . s 2 s 1 s 0 Elements generated by { s 0 , s 1 , s 2 } correspond to alcoves here. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 12 / 24

  56. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . H 1 H 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  57. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. H 1 H 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  58. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 Coordinates: 3 1 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  59. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a Coordinates: (short) product of generators. 3 1 1 Word: s 0 s 1 s 2 s 1 s 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  60. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a Coordinates: (short) product of generators. 3 1 ◮ One-line notation. Similar to 1 writing finite permutations as 312. Word: s 0 s 1 s 2 s 1 s 0 Permutation: ( − 3 , 2 , 7) Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  61. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a Abacus diagram: (short) product of generators. - 8 - 7 - 6 ◮ One-line notation. Similar to - 5 - 4 - 3 writing finite permutations as 312. - 2 - 1 0 ◮ Abacus diagram. Columns of numbers. 1 2 3 4 5 6 7 8 9 10 11 12 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  62. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a Core partition: (short) product of generators. 0 1 2 0 ◮ One-line notation. Similar to 2 0 1 writing finite permutations as 312. 0 ◮ Abacus diagram. Columns of numbers. ◮ Core partition. Hook length condition. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  63. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a Core partition: (short) product of generators. 0 1 2 0 ◮ One-line notation. Similar to 2 0 1 writing finite permutations as 312. 0 ◮ Abacus diagram. Columns of numbers. Bounded partition: ◮ Core partition. Hook length condition. 0 1 2 ◮ Bounded partition. Part size bounded. 1 0 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  64. Introduction Finite Groups Infinite Groups Research Combinatorics of affine permutations Many ways to reference elements in � H 2 S n . ◮ Geometry. Point to the alcove. ◮ Alcove coordinates. Keep track of how many hyperplanes of each type H 1 you have crossed to get to your alcove. H 0 ◮ Word. Write the element as a (short) product of generators. They all play nicely ◮ One-line notation. Similar to with each other. writing finite permutations as 312. ◮ Abacus diagram. Columns of numbers. ◮ Core partition. Hook length condition. ◮ Bounded partition. Part size bounded. ◮ Others! Lattice path, order ideal, etc. Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 13 / 24

  65. Introduction Finite Groups Infinite Groups Research Affine permutations (Finite) n -Permutations S n ◮ Visually: ... s 1 s 2 s 3 s n - 2 s n - 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 14 / 24

  66. Introduction Finite Groups Infinite Groups Research Affine permutations (Finite) n -Permutations S n ◮ Visually: ... s 1 s 2 s 3 s n - 2 s n - 1 s 0 Affine n -Permutations � S n ◮ Generators: { s 0 , s 1 , . . . , s n − 1 } ◮ s 0 has a braid relation with s 1 and s n − 1 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 14 / 24

  67. Introduction Finite Groups Infinite Groups Research Affine permutations (Finite) n -Permutations S n ◮ Visually: ... s 1 s 2 s 3 s n - 2 s n - 1 s 0 Affine n -Permutations � S n ◮ Generators: { s 0 , s 1 , . . . , s n − 1 } ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact one-line notation? Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 14 / 24

  68. Introduction Finite Groups Infinite Groups Research Affine permutations (Finite) n -Permutations S n ◮ Visually: ... s 1 s 2 s 3 s n - 2 s n - 1 s 0 Affine n -Permutations � S n ◮ Generators: { s 0 , s 1 , . . . , s n − 1 } ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact one-line notation? ◮ Perhaps interchanges 1 and n ? Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 14 / 24

  69. Introduction Finite Groups Infinite Groups Research Affine permutations (Finite) n -Permutations S n ◮ Visually: ... s 1 s 2 s 3 s n - 2 s n - 1 s 0 Affine n -Permutations � S n ◮ Generators: { s 0 , s 1 , . . . , s n − 1 } ◮ s 0 has a braid relation with s 1 and s n − 1 ◮ How does this impact one-line notation? ◮ Perhaps interchanges 1 and n ? ◮ Not quite! (Would add a relation.) Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 14 / 24

  70. Introduction Finite Groups Infinite Groups Research Window notation Affine n -Permutations � (G. Lusztig 1983, H. Eriksson, 1994) S n w ∈ � Write an element � S n in 1-line notation as a permutation of Z . Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 15 / 24

  71. Introduction Finite Groups Infinite Groups Research Window notation Affine n -Permutations � (G. Lusztig 1983, H. Eriksson, 1994) S n 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 15 / 24

  72. Introduction Finite Groups Infinite Groups Research Window notation Affine n -Permutations � (G. Lusztig 1983, H. Eriksson, 1994) S n 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 15 / 24

  73. Introduction Finite Groups Infinite Groups Research Window notation Affine n -Permutations � (G. Lusztig 1983, H. Eriksson, 1994) S n 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 15 / 24

  74. Introduction Finite Groups Infinite Groups Research Window notation Affine n -Permutations � (G. Lusztig 1983, H. Eriksson, 1994) S n 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 15 / 24

  75. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  76. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w n ], ◮ Place integers in n runners . - 15 - 14 - 13 - 12 - 11 - 10 - 9 - 8 ◮ Circled: beads . Empty: gaps - 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  77. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w n ], ◮ Place integers in n runners . - 15 - 14 - 13 - 12 - 11 - 10 - 9 - 8 ◮ Circled: beads . Empty: gaps - 7 - 6 - 5 - 4 ◮ Create an abacus where each - 3 - 2 - 1 0 runner has a lowest bead at w i . 1 2 3 4 5 6 7 8 9 10 11 12 Example: [ − 4 , − 3 , 7 , 10] 13 14 15 16 17 18 19 20 Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  78. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w n ], ◮ Place integers in n runners . - 15 - 14 - 13 - 12 - 11 - 10 - 9 - 8 ◮ Circled: beads . Empty: gaps - 7 - 6 - 5 - 4 ◮ Create an abacus where each - 3 - 2 - 1 0 runner has a lowest bead at w i . 1 2 3 4 5 6 7 8 9 10 11 12 Example: [ − 4 , − 3 , 7 , 10] 13 14 15 16 17 18 19 20 ◮ Generators act nicely. ◮ s i interchanges runners i ↔ i + 1. ◮ s 0 interchanges runners 1 and n (with shifts) Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  79. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w n ], ◮ Place integers in n runners . - 15 - 14 - 13 - 12 - 15 - 14 - 13 - 12 - 11 - 10 - 9 - 8 - 11 - 10 - 9 - 8 ◮ Circled: beads . Empty: gaps - 7 - 6 - 5 - 4 - 7 - 6 - 5 - 4 ◮ Create an abacus where each - 3 - 2 - 1 0 - 3 - 2 - 1 0 runner has a lowest bead at w i . s 1 1 2 3 4 1 2 3 4 → 5 6 7 8 5 6 7 8 9 10 11 12 9 10 11 12 Example: [ − 4 , − 3 , 7 , 10] 13 14 15 16 13 14 15 16 17 18 19 20 17 18 19 20 ◮ Generators act nicely. ◮ s i interchanges runners i ↔ i + 1. ( s 1 : 1 ↔ 2) ◮ s 0 interchanges runners 1 and n (with shifts) Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  80. Introduction Finite Groups Infinite Groups Research An abacus model for affine permutations (James and Kerber, 1981) Given an affine permutation [ w 1 , . . . , w n ], ◮ Place integers in n runners . - 15 - 14 - 13 - 12 - 15 - 14 - 13 - 12 - 15 - 14 - 13 - 12 - 11 - 10 - 9 - 8 - 11 - 10 - 9 - 8 - 11 - 10 - 9 - 8 ◮ Circled: beads . Empty: gaps - 7 - 6 - 5 - 4 - 7 - 6 - 5 - 4 - 7 - 6 - 5 - 4 ◮ Create an abacus where each - 3 - 2 - 1 0 - 3 - 2 - 1 0 - 3 - 2 - 1 0 runner has a lowest bead at w i . s 1 s 0 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 Example: [ − 4 , − 3 , 7 , 10] 13 14 15 16 13 14 15 16 13 14 15 16 17 18 19 20 17 18 19 20 17 18 19 20 ◮ Generators act nicely. ◮ s i interchanges runners i ↔ i + 1. ( s 1 : 1 ↔ 2) ◮ s 0 interchanges runners 1 and n (with shifts) ( s 0 : 1 shift ↔ 4) Combinatorial interpretations in affine Coxeter groups QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 16 / 24

  81. Introduction Finite Groups Infinite Groups Research Core partitions For an integer partition λ = ( λ 1 , . . . , λ k ) drawn as a Young 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 QCC Colloquium Christopher R. H. Hanusa Queens College, CUNY September 12, 2012 17 / 24

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