Coxeter groups and palindromic Poincar e polynomials Edward - PowerPoint PPT Presentation

Coxeter groups and palindromic Poincar´ e polynomials Edward Richmond (joint work with W. Slofstra) University of British Columbia January 10, 2013 Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 1 / 1

Coxeter groups and Poincar´ e polynomials Let W be a Coxeter group with finite simple reflection set S . By definition, W is the group generated by S where for any s, t ∈ S, s 2 = e ( st ) m st = e for some m st ∈ { 2 , 3 , . . . , ∞} . and Examples: The symmetric group W = S n , with S = { s 1 , . . . , s n − 1 } and ( s i s i +1 ) 3 = ( s i s j ) 2 = e where | i − j | > 1 . The crystallographic Coxeter groups where W is the Weyl group of a Lie group or Kac Moody group G. Here we have m st ∈ { 2 , 3 , 4 , 6 , ∞} . Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 2 / 1

Coxeter groups and Poincar´ e polynomials Length: For any w ∈ W, the length ℓ ( w ) is the smallest number of simple reflections needed to express w. Any expression w = s i 1 · · · s i ℓ ( w ) is called a reduced word of w. Bruhat partial order: For any w, u ∈ W, we say that u ≤ w if there exist reduced words u = s j 1 · · · s j ℓ ( u ) and w = s i 1 · · · s i ℓ ( w ) where j 1 , . . . , j ℓ ( u ) is a subsequence of i 1 , . . . , i ℓ ( w ) . Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 3 / 1

Coxeter groups and Poincar´ e polynomials The Poincar´ e polynomial: For any w ∈ W, define � q ℓ ( x ) . P w ( q ) := x ≤ w Example: W = S 3 and w = s 1 s 2 s 1 . s 1 s 2 s 1 s 1 s 2 s 2 s 1 s 1 s 2 e P w ( q ) = 1 + 2 q + 2 q 2 + q 3 Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 4 / 1

Coxeter groups and Poincar´ e polynomials Example: The group W = � s 1 , s 2 , s 3 | s 2 i = e � and w = s 1 s 2 s 3 s 1 . s 1 s 2 s 3 s 1 s 1 s 2 s 1 s 1 s 2 s 3 s 2 s 3 s 1 s 1 s 3 s 1 s 1 s 2 s 2 s 1 s 2 s 3 s 1 s 3 s 3 s 1 s 2 s 1 s 3 e P w ( q ) = 1 + 3 q + 5 q 2 + 4 q 3 + q 4 Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 5 / 1

Coxeter groups and Poincar´ e polynomials Question: When is P w ( q ) a palindromic polynomial? i =0 a i q i is palindromic if a i = a ℓ − i Definition: A polynomial � ℓ ∀ i. Example: W = S 3 and w = s 1 s 2 s 1 . P w ( q ) = 1 + 2 q + 2 q 2 + q 3 In this case, P w ( q ) is palindromic! Example: W = � s 1 , s 2 , s 3 | s 2 i = e � and w = s 1 s 2 s 3 s 1 . P w ( q ) = 1 + 3 q + 5 q 2 + 4 q 3 + q 4 In this case, P w ( q ) is not palindromic! Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 6 / 1

Coxeter groups and Poincar´ e polynomials Motivation from Algebraic Geometry: When W is the Weyl group of some Kac-Moody group G (i.e cystallographic), then each w ∈ W corresponds to a Schubert variety X w in the flag variety G/B. It is well known that P w ( q 2 ) = � dim H i ( X w , C ) q i . i ≥ 0 Theorem: Carrell-Peterson ’94 Let W be the Weyl group of some Kac-Moody group G. Then X w is rationally smooth if and only if P w ( q ) is palindromic. Suppose G is simply laced of finite type. Then X w is smooth if and only if X w is rationally smooth. Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 7 / 1

Coxeter groups and Poincar´ e polynomials History of characterizing palindromic Poincar´ e polynomials: For W of a classical type (ABCD), P w ( q ) is palindromic if and only if w avoids a certain list of patterns (Lakshmibai-Sandhya ’90, Billey ’98). For W of finite Lie type, P w ( q ) is palindromic if and only if the inversion set of w avoids a certain list of root system patterns (Billey-Postnikov ’05). Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 8 / 1

Coxeter groups and Poincar´ e polynomials Weaker notion of palindromic: i =0 a i q i is k -palindromic if a i = a ℓ − i Definition: A polynomial � ℓ ∀ i < k. Example: 1 + q + 2 q 2 + 3 q 3 + q 4 + q 5 is 2-palindromic, but not 3-palindromic. Observation: Billey-Postnikov ’05 For W = S n , P w ( q ) is ( n − 1) -palindromic if and only if P w ( q ) is palindromic. Question: Is this a good criterion for detecting palindromic Poincar´ e polynomials for general Coxeter groups? Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 9 / 1

Coxeter groups and Poincar´ e polynomials Theorem 1: Slofstra-R. ’12 Let W be a Coxeter group with generating set S. Suppose that m st � = 2 ∀ s, t ∈ S. Then P w ( q ) is 4-palindromic if and only if P w ( q ) is palindromic. Suppose that m st � = 2 , 3 ∀ s, t ∈ S. Then P w ( q ) is 2-palindromic if and only if P w ( q ) is palindromic. Example: Let W = � s 1 , s 2 , s 3 , s 4 | s 2 i = e � and w = s 1 s 2 s 1 s 3 s 1 s 3 s 4 s 3 . i =0 a i q i is palindromic since We observe that P w ( q ) = � 8 a 0 = a 8 = 1 and a 1 = a 7 = 4 . In particular, P w ( q ) = 1 + 4 q + 9 q 2 + 14 q 3 + 16 q 4 + 14 q 5 + 9 q 6 + 4 q 7 + q 8 . Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 10 / 1

Enumeration results The theorem follows from a much stronger result where we can explicitly factor Poincar´ e polynomials given that they are 2-palindromic. Enumeration results: We can explicitly enumerate the number of palindromic elements in uniform Coxeter groups. For m, n ∈ Z + , let W ( m, n ) denote the Coxeter group with | S | = n and m s,t = m ∀ s, t ∈ S. Define the generating series q k t n � Φ m ( q, t ) := P n,k n ! n,k ≥ 0 where P n,k denotes the number of w ∈ W ( m, n ) of length k with a palindromic Poincar´ e polynomial. Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 11 / 1

Enumeration results Corollary: Slofstra-R. ’12 The generating series for the number of palindromic elements is exp( t ) Φ m ( q, t ) = 1 − φ m ( q, t ) where (2 q − 2 q 3 ) t − (3 q 3 + q 5 ) t 2  for m = 3  2 − 2 q 2 − 4 q 2 t        2 qt − 3 q m t 2 − q m +2 [ m − 3] q t 3    φ m ( q, t ) = for 4 ≤ m < ∞ 2 − 2 q 2 t ([ m − 2] q + q m − 3 )       qt − q 2 t    for m = ∞ .   1 − q − q 2 t Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 12 / 1

Enumeration results Example: Φ 4 ( q, t ) = 1 + (1 + q ) t + (1 + 2 q + 2 q 2 + 2 q 3 + q 4 ) t 2 2 + (1 + 3 q + 6 q 2 + 12 q 3 + 15 q 4 + 12 q 5 + 12 q 6 + 6 q 7 ) t 3 6 + (1 + 4 q + 12 q 2 + 36 q 3 + 78 q 4 + 120 q 5 + 156 q 6 + 168 q 7 + 150 q 8 + 120 q 9 + 48 q 10 ) t 4 24 + O ( t 5 ) . Evaluating Φ m (1 , t ) gives the following table on the total number of palindromic elements in W ( m, n ) . m � n 1 2 3 4 5 6 7 4 2 8 67 893 15596 330082 8165963 5 2 10 115 2057 47356 1314292 42584795 6 2 12 175 3893 110436 3768982 150113447 7 2 14 247 6545 219956 8884312 418725119 8 2 16 331 10157 393916 18351562 997538291 Slofstra-Richmond* (UBC) Coxeter groups and Poincar´ e polynomials January 10, 2013 13 / 1