permutation statistics in the affine Coxeter group of type A Eli - PowerPoint PPT Presentation

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition permutation statistics in the affine Coxeter group of type A Eli Bagno Jerusalem College of Technology Riccardo Biagioli University of Lyon 1 Robert Schwarz Bar Ilan University permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The length function on S n For σ ∈ S n (symmetric group on n letters) the length of σ ( ℓ ( σ ) ) is defined by min { r ∈ N | σ = s i 1 · · · s i r for some i 1 , . . . , i r ∈ [ n − 1 ] } For example σ = 3142 = s 2 s 3 s 1 , ℓ ( σ ) = 3 . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Statistics on S n The length is equal to the number of inversions inv ( σ ) := |{ ( i , j ) | i < j , σ ( i ) > σ ( j ) }| The descent set of σ := σ ( 1 ) · · · σ ( n ) ∈ S n is Des ( σ ) := { i ∈ [ n − 1 ] | σ ( i ) > σ ( i + 1 ) } , and des ( σ ) := | Des ( σ ) | is the descent number . The major index of σ ∈ S n is � maj ( σ ) := i i ∈ Des ( σ ) permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Example Ex. If σ = 623514 ∈ S 6 then inv ( σ ) = 9 ; Des ( σ ) = { 1 , 4 } des ( σ ) = 2 and maj ( σ ) = 1 + 4 = 5 . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Equidistribution over S n Theorem [MacMahon, 1915] n − 1 q maj ( β ) = q ℓ ( β ) = � � � ( 1 + q + . . . + q i ) . β ∈ S n β ∈ S n i = 0 A statistic equidistributed with the length is said to be Mahonian permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Is there any analogue for affine Coxeter groups? Since these groups are infinite, there is very little permutation statistics on them in the literature. Couple of works, by A. Bjiorner, F . Brenti, V.Reiner‘. Recently, Clark and Ehrenborg introduced an excedance statistics for the affine Coxeter group of type A. They used enumeration of lattice points of a skew version of the root polytope of type A . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The Coxeter affine group of type A ˜ S n is the group of all bijections π : Z → Z satisfying the following two identities: π ( x + n ) = π ( x ) + n , n � n + 1 � � π ( x ) = 2 x = 1 with composition as group operation. permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition Integer notation The values of π on { 1 , . . . , n } determine the other values so we write π = [ π ( 1 ) , . . . , π ( n )] = [ π 1 , . . . , π n ] . Example: n = 5 We represent the affine permutation π = . . . − 1 0 1 2 3 4 5 6 7 . . . − 7 − 1 − 2 12 1 0 17 6 4 by π = [ 1 , − 1 , 0 , − 2 , 17 ] ∈ ˜ S 5 permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition Colored notation For p ∈ Z , write p = c · n + a with a ∈ { 1 , . . . , n } and denote p = a c . So π = a c 1 1 a c 2 2 . . . a c n n , where π i = c i · n + a i . Denote also: | π | = a 1 · · · a n ∈ S n . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition Example: n = 5, π = [ 1 , − 1 , 0 , − 2 , 17 ] = 1 0 4 − 1 5 − 1 3 − 1 2 3 − 2 = ( − 1 ) · 5 + 3 | π | = 14532 ∈ S 5 permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition Define: n csum ( π ) = csum ( a c 1 1 . . . a c n � n ) = c i . i = 1 Note that for each π ∈ ˜ S n , csum ( π ) = 0. Hence, the group ˜ S n can be seen as the subgroup of the wreath product Z ≀ S n = Z n ⋊ S n : ˜ S n = { π ∈ Z ≀ S n | csum ( π ) = 0 } The numbers r 1 , . . . , r n are called the colors of π . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The coxeter generators The affine group ˜ S n is a Coxeter group Generators: S = { s 0 , s 1 , . . . , s n − 1 } Relations: s 2 i = 1 , ( i ∈ { 0 , . . . , n − 1 } ) s i s i + 1 s i = s i + 1 s i s i + 1 , ( 0 ≤ i < n ) s i s j = s j s i , ( 0 ≤ i < j < n , | j − i | > 1 ) permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The length generating function The length of an element π ∈ ˜ S n : ℓ ( π ) = min { r ∈ N : π = s i 1 · · · s i r , for some i 1 , ..., i r ∈ [ 0 , n − 1 ] } . Theorem: (Bjorner, Brenti, Shi) � π ( j ) − π ( i ) � � � � ℓ ( π ) = � . � � n � 1 ≤ i < j ≤ n The generating function of the length is equal to [ n ] q q ℓ ( π ) = � ( 1 − q ) n − 1 , π ∈ ˜ S n where [ n ] q = 1 − q n 1 − q . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The O.G.S. decomposition G is a group. Let S = { t 0 , . . . , t n − 1 } ⊂ G s.t. each g ∈ G can be written uniquely in the form g = t k n − 1 1 t k 0 n − 1 · · · t k 1 0 where For torsion free generators: k i ∈ Z . For generators of order u i : 0 ≤ k i ≤ m i − 1 for some m i | u i . S is called an Ordered Generating System (O.G.S) We say that G has an O.G.S decomposition. permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The O.G.S. decomposition Clearly, every finite solvable group has an O.G.S given by its cyclic decomposition factors. Theorem:(Shwartz) The simple groups A n (alternating), PSL ( n , Q ) , PSp ( 2 n , Q ) (symplectic) and all the 5 sporadic Mathieu groups have O.G.S. decomposition. The symmetric group and the colored permutation groups also have an O.G.S decomposition. permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition An O.G.S decomposition for ˜ S n We present an O.G.S decomposition for ˜ S n : 1 − 1 23 . . . n 1 t 1 = 2 − 1 13 . . . n 1 t 2 = . . . ( n − 1 ) − 1 12 . . . ( n − 2 ) n 1 t n − 1 = n − 1 12 . . . ( n − 1 ) 1 . = t 0 Define for each π = t k n − 1 n − 1 · · · t k 0 0 , n − 1 � α ( π ) = k i . i = 0 permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The action of t 0 in multiplying from the right Right shift all the places. Subtract 1 color from the first place. Add 1 color to the last place. Example: 1 1 2 − 2 3 1 4 0 5 0 t 0 �→ 5 − 1 1 1 2 − 2 3 1 4 1 . permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The action of t ± 1 in multiplying from the right 1 t 1 Subtract 1 color from the first place. Add 1 color to the last place. Example t 1 ( 1 1 2 − 2 3 1 4 0 5 0 ) �→ ( 1 0 2 − 2 3 1 4 0 5 1 ) . t − 1 1 Add 1 color to the first place. Subtract 1 color from the last place. Example t − 1 ( 5 1 2 − 2 3 1 4 0 5 0 ) �→ ( 1 2 2 − 2 3 1 4 0 5 − 1 ) . 1 permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A The Coxeter genrators An algorithm for the O.G.S. decomposition The O.G.S. decomposition The action of the t i , ( i > 1 ) in multiplying from the right t i : Right shift the first i places. Subtract 1 color from the first place. Add one color to the last place. All other places are staying fixed. a r 1 1 a r 2 2 · · · a r i n �→ a r i − 1 a r 1 1 · · · a r i − 1 i · · · a r n i − 1 · · · a r n + 1 . n i t − 1 : i Left shift the first i places. Add 1 color to the i -th place. subtract one color from the last place. All other places are staying fixed. a r 1 1 a r 2 2 · · · a r i n �→ a r 2 2 a r 3 3 · · · a r i i a r 1 + 1 i · · · a r n · · · a r n − 1 n 1 permutation statistics in the affine Coxeter group of type A