The shortest path poset of finite Coxeter Groups Sal A. Blanco - PowerPoint PPT Presentation

The shortest path poset of finite Coxeter Groups Saúl A. Blanco Cornell University Fall Eastern Section Meeting of the AMS Penn State, October 24–25

Coxeter Groups Groups with presentation � S | ( ss ′ ) m ( s , s ′ ) = e , for all s , s ′ ∈ S � where ◮ m ( s , s ) = 1 ◮ m ( s , s ′ ) = m ( s ′ , s ) ≥ 2 for s � = s ′ ◮ m ( s , s ′ ) = ∞ means that there is no relation between s and s ′ .

Examples. ◮ Z 2 = � s | s 2 = 1 � . ◮ The dihedral group of order 2 m . I 2 ( m ) = � s 1 , s 2 | ( s 1 s 2 ) m = ( s 2 s 1 ) m = s 2 1 = s 2 2 = 1 � . When m ≥ 3, this is the group of symmetries of the m -gon. ◮ The symmetric group. A n − 1 = S n = � s 1 , s 2 , . . . , s n − 1 | ( s i s j ) m ( s i , s j ) � , where s i = ( i , i + 1 ) , m ( s i , s i + 1 ) = 3 and otherwise m ( s i , s j ) = 2 for i < j .

Basic Definitions ◮ Each w ∈ W can be expressed as w = s 1 s 2 . . . s n with s i ∈ S . If n is minimal, then s 1 s 2 . . . s n is a reduced expression for w . In this case, we define the length function by ℓ ( w ) = n . ◮ T ( W ) = { wsw − 1 | w ∈ W , s ∈ S } is the set of reflections of ( W , S ) . ◮ Bruhat Order: Let v , w ∈ W . We say that v ≤ w if and only if there exist t 1 , . . . , t k ∈ T so that vt 1 t 2 · · · t k = w with ℓ ( vt 1 ) > ℓ ( v ) and ℓ ( vt 1 · · · t i ) > ℓ ( vt 1 · · · t i − 1 ) for i > 1. ◮ If W is finite, then there exists a maximal-length word w W 0 ; that is, ℓ ( w ) ≤ ℓ ( w W 0 ) for all w ∈ W . ◮ If | W | < ∞ , then ℓ ( w W 0 ) = | T ( W ) | .

Bruhat Graph The directed graph ( V , E ) consisting of V = W and ( u , v ) ∈ E if ℓ ( u ) < ℓ ( v ) and there exists t ∈ T with ut = v is called the Bruhat graph. For example, consider S 3 with generators s 1 = ( 1 , 2 ) , s 2 = ( 2 , 3 ) , with labeling1 → s 1 , 2 → s 1 s 2 s 1 , 3 → s 2 s 2 s 1 s 2 = s 1 s 2 s 1 1 3 s 1 s 2 s 2 s 1 2 2 1 3 s 2 s 1 2 3 1 e

Reflection Order A reflection order Is a total order < T on the reflections of W so that for any dihedral reflection subgroup W ′ (i.e, W ′ has two generators, x , y ∈ T ) , then either x < T xyx < T xyxyx < T . . . < T yxyxy < T yxy < T y or y < T yxy < T yxyxy < T . . . < T xyxyx < T xyx < T x where x and y are the generators of W ′ .

Complete cd -index Fix a reflection ordering < T . Consider a chain (path) C in the Bruhat graph of [ u , v ] labeled by reflections, say C = ( t 1 , t 2 , . . . , t k ) The descent set of C is D ( C ) = { i ∈ [ k − 1 ] | t i + 1 < T t i } The complete cd -index encodes the descent sets of all the Bruhat paths.

Complete cd -index The encoding is done as follows: Let ∆ = ( t 1 , t 2 , . . . , t k ) be a path of length k from u to v . Then define w (∆) = x 1 x 2 · · · x k − 1 where � a if t i < T t i + 1 (for ascent) x i = if t i + 1 < T t i b Now consider the polynomial � ∆ w (∆) . Set c = a + b d = ab + ba After the substitution, � ∆ w (∆) becomes a polynomial with variables c and d . This is denoted by � ψ u , v , and it is called the complete cd -index of [ u , v ] .

Example Consider S 3 with generators s 1 = ( 1 , 2 ) and s 2 = ( 2 , 3 ) , and reflection ordering s 1 = ( 1 , 2 ) < T s 1 s 2 s 1 = ( 1 , 3 ) < T s 2 = ( 2 , 3 ) . s 1 < T s 1 s 2 s 1 < T s 2 s 2 s 1 s 2 = s 1 s 2 s 1 1 3 a 2 123 s 1 s 2 s 2 s 1 131 ab 313 ba 2 2 1 3 b 2 321 s 2 s 1 2 2 1 3 1 � ψ e,s 1 s 2 s 1 = c 2 + 1 e

A bigger example ψ 12435 , 53142 = c 5 + 6 cdc 2 + 6 c 2 dc + 3 dc 3 + 3 c 3 d + 7 cd 2 + � + 7 d 2 c + 6 dcd + c 3 + 2 dc + 2 cd

Shortest Path Poset of W If W is a finite Coxeter group, we can form a poset SP ( W ) with the shortest paths of W . For example, consider the Bruhat graph of B 2 (signed permutations of two elements) 1 2 2 1 1 2 1 2 2 1 2 1 2 2 1 2 2 1 1 2

SP ( W ) is a gra The absolute length of w ∈ W is the minimal number of reflections t 1 , . . . , t k so that t 1 t 2 · · · t k = w . We write ℓ T ( w ) = k . s 1 s 2 s 1 s 2 s 1 s 1 s 2 s 2 s 1 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 e e Bruhat Order for A 2 Absolute Order for A 2

SP ( A n − 1 ) How to describe the shortest paths from e to w A n − 1 = n n − 1 . . . 2 1?. 0 Let r i = ( i n + 1 − i ) and k = ⌊ n 2 ⌋ . Then Theorem If t 1 t 2 · · · t k = w A n − 1 then 0 ◮ { t 1 , t 2 , . . . , t k } = { r 1 , r 2 , . . . , r k } ◮ t i t j = t j t i for all i , j ◮ ( t σ ( 1 ) , t σ ( 2 ) , . . . , t σ ( k ) ) is a path in B ( A n − 1 ) for all σ ∈ A n − 1 . Corollary SP ( A n − 1 ) ∼ = Boolean ( k ) , the Boolean poset of rank k (poset of subsets of { 1 , . . . , k } ordered by inclusion).

Example: B 2 {1,2} {1} {2} SP ( B 2 ) is formed by two copies of Boolean ( 2 ) that share the smallest and biggest elements.

In general, we have Theorem Let W be finite Coxeter group, w 0 the longest element in W, and ℓ 0 = ℓ T ( w 0 ) . If t 1 t 2 · · · t ℓ 0 = w 0 then (a) t i t j = t j t i for 1 ≤ i , j ≤ ℓ 0 . In particular t τ ( 1 ) t τ ( 2 ) · · · t τ ( ℓ 0 ) = w 0 for all τ ∈ A ℓ 0 − 1 . (b) ( t τ ( 1 ) , t τ ( 2 ) , . . . , t τ ( ℓ 0 ) ) is a path in the Bruhat graph of W for all τ ∈ A ℓ 0 − 1 Corollary ( SP ( W ) ) SP ( W ) is formed by α W Boolean posets of rank ℓ 0 (that share the smallest and biggest elements).

W rank ( SP ( W )) # of Boolean posets ⌊ n A n 2 ⌋ 1 B n n b n D n n if n is even; n − 1 if n is odd d n m I 2 ( m ) 2 m even; 1 m odd 2 m even; 1 m odd F 4 2 1 H 3 3 5 H 4 4 75 E 6 4 3 E 7 7 135 E 8 8 2025 ⌊ n � n − 2 i � 2 ⌋ j − 1 � � 1 b n = 1 + j ! 2 j = 1 i = 0 ⌊ m � n − 2 i � 2 ⌋− 1 � 1 d n = , m = n if n is even. Otherwise m = n − 1. ⌊ m 2 ⌋ ! 2 i = 0

cd -index of Boolean ( k ) Let ψ ( Boolean ( k )) be the cd -index of Boolean ( k ) (that is, the regular cd -index of the Eulerian poset Boolean ( k ) . Then Ehrenborg and Readdy show that ψ ( Boolean ( 1 ))= 1 ψ ( Boolean ( k ))= ψ ( Boolean ( k − 1 )) · c + G ( ψ ( Boolean ( k − 1 )) G is the derivation (derivation means G ( xy ) = xG ( y ) + G ( x ) y ) G ( c ) = d and G ( d ) = cd . For example ψ ( Boolean ( 2 )) = c ψ ( Boolean ( 3 )) = c 2 + d ψ ( Boolean ( 4 )) = c 3 + 2 ( cd + dc ) Theorem The lowest-degree terms of � ψ e , w 0 are given by α W ψ ( Boolean ( ℓ T ( w 0 ))) for some α W ∈ Z .

Corollary The lowest-degree terms of � ψ e , w 0 are minimized (component-wise) by ψ ( Boolean ( ℓ 0 )) . This corollary is true for the lowest degree terms of ψ e , v if [ c ℓ 0 − 1 ] = 1, where [ c k ] is denotes the coefficient of c k in ψ e , v . Conjecture: Corollary holds for � ψ u , v .