The excess zero graph of a Coxeter group

Let 𝑊 be a Coxeter group with length function ℓ and set of non-identity involutions I ⁢ ( W ) . The excess zero graph E 0 ⁢ ( W ) has vertex set I ⁢ ( W ) with x , y ∈ I ⁢ ( W ) adjacent whenever ℓ ⁢ ( x ⁢ y ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) .

Let 𝑊 be a Coxeter group of rank at least 2.

1. If 𝑊 is finite, then E 0 ⁢ ( W ) has two connected components: I ⁢ ( W ) ∖ { w 0 } and { w 0 } . If 𝑊 is infinite, then E 0 ⁢ ( W ) is connected.

2. Let E ̂ 0 ⁢ ( W ) be the connected component of E 0 ⁢ ( W ) not containing w 0 . Then the diameter of E ̂ 0 ⁢ ( W ) is at most 3.

Suppose that 𝑊 is a Coxeter group of finite rank and let 𝑟 be a fundamental reflection of 𝑊. Then the sets Δ 1 ⁢ ( r ) and I ⁢ ( W ) ∖ ( { r } ∪ Δ 1 ⁢ ( r ) ) have the same cardinality.

For a finite Coxeter group 𝑊 and 𝑟 a fundamental reflection,

Moreover, for x ∈ I ⁢ ( W ) ∖ R ,

Let m ≥ 2 and n ≥ 2 ⁢ m . Then

Let 𝑊 be a finite irreducible Coxeter group of rank n ≥ 2 with R = { r 1 , r 2 , … , r n } .

1. Suppose w 0 ∈ Z ⁢ ( W ) . (This occurs if and only if 𝑊 is of type B n , type D n with 𝑛 even, type F 4 , H 3 , H 4 , E 7 or E 8 , or type I 2 ⁢ ( m ) with 𝑚 even.) Then P ⁢ ( W ) = { w 0 ⁢ r ∣ r ∈ R } .

2. If 𝑊 is of type A n and n ≥ 2 , setting J = { 1 , 2 , … , ⌈ n 2 ⌉ } , we have

   P ⁢ ( W ) = { w 0 ⁢ [ i ↗ ( n + 1 − i ) ] , w 0 ⁢ [ ( n + 1 − i ) ↘ i ] ∣ i ∈ J } .

3. If 𝑊 is of type D n , where 𝑛 is odd, then

   P ⁢ ( W ) = { w 0 ⁢ r i , w 0 ⁢ r n ⁢ r n − 2 ⁢ r n − 1 , w 0 ⁢ r n − 1 ⁢ r n − 2 ⁢ r n ∣ i ∈ { 1 , … , n − 2 } } .

4. If 𝑊 is of type E 6 , then

   P ⁢ ( W ) = { w 0 ⁢ r 2 , w 0 ⁢ r 4 , w 0 ⁢ r 5 ⁢ r 4 ⁢ r 3 , w 0 ⁢ r 3 ⁢ r 4 ⁢ r 5 , w 0 ⁢ r 6 ⁢ r 5 ⁢ r 4 ⁢ r 3 ⁢ r 1 , w 0 ⁢ r 1 ⁢ r 3 ⁢ r 4 ⁢ r 5 ⁢ r 6 } .

5. If 𝑊 is of type I 2 ⁢ ( m ) , with 𝑚 odd and m ≥ 5 , then

   P ⁢ ( W ) = { w 0 ⁢ r 1 ⁢ r 2 , w 0 ⁢ r 2 ⁢ r 1 } .

If 𝑊 is a finite irreducible Coxeter group of rank 𝑛, then

Lemma 2.1 ([11, §5.2])

For w ∈ W and r ∈ R , either

Suppose that 𝑊 is a Coxeter group with x , y ∈ W . Then the following hold:

1. ℓ ⁢ ( x ) = | N ⁢ ( x ) | ,

2. N ⁢ ( x ⁢ y ) = [ N ⁢ ( y ) ∖ ( − y − 1 ⁢ N ⁢ ( x ) ) ] ∪ ̇ y − 1 ⁢ ( N ⁢ ( x ) ∖ N ⁢ ( y − 1 ) ) ,

3. ℓ ⁢ ( x ⁢ y ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) − 2 ⁢ | N ⁢ ( x ) ∩ N ⁢ ( y − 1 ) | .

Proof

For (i) and (ii), see [11, §5.6 Proposition (b)] and [5, Lemma 2.2] respectively. Part (iii) follows from (ii). ∎

Lemma 2.3 ([11, §5.8])

Let w = r 1 ⁢ ⋯ ⁢ r k (not necessarily reduced), where each r i is a simple reflection. If ℓ ⁢ ( w ⁢ r ) < ℓ ⁢ ( w ) for some simple reflection 𝑟, then there exists an index 𝑖 for which w ⁢ r = r 1 ⁢ ⋯ ⁢ r ̂ i ⁢ ⋯ ⁢ r k (and thus w = r 1 ⁢ ⋯ ⁢ r ̂ i ⁢ ⋯ ⁢ r k ⁢ r , with a factor 𝑟 exchanged for a factor r i ). In particular, 𝑤 has a reduced expression ending in 𝑟 if and only if ℓ ⁢ ( w ⁢ r ) < ℓ ⁢ ( w ) .

Lemma 2.4 ([3, Propostion 2.1.1])

Suppose that 𝑊 is a Coxeter group and J ⊆ R . Then the following hold.

1. For each w ∈ W , there is a unique pair y ∈ W J and x ∈ X J such that w = y ⁢ x . Furthermore, ℓ ⁢ ( y ⁢ x ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) . In particular, X J is a complete set of right coset representatives of W J in 𝑊.

2. The set { x − 1 ∣ x ∈ X J } is equal to X J , and X J is a complete set of left coset representatives of W J in 𝑊.

Suppose that 𝑊 is a Coxeter group of type D n . Let J : = R ∖ { [ n ] } and let x ∈ X J be a non-identity element. Then 𝑥 has a reduced expression x = a ⁢ b , where either

1. a ∈ { [ n ] , [ n , n − 2 , n − 1 ] } and b ∈ W { 1 , … , n − 2 } ; or

2. a = [ n , n − 2 , n − 3 , n − 1 , n − 2 , n ] .

Proof

We argue by induction on 𝑛, starting with n = 4 . Employing [3, Algorithm B, §2.1] gives

Bearing in mind that [ 1 ] and [ 3 ] commute, we see that either alternative (i) holds or x = [ 4 , 2 , 1 , 3 , 2 , 4 ] , which gives (ii). Now assume that n ≥ 5 . Set K = R ∖ { [ 1 ] } and L = R ∖ { [ 1 ] , [ n − 1 ] } . Then W K is of type D n − 1 , with W L a standard parabolic subgroup of W K of type A n − 2 . Let x ∈ X J .

Appealing to the Mackey decomposition [3, Lemma 2.1.9],

Again using [3, Algorithm B, §2.1]

and hence

Then we see that J d 2 ∩ K = L and hence

If x ∈ X J ∩ K K , then, as W K is of type D n − 1 (with [ n − 1 ] , [ n ] in X J ∩ K K ), induction yields that either (i) or (ii) hold. Now suppose that x ∈ d 2 ⁢ X L K . Therefore, x = d 2 ⁢ e for some e ∈ X L K . Again applying induction, this time to X L K , we deduce that we have a reduced expression e = a ⁢ b , where either a ∈ { [ n − 1 ] , [ n − 1 , n − 2 , n ] } with 𝑏 having neither appearances of [ n − 1 ] nor [ n ] , or

Assume the first possibility holds, so a ∈ { [ n − 1 ] , [ n − 1 , n − 2 , n ] } . If a = [ n − 1 ] , then

which is of the desired form. Meanwhile, if a = [ n − 1 , n − 2 , n ] , then

which is alternative (ii). In the remaining case, we have

again yielding alternative (ii), and this completes the proof of the theorem. ∎

Let x ∈ I ⁢ ( W ) . Either 𝑥 is adjacent in E 0 ⁢ ( W ) to some r ∈ R , or 𝑊 is finite and x = w 0 .

Proof

Assume, for a contradiction, that 𝑥 is not adjacent to any 𝑟 in 𝑅. Then ℓ ⁢ ( x ⁢ r ) = ℓ ⁢ ( x ) − 1 , and so α r ∈ N ⁢ ( x ) for all r ∈ R . Since

it follows that Φ + ⊆ N ⁢ ( x ) . If 𝑊 is infinite, then Φ + is infinite, whereas N ⁢ ( x ) is finite, a contradiction. Therefore, it follows that 𝑊 is finite, and now Φ + ⊆ N ⁢ ( x ) forces Φ + = N ⁢ ( x ) , which implies x = w 0 . ∎

Proof of Theorem 1.2

If 𝑊 has a longest element w 0 (that is, if 𝑊 is finite), observe that ℓ ⁢ ( x ⁢ w 0 ) = ℓ ⁢ ( x ) + ℓ ⁢ ( w 0 ) only when x = 1 , and so { w 0 } is a connected component of E 0 ⁢ ( W ) . Let 𝑥 be any element of I ⁢ ( W ) other than (where it exists) w 0 . By Lemma 3.1, 𝑥 is adjacent to some r ∈ R . If 𝑟 and 𝑠 are distinct elements of 𝑅, then ℓ ⁢ ( r ⁢ s ) = 2 = ℓ ⁢ ( r ) + ℓ ⁢ ( s ) and so 𝑟 and 𝑠 are adjacent. Hence E ̂ 0 ⁢ ( W ) is connected with diameter at most 3. ∎

Let 𝑊 be a Coxeter group of finite rank at least 3. If there exist distinct fundamental reflections 𝑟 and 𝑠 such that W R ∖ { r } and W R ∖ { s } are both finite, then the diameter of E ̂ 0 ⁢ ( W ) is 3.

Proof

Let X = W R ∖ { r } and Y = W R ∖ { s } with longest elements 𝑥 and 𝑦 respectively. Then x , y ∈ I ⁢ ( W ) . For all t ∈ R ∖ { r } , it follows from the maximal length of 𝑥 that ℓ ⁢ ( x ⁢ t ) < ℓ ⁢ ( x ) . Hence if 𝑧 is adjacent to 𝑥 in E ̂ 0 ⁢ ( W ) , then every reduced expression for 𝑧 starts with 𝑟. Similarly, if 𝑧 is adjacent to 𝑦 in E ̂ 0 ⁢ ( W ) , every reduced expression for 𝑧 starts with 𝑠. It follows that 𝑥 and 𝑦 have no common neighbours. Finally, let t ∈ R ∖ { r , s } . By the Exchange Condition, there exists a reduced expression for 𝑦 starting in 𝑡, and so 𝑥 and 𝑦 are not adjacent. Hence the diameter of E ̂ 0 ⁢ ( W ) is 3. ∎

Let 𝑊 be an infinite Coxeter group. Then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) ∈ { 2 , 3 } . If there exists r ∈ R such that m r ⁢ s = ∞ for all s ∈ R ∖ { r } , or if 𝑊 has infinite rank, then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 2 .

Proof

Since 𝑊 is infinite, 𝑊 has rank at least 2, and so there exist distinct r , s in 𝑅. By Theorem 1.2, E ̂ 0 ⁢ ( W ) is connected with diameter at most 3. If 𝑟 commutes with 𝑠, then r ⁢ s is an involution in E ̂ 0 ⁢ ( W ) which is not adjacent to 𝑟. Otherwise, r ⁢ s ⁢ r is an involution in E ̂ 0 ⁢ ( W ) which is not adjacent to 𝑟. Therefore, Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) ≥ 2 . Hence Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) ∈ { 2 , 3 } .

Now, suppose there exists r ∈ R such that m r ⁢ s = ∞ for all s ∈ R ∖ { r } . For any element 𝑤 of 𝑊, if 𝑤 has a reduced expression ending in 𝑟, then all reduced expressions for 𝑤 end in 𝑟. To see this, recall that any reduced expression for 𝑤 can be obtained from any other by use of braid relations, and there are no braid relations involving 𝑟, so there is no braid relation that will replace the 𝑟 at the end of 𝑤 with a different fundamental reflection. Moreover, if 𝑤 is an involution with a reduced expression ending in 𝑟, then since w − 1 = w , every reduced expression for 𝑤 also begins with 𝑟. Let 𝑥 and 𝑦 be involutions and let s ∈ R ∖ { r } . If both 𝑥 and 𝑦 have reduced expressions ending with 𝑟, then these reduced expressions also begin with 𝑟, and all reduced expressions for 𝑥 and for 𝑦 both begin and end with 𝑟. Therefore, both 𝑥 and 𝑦 are adjacent to 𝑠 and are thus distance at most 2 apart. Meanwhile, if neither 𝑥 nor 𝑦 have a reduced expression beginning with 𝑟, then both 𝑥 and 𝑦 are adjacent to 𝑟. If 𝑥 has a reduced expression ending in 𝑟 and 𝑦 does not (or vice versa), then ℓ ⁢ ( x ⁢ y ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) , because in order for any cancellation to occur, at least one braid relation involving 𝑟 would have to be used, in order to juxtapose and cancel any pairs of fundamental reflections from the expressions for 𝑥 and 𝑦. Therefore, 𝑥 and 𝑦 are adjacent. Hence Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 2 .

The final case to consider is where 𝑊 has infinite rank. Since any element of 𝑊 has finite length, any reduced expression involves at most finitely many fundamental reflections. Thus, for any involutions 𝑥 and 𝑦, there is a fundamental reflection 𝑟 that does not appear in any reduced expression for 𝑥 or 𝑦. Clearly, 𝑥 and 𝑦 are both adjacent to 𝑟 in E ̂ 0 ⁢ ( W ) , and so, again, Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 2 . ∎

Let 𝑊 be a Coxeter group of rank at least 2.

1. If 𝑊 is a finite Coxeter group, then

   Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = { 1 if ⁢ W = W ⁢ ( A 2 ) ⁢ or ⁢ W ⁢ ( A 1 × A 1 ) , 3 otherwise .

2. If 𝑊 is an affine Coxeter group, then

   Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = { 2 if ⁢ W = W ⁢ ( A ̃ 1 ) , 3 otherwise .

3. If 𝑊 is compact hyperbolic, then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 .

Proof

Suppose that 𝑊 is finite, affine or compact hyperbolic. If rank ⁢ ( W ) ≥ 3 , then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 by Lemma 3.2. Hence we may assume that rank ⁢ ( W ) = 2 .

(i) Let 𝑊 be finite of rank 2 with generators 𝑟 and 𝑠 such that o ⁢ ( r ⁢ s ) = m . If m = 2 , then 𝑊 is of type A 1 × A 1 , and if m = 3 , then 𝑊 is of type A 2 . In both cases, I ⁢ ( W ) ∖ { w 0 } = { r , s } , and 𝑟 is adjacent to 𝑠 in E ̂ 0 ⁢ ( W ) . Thus the diameter is 1.

Suppose m > 3 . If 𝑚 is even, then w 0 is central and so w 0 ⁢ r is an involution. Since N ⁢ ( w 0 ⁢ r ) = Φ + ∖ { α r } , it follows that w 0 ⁢ r is adjacent only to 𝑟 in E ̂ 0 ⁢ ( W ) . Similarly, w 0 ⁢ s is adjacent only to 𝑠. Therefore, w 0 ⁢ r and w 0 ⁢ s share no common neighbours and Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 . If 𝑚 is odd, then every involution in 𝑊 is a reflection. Since ℓ ⁢ ( ( r ⁢ s ) ( m − 3 ) / 2 ⁢ r ) = m − 2 , any involution adjacent to ( r ⁢ s ) ( m − 3 ) / 2 ⁢ r in E ̂ 0 ⁢ ( W ) has length at most 2. But all reflections have odd length. Thus ( r ⁢ s ) ( m − 3 ) / 2 ⁢ r is adjacent only to 𝑠, and similarly ( s ⁢ r ) ( m − 3 ) / 2 ⁢ s is adjacent only to 𝑟. As before, this implies that Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 .

(ii) The only affine Coxeter group with rank less than 3 is W ⁢ ( A ̃ 1 ) , which has Dynkin diagram . By Lemma 3.3, Diam ⁢ ( E 0 ⁢ ( W ⁢ ( A ̃ 1 ) ) ) = 2 .

(iii) All compact hyperbolic Coxeter groups have rank at least 3. ∎

Suppose that 𝑊 is infinite and reducible with W = W 1 × ⋯ × W m , where each W i is irreducible and m > 1 .

1. If Diam ⁢ ( E ̂ 0 ⁢ ( W i ) ) = 2 for all i ∈ { 1 , … , m } , then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 2 .

2. If Diam ⁢ ( E ̂ 0 ⁢ ( W i ) ) = 3 for all i ∈ { 1 , … , m } , then Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 .

Proof

(i) By Theorem 3.4 (i), each W i is infinite, and so E ̂ 0 ⁢ ( W i ) = E 0 ⁢ ( W i ) by Theorem 1.2. Let x = ( x 1 , … , x m ) , y = ( y 1 , … , y m ) ∈ I ⁢ ( W ) . If x i is adjacent to y i in E 0 ⁢ ( W i ) for all i ∈ { 1 , … , m } , then 𝑥 is adjacent to 𝑦 in E 0 ⁢ ( W ) . Hence assume that there exists i ∈ { 1 , … , m } such that x i and y i are not adjacent in E 0 ⁢ ( W i ) . Since Diam ⁢ ( E 0 ⁢ ( W i ) ) = 2 , there exists z i ∈ I ⁢ ( W i ) such that x i and y i are both adjacent to z i . Hence 𝑥 and 𝑦 are both adjacent to the element of I ⁢ ( W ) with z i in its 𝑖-th position and the identity elsewhere.

(ii) For all i ∈ { 1 , … , m } , there exist x i , y i ∈ I ⁢ ( W i ) such that d ⁢ ( x i , y i ) = 3 . Let x : = ( x 1 , … , x m ) , y : = ( y 1 , … , y m ) ∈ I ( W ) . Then clearly 𝑥 and 𝑦 are not adjacent. Suppose there exists z = ( z 1 , … , z m ) ∈ I ⁢ ( W ) adjacent to both 𝑥 and 𝑦. Since ℓ ⁢ ( x i ⁢ z i ) ≤ ℓ ⁢ ( x i ) + ℓ ⁢ ( z i ) , it follows that ℓ ⁢ ( x ⁢ z ) = ℓ ⁢ ( x ) + ℓ ⁢ ( z ) if and only if ℓ ⁢ ( x i ⁢ z i ) = ℓ ⁢ ( x i ) + ℓ ⁢ ( z i ) for all i ∈ { 1 , … , m } . In order for 𝑧 to be an involution, either z i ∈ I ⁢ ( W i ) or z i = 1 . The first is impossible because d ⁢ ( x i , y i ) = 3 . Therefore, we have z i = 1 for all 𝑖, and so z = 1 ∉ I ⁢ ( W ) , a contradiction. Thus Diam ⁢ ( E ̂ 0 ⁢ ( W ) ) = 3 . ∎

For n ≥ 2 , let W n be the Coxeter group given by

Then Diam ⁢ ( E ̂ 0 ⁢ ( W n ) ) = 2 . Furthermore, if n = 2 , then no pair of adjacent involutions in E ̂ 0 ⁢ ( W n ) has a neighbour in common. If n > 2 , then every pair of adjacent involutions has a neighbour in common.