Reminders on the strong Bruhat order
Let ( W , S ) (W,S) ( W , S ) [1 ] , [5 ] , [15 ] , [12 ] , [8 ] for definitions and properties of Coxeter groups. The length of an element w ∈ W w \in W w ∈ W ℓ ( w ) \ell(w) ℓ ( w )
We shall use the strong Bruhat order on W W W W W W [1 ] , and that is defined but left nameless in [12 §2.1] . One of the many ways to define it is as follows:
Proposition 1.1. Let u , v ∈ W u,v \in W u , v ∈ W s 1 s 2 ⋯ s m s_1s_2 \cdots s_m s 1 s 2 ⋯ s m v v v u ≤ v u \le v u ≤ v u u u i 1 < i 2 < ⋯ < i k i_1 < i_2 < \cdots< i_k i 1 < i 2 < ⋯ < i k { 1 , 2 , … , m } \{1,2,\ldots,m\} { 1 , 2 , … , m } u = s i 1 s i 2 ⋯ s i k u=s_{i_1}s_{i_2}\cdots s_{i_k} u = s i 1 s i 2 ⋯ s i k
This indeed can be used as a definition of ≤ \le ≤ u ≤ v u \le v u ≤ v [12 Proposition 2.4 (iii)] or [1 Theorem 2.2.2] or [10 §5.10] for proofs.
Henceforth, inequality signs such as ≤ \le ≤ < < < ≥ \ge ≥ > > > W W W
Proposition 1.2. Let u , v ∈ W u,v \in W u , v ∈ W
(a) If u ≤ v u \le v u ≤ v ℓ ( u ) ≤ ℓ ( v ) \ell(u) \le\ell(v) ℓ ( u ) ≤ ℓ ( v )
(b) If u < v u < v u < v ℓ ( u ) < ℓ ( v ) \ell(u) < \ell(v) ℓ ( u ) < ℓ ( v )
(c) We have u ≤ v u \le v u ≤ v u − 1 ≤ v − 1 u^{-1} \le v^{-1} u − 1 ≤ v − 1
(d) We have u < v u < v u < v u − 1 < v − 1 u^{-1} < v^{-1} u − 1 < v − 1
Proof. All four parts follow easily from Proposition 1.1. (For parts (a) and (b) , use the trivial observation that ℓ ( s i 1 s i 2 ⋯ s i k ) ≤ k \ell(s_{i_1}s_{i_2}\cdots s_{i_k}) \le k ℓ ( s i 1 s i 2 ⋯ s i k ) ≤ k i 1 < i 2 < ⋯ < i k i_1 < i_2 < \cdots< i_k i 1 < i 2 < ⋯ < i k (c) and (d) , observe that if s 1 s 2 ⋯ s m s_1s_2\cdots s_m s 1 s 2 ⋯ s m w ∈ W w \in W w ∈ W s m s m − 1 ⋯ s 1 s_ms_{m-1}\cdots s_1 s m s m − 1 ⋯ s 1 w − 1 w^{-1} w − 1
For the next proposition, we recall the notion of a reflection. The elements of S S S simple reflections (of W W W W W W reflections (of W W W T T T
Proposition 1.3. Let t ∈ W t \in W t ∈ W w ∈ W w \in W w ∈ W
(a) If ℓ ( w ) < ℓ ( t w ) \ell(w) < \ell(tw) ℓ ( w ) < ℓ ( tw ) w < t w w < tw w < tw
(b) If ℓ ( w ) < ℓ ( w t ) \ell(w) < \ell(wt) ℓ ( w ) < ℓ ( w t ) w < w t w < wt w < w t
Proof. (a) This follows immediately from the definition of the strong Bruhat order given in [1 Definition 2.1.1] . Indeed, w − 1 t w w^{-1}tw w − 1 tw t t t ℓ ( w ) < ℓ ( t w ) \ell(w) < \ell(tw) ℓ ( w ) < ℓ ( tw ) w → w − 1 t w t w w \xrightarrow{w^{-1}tw} tw w w − 1 tw tw [1 Definition 2.1.1] , and therefore w < t w w < tw w < tw (a) .
Alternatively, Proposition 1.3 (a) can also be proved using the definition of the strong Bruhat order given in [12 §2.1] . Namely, we observe that t 2 = 1 t^2=1 t 2 = 1 u : = t w u := tw u := tw t u = t t 2 ⏟ = 1 w = w tu = t\underbrace{t^2}_{=1}w = w t u = t = 1 t 2 w = w u = s 1 s 2 ⋯ s q u = s_1s_2\cdots s_q u = s 1 s 2 ⋯ s q u u u ℓ ( w ) < ℓ ( t w ) \ell(w) < \ell(tw) ℓ ( w ) < ℓ ( tw ) t u = w tu = w t u = w ℓ ( t u ) = ℓ ( w ) < ℓ ( t w ) = ℓ ( u ) \ell(tu) = \ell(w) < \ell(tw) = \ell(u) ℓ ( t u ) = ℓ ( w ) < ℓ ( tw ) = ℓ ( u ) t w = u tw = u tw = u [12 Lemma 2.2, (iii) $\Longrightarrow$ (ii)] (applied to u u u w w w i ∈ { 1 , 2 , … , q } i \in\{1,2,\ldots,q\} i ∈ { 1 , 2 , … , q } i i i
w = t ⏟ = s 1 s 2 ⋯ s i ⋯ s 2 s 1 = ( s 1 s 2 ⋯ s i − 1 ) ( s i s i − 1 ⋯ s 1 ) = ( s 1 s 2 ⋯ s i ) ( s i + 1 s i + 2 ⋯ s q ) = ( s 1 s 2 ⋯ s i − 1 ) ( s i s i − 1 ⋯ s 1 ) ⋅ ( s 1 s 2 ⋯ s i ) ( s i + 1 s i + 2 ⋯ s q ) = s 1 s 2 ⋯ s i − 1 s i + 1 s i + 2 ⋯ s q . \begin{aligned}
w &= \underbrace{t}_{=s_1s_2\cdots s_i\cdots s_2s_1} \\
&= (s_1s_2\cdots s_{i-1})(s_is_{i-1}\cdots s_1)
= (s_1s_2\cdots s_i)(s_{i+1}s_{i+2}\cdots s_q) \\
&= (s_1s_2\cdots s_{i-1})(s_is_{i-1}\cdots s_1)\cdot(s_1s_2\cdots s_i)(s_{i+1}s_{i+2}\cdots s_q) \\
&= s_1s_2\cdots s_{i-1}s_{i+1}s_{i+2}\cdots s_q.
\end{aligned} w = = s 1 s 2 ⋯ s i ⋯ s 2 s 1 t = ( s 1 s 2 ⋯ s i − 1 ) ( s i s i − 1 ⋯ s 1 ) = ( s 1 s 2 ⋯ s i ) ( s i + 1 s i + 2 ⋯ s q ) = ( s 1 s 2 ⋯ s i − 1 ) ( s i s i − 1 ⋯ s 1 ) ⋅ ( s 1 s 2 ⋯ s i ) ( s i + 1 s i + 2 ⋯ s q ) = s 1 s 2 ⋯ s i − 1 s i + 1 s i + 2 ⋯ s q . Since this is clearly a subproduct of the reduced expression s 1 s 2 ⋯ s q s_1s_2\cdots s_q s 1 s 2 ⋯ s q u u u w ≤ u w \leq u w ≤ u w ≤ u = t w w \leq u = tw w ≤ u = tw w ≠ t w w \ne tw w = tw t ≠ 1 t \ne1 t = 1 w < t w w < tw w < tw
(b) This can be easily derived from part (a) using Proposition 1.2 (d). Namely: Assume that ℓ ( w ) < ℓ ( w t ) \ell(w) < \ell(wt) ℓ ( w ) < ℓ ( w t ) u ∈ W u \in W u ∈ W ℓ ( u − 1 ) = ℓ ( u ) \ell(u^{-1}) = \ell(u) ℓ ( u − 1 ) = ℓ ( u ) ℓ ( w − 1 ) = ℓ ( w ) \ell(w^{-1}) = \ell(w) ℓ ( w − 1 ) = ℓ ( w ) ℓ ( ( w t ) − 1 ) = ℓ ( w t ) \ell((wt)^{-1}) = \ell(wt) ℓ (( w t ) − 1 ) = ℓ ( w t ) t t t t − 1 = t t^{-1} = t t − 1 = t ( w t ) − 1 = t − 1 w − 1 = t w − 1 (wt)^{-1} = t^{-1}w^{-1} = tw^{-1} ( w t ) − 1 = t − 1 w − 1 = t w − 1
Thus, we can rewrite ℓ ( ( w t ) − 1 ) = ℓ ( w t ) \ell((wt)^{-1}) = \ell(wt) ℓ (( w t ) − 1 ) = ℓ ( w t ) ℓ ( t w − 1 ) = ℓ ( w t ) \ell(tw^{-1}) = \ell(wt) ℓ ( t w − 1 ) = ℓ ( w t )
ℓ ( w − 1 ) = ℓ ( w ) < ℓ ( w t ) = ℓ ( t w − 1 ) . \ell(w^{-1}) = \ell(w) < \ell(wt) = \ell(tw^{-1}). ℓ ( w − 1 ) = ℓ ( w ) < ℓ ( w t ) = ℓ ( t w − 1 ) . Thus, Proposition 1.3 (a) (applied to w − 1 w^{-1} w − 1 w w w w − 1 < t w − 1 = ( w t ) − 1 w^{-1} < tw^{-1} = (wt)^{-1} w − 1 < t w − 1 = ( w t ) − 1 u = w u = w u = w v = w t v = wt v = w t w < w t w < wt w < w t
The last property of the Bruhat order that we will need is a known lemma [1 Lemma 2.2.10] :
Lemma 1.4. Let x , y ∈ W x,y \in W x , y ∈ W t ∈ W t \in W t ∈ W x < x t x < xt x < x t y < t y y < ty y < t y x y < x t y xy < xty x y < x t y
Proof. We have x < x t x < xt x < x t ℓ ( x ) < ℓ ( x t ) \ell(x) < \ell(xt) ℓ ( x ) < ℓ ( x t ) ℓ ( x t ) > ℓ ( x ) \ell(xt) > \ell(x) ℓ ( x t ) > ℓ ( x ) y < t y y < ty y < t y ℓ ( t y ) > ℓ ( y ) \ell(ty) > \ell(y) ℓ ( t y ) > ℓ ( y )
Set t ′ : = x t x − 1 t' := xtx^{-1} t ′ := x t x − 1 x t = t ′ x xt = t'x x t = t ′ x x t y = t ′ x y xty = t'xy x t y = t ′ x y x y < t ′ x y xy < t'xy x y < t ′ x y t ′ t' t ′ t t t x y xy x y t ′ t' t ′ w w w t t t ℓ ( x y ) < ℓ ( t ′ x y ) \ell(xy) < \ell(t'xy) ℓ ( x y ) < ℓ ( t ′ x y ) [5 errata, Lemma 4.2.4] shows that ℓ ( x t y ) > ℓ ( x y ) \ell(xty) > \ell(xy) ℓ ( x t y ) > ℓ ( x y ) ℓ ( x y ) < ℓ ( x t y ) = ℓ ( t ′ x y ) \ell(xy) < \ell(xty) = \ell(t'xy) ℓ ( x y ) < ℓ ( x t y ) = ℓ ( t ′ x y ) x t = t ′ x xt = t'x x t = t ′ x
For any u , v ∈ W u,v \in W u , v ∈ W u ≤ v u \leq v u ≤ v [ u , v ] [u,v] [ u , v ]
{ x ∈ W ∣ u ≤ x ≤ v } . \{x \in W \mid u \leq x \leq v\}. { x ∈ W ∣ u ≤ x ≤ v } . This set is called a Bruhat interval .
If X X X W W W m ∈ X m \in X m ∈ X Bruhat minimum of X X X x ∈ X x \in X x ∈ X m ≤ x m \leq x m ≤ x X X X W W W X X X m 1 m_1 m 1 m 2 m_2 m 2 X X X m 1 ≤ m 2 m_1 \leq m_2 m 1 ≤ m 2 m 2 ≤ m 1 m_2 \leq m_1 m 2 ≤ m 1 m 1 = m 2 m_1 = m_2 m 1 = m 2 X X X m m m X X X x ≠ m x \ne m x = m X X X m < x m < x m < x ℓ ( m ) < ℓ ( x ) \ell(m) < \ell(x) ℓ ( m ) < ℓ ( x ) Proposition 1.2 (b)).
All the same applies to Bruhat maxima: If X X X W W W M ∈ X M \in X M ∈ X Bruhat maximum of X X X x ∈ X x \in X x ∈ X M ≥ x M \geq x M ≥ x M M M X X X
Any Bruhat interval [ u , v ] [u,v] [ u , v ] u u u v v v W W W
The theorems
If I I I S S S W I W_I W I W W W I I I [1 Proposition 2.4.1 (i) and (ii)] or [12 §9.4 and §9.6] ) that W I W_I W I I I I W W W parabolic subgroup of W W W [10 ] , use this word more widely for conjugates of such subgroups).
Given two subsets I I I J J J S S S W I w W J W_IwW_J W I w W J W W W parabolic double cosets ; they have been studied by various authors from the viewpoints of descent algebras ([16 ] ), Hecke algebras ([4 ] ), row-semistandard tableaux ([18 ] ) and contingency tables ([9 ] ). Recently, the total number of distinct parabolic double cosets in a finite Coxeter group was computed ([2 ] ).
A classical result (essentially a consequence of [3 Exercise IV.§1.3] ) says that each parabolic double coset W I w W J W_IwW_J W I w W J I I I J J J S S S W I W_I W I W J W_J W J W W W W I W_I W I W J W_J W J W I w W J W_IwW_J W I w W J [2 Proposition 2.7 (c)] and [11 ] ), but proofs are not easy to find in the literature, so we will include a proof for the sake of completeness:
Theorem 2.1. Let I I I J J J S S S W I W_I W I W J W_J W J w ∈ W w \in W w ∈ W W I w W J W_IwW_J W I w W J
In 2024, the MathOverflow user “fool rabbit” conjectured1 W I u W J v W K W_IuW_JvW_K W I u W J v W K I , J , K I,J,K I , J , K S S S W W W 2
Theorem 2.2. Let I , J , K I,J,K I , J , K S S S W I , W J , W K W_I,W_J,W_K W I , W J , W K u , v ∈ W u,v \in W u , v ∈ W W I u W J v W K W_IuW_JvW_K W I u W J v W K W W W
Theorem 2.3. Let I , J , K I,J,K I , J , K S S S W I , W J , W K W_I,W_J,W_K W I , W J , W K u , v ∈ W u,v \in W u , v ∈ W W I u W J v W K W_IuW_JvW_K W I u W J v W K W W W
However, triple cosets are not as well-behaved as double cosets. In particular, even if W W W Theorem 2.1 ). For example, in the symmetric group S 3 S_3 S 3 A 2 A_2 A 2 W ∅ s 2 W { s 1 } s 2 W ∅ = s 2 W { s 1 } s 2 = { 1 , w 0 } W_{\varnothing}s_2W_{\{s_1\}}s_2W_{\varnothing}=s_2W_{\{s_1\}}s_2=\{1,w_0\} W ∅ s 2 W { s 1 } s 2 W ∅ = s 2 W { s 1 } s 2 = { 1 , w 0 }
One could also define “parabolic quadruple cosets” W I u W J v W K w W L W_IuW_JvW_KwW_L W I u W J v W K w W L Theorem 2.2 . For instance, if W W W S 3 S_3 S 3 W ∅ s 1 W { s 2 } 1 W { s 1 } s 2 W ∅ W_{\varnothing}s_1W_{\{s_2\}}1W_{\{s_1\}}s_2W_{\varnothing} W ∅ s 1 W { s 2 } 1 W { s 1 } s 2 W ∅ s 1 s_1 s 1 s 2 s_2 s 2
Double cosets revisited
We will use some classical properties of double cosets in Coxeter groups:
Proposition 3.1. Let P P P Q Q Q S S S w ∈ W w \in W w ∈ W
(a) The double coset W P w W Q W_PwW_Q W P w W Q
Let us denote this element by m m m
(b) Each u ∈ W P w W Q u \in W_PwW_Q u ∈ W P w W Q u = x m y u=xmy u = x m y x ∈ W P x \in W_P x ∈ W P y ∈ W Q y \in W_Q y ∈ W Q
(c) The element m m m W P w W Q W_PwW_Q W P w W Q
(d) Let x ∈ W P x \in W_P x ∈ W P y ∈ W Q y \in W_Q y ∈ W Q ℓ ( x m y ) ≠ ℓ ( x ) + ℓ ( m ) + ℓ ( y ) \ell(xmy) \ne\ell(x)+\ell(m)+\ell(y) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) x ′ ∈ W P x' \in W_P x ′ ∈ W P y ′ ∈ W Q y' \in W_Q y ′ ∈ W Q x ′ < x x' < x x ′ < x y ′ < y y' < y y ′ < y x m y = x ′ m y ′ xmy=x'my' x m y = x ′ m y ′
Most of this proposition is well-known (see, e.g., [4 ] and [?, Proposition 8.3]), and in particular its part (a) appears in [12 Proposition 9.15 (a)] . However, we shall give a self-contained proof using nothing but the above definition of the Bruhat order and the well-known deletion property of Coxeter groups:
Theorem 3.2 (Deletion property). Let w ∈ W w \in W w ∈ W s 1 , s 2 , … , s k ∈ S s_1,s_2,\ldots,s_k \in S s 1 , s 2 , … , s k ∈ S w = s 1 s 2 ⋯ s k w=s_1s_2\cdots s_k w = s 1 s 2 ⋯ s k not a reduced expression for w w w ℓ ( w ) < k \ell(w)<k ℓ ( w ) < k s 1 s 2 ⋯ s k s_1s_2\cdots s_k s 1 s 2 ⋯ s k w w w i < j i<j i < j { 1 , 2 , … , k } \{1,2,\ldots,k\} { 1 , 2 , … , k } w = s 1 s 2 ⋯ s i ^ ⋯ s j ^ ⋯ s k w=s_1s_2\cdots\hat{s_i}\cdots\hat{s_j}\cdots s_k w = s 1 s 2 ⋯ s i ^ ⋯ s j ^ ⋯ s k
Proof. See [1 Proposition 1.4.7] or [5 Theorem 3.3.4 (iii) + Theorem 3.2.16 (E) $\Longrightarrow$ (D)] or [15 (a) $\Longrightarrow$ (d)] or [8 Theorem 1.2.5] or [14 Corollary 1.23] or [Werner12, Satz 2.15]. □
Proof of Proposition 3.1. We shall postpone the proof of part (a) for now. Of course, this complicates our life, since the uniqueness of the element m m m m m m W P w W Q W_PwW_Q W P w W Q m m m W P w W Q W_PwW_Q W P w W Q
(b) Let u ∈ W P w W Q u \in W_PwW_Q u ∈ W P w W Q m ∈ W P w W Q m \in W_PwW_Q m ∈ W P w W Q W P w W Q = W P m W Q W_PwW_Q=W_PmW_Q W P w W Q = W P m W Q u ∈ W P w W Q = W P m W Q u \in W_PwW_Q=W_PmW_Q u ∈ W P w W Q = W P m W Q u = x m y u=xmy u = x m y u u u x ∈ W P x \in W_P x ∈ W P y ∈ W Q y \in W_Q y ∈ W Q
(d) In the following, the “hat” symbol ( ^ \hat{\ } ^ a 1 , a 2 , … , a k a_1,a_2,\ldots,a_k a 1 , a 2 , … , a k i , j ∈ { 1 , 2 , … , k } i,j \in\{1,2,\ldots,k\} i , j ∈ { 1 , 2 , … , k } i < j i<j i < j a 1 a 2 ⋯ a i ^ ⋯ a j ^ ⋯ a k a_1a_2\cdots\hat{a_i}\cdots\hat{a_j}\cdots a_k a 1 a 2 ⋯ a i ^ ⋯ a j ^ ⋯ a k
a 1 a 2 ⋯ a i − 1 a i + 1 a i + 2 ⋯ a j − 1 a j + 1 a j + 2 ⋯ a k a_1a_2\cdots a_{i-1}a_{i+1}a_{i+2}\cdots a_{j-1}a_{j+1}a_{j+2}\cdots a_k a 1 a 2 ⋯ a i − 1 a i + 1 a i + 2 ⋯ a j − 1 a j + 1 a j + 2 ⋯ a k (that is, the product a 1 a 2 ⋯ a k a_1a_2\cdots a_k a 1 a 2 ⋯ a k a i a_i a i a j a_j a j
Let p 1 p 2 ⋯ p i p_1p_2\cdots p_i p 1 p 2 ⋯ p i q 1 q 2 ⋯ q j q_1q_2\cdots q_j q 1 q 2 ⋯ q j m 1 m 2 ⋯ m ℓ m_1m_2\cdots m_\ell m 1 m 2 ⋯ m ℓ x x x y y y m m m p k p_k p k P P P x ∈ W P x \in W_P x ∈ W P q k q_k q k Q Q Q y ∈ W Q y \in W_Q y ∈ W Q ℓ ( x ) = i \ell(x)=i ℓ ( x ) = i ℓ ( y ) = j \ell(y)=j ℓ ( y ) = j ℓ ( m ) = ℓ \ell(m)=\ell ℓ ( m ) = ℓ x = p 1 p 2 ⋯ p i x=p_1p_2\cdots p_i x = p 1 p 2 ⋯ p i y = q 1 q 2 ⋯ q j y=q_1q_2\cdots q_j y = q 1 q 2 ⋯ q j m = m 1 m 2 ⋯ m ℓ m=m_1m_2\cdots m_\ell m = m 1 m 2 ⋯ m ℓ
x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j . (1) xmy=p_1p_2\cdots p_im_1m_2\cdots m_\ell q_1q_2\cdots q_j. \tag*{(1)} x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j . (1) This cannot be a reduced expression for x m y xmy x m y
ℓ ( x m y ) = i + ℓ + j = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) , \ell(xmy)=i+\ell+j=\ell(x)+\ell(m)+\ell(y), ℓ ( x m y ) = i + ℓ + j = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) , which would contradict our condition ℓ ( x m y ) ≠ ℓ ( x ) + ℓ ( m ) + ℓ ( y ) \ell(xmy)\ne\ell(x)+\ell(m)+\ell(y) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j Theorem 3.2 ), we can remove two of the factors
p 1 , p 2 , … , p i , m 1 , m 2 , … , m ℓ , q 1 , q 2 , … , q j p_1,p_2,\ldots,p_i,m_1,m_2,\ldots,m_\ell,q_1,q_2,\ldots,q_j p 1 , p 2 , … , p i , m 1 , m 2 , … , m ℓ , q 1 , q 2 , … , q j from this product without changing the value of this product. Let us refer to these two factors as the redundant factors (since their removal leaves the product unchanged). We are in one of the following five cases:
Case 1: Both redundant factors have the forms m a m_a m a m b m_b m b a , b ∈ { 1 , 2 , … , ℓ } a,b\in\{1,2,\ldots,\ell\} a , b ∈ { 1 , 2 , … , ℓ }
Case 2: Both redundant factors have the forms p a p_a p a p b p_b p b a , b ∈ { 1 , 2 , … , i } a,b\in\{1,2,\ldots,i\} a , b ∈ { 1 , 2 , … , i }
Case 3: Both redundant factors have the forms q a q_a q a q b q_b q b a , b ∈ { 1 , 2 , … , j } a,b\in\{1,2,\ldots,j\} a , b ∈ { 1 , 2 , … , j }
Case 4: Only one of the redundant factors has the form m a m_a m a a ∈ { 1 , 2 , … , ℓ } a\in\{1,2,\ldots,\ell\} a ∈ { 1 , 2 , … , ℓ } p b p_b p b q b q_b q b
Case 5: One of the redundant factors has the form p a p_a p a a ∈ { 1 , 2 , … , i } a\in\{1,2,\ldots,i\} a ∈ { 1 , 2 , … , i } q b q_b q b b ∈ { 1 , 2 , … , j } b\in\{1,2,\ldots,j\} b ∈ { 1 , 2 , … , j }
We shall now show that Cases 1, 2, 3 and 4 are all impossible. Indeed:
Assume that we are in Case 1. Thus, both redundant factors have the forms m a m_a m a m b m_b m b a , b ∈ { 1 , 2 , … , ℓ } a,b\in\{1,2,\ldots,\ell\} a , b ∈ { 1 , 2 , … , ℓ } a , b a,b a , b a < b a<b a < b
p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m a ^ ⋯ m b ^ ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j
=
p_1p_2\cdots p_i m_1m_2\cdots\widehat{m_a}\cdots\widehat{m_b}\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m a ⋯ m b ⋯ m ℓ q 1 q 2 ⋯ q j (since the removal of the redundant factors leaves the product p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j
m 1 m 2 ⋯ m ℓ = m 1 m 2 ⋯ m a ^ ⋯ m b ^ ⋯ m ℓ , m_1m_2\cdots m_\ell=m_1m_2\cdots\widehat{m_a}\cdots\widehat{m_b}\cdots m_\ell, m 1 m 2 ⋯ m ℓ = m 1 m 2 ⋯ m a ⋯ m b ⋯ m ℓ , which shows that m 1 m 2 ⋯ m ℓ m_1m_2\cdots m_\ell m 1 m 2 ⋯ m ℓ m 1 m 2 ⋯ m ℓ m_1m_2\cdots m_\ell m 1 m 2 ⋯ m ℓ
Assume that we are in Case 4. Thus, only one of the redundant factors has the form m a m_a m a a ∈ { 1 , 2 , … , ℓ } a\in\{1,2,\ldots,\ell\} a ∈ { 1 , 2 , … , ℓ } m b m_b m b p b p_b p b q b q_b q b b b b p b p_b p b a a a b b b
p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p b ^ ⋯ p i m 1 m 2 ⋯ m a ^ ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j
=
p_1p_2\cdots\widehat{p_b}\cdots p_i m_1m_2\cdots\widehat{m_a}\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p b ⋯ p i m 1 m 2 ⋯ m a ⋯ m ℓ q 1 q 2 ⋯ q j (since the removal of the redundant factors leaves the product p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j
x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p b ^ ⋯ p i ⏟ ∈ W P m 1 m 2 ⋯ m a ^ ⋯ m ℓ q 1 q 2 ⋯ q j ⏟ ∈ W Q \begin{aligned}
xmy &= p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j \\
&= \underbrace{p_1p_2\cdots\widehat{p_b}\cdots p_i}_{\in W_P}m_1m_2\cdots\widehat{m_a}\cdots m_\ell\underbrace{q_1q_2\cdots q_j}_{\in W_Q}
\end{aligned} x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = ∈ W P p 1 p 2 ⋯ p b ⋯ p i m 1 m 2 ⋯ m a ⋯ m ℓ ∈ W Q q 1 q 2 ⋯ q j (since all the p k p_k p k P P P
∈ W P ( m 1 m 2 ⋯ m a ^ ⋯ m ℓ ) W Q , \in W_P\left(m_1m_2\cdots\widehat{m_a}\cdots m_\ell\right)W_Q, ∈ W P ( m 1 m 2 ⋯ m a ⋯ m ℓ ) W Q , (since all the q k q_k q k Q Q Q
so that
W P ( m 1 m 2 ⋯ m a ^ ⋯ m ℓ ) W Q = W P x ⏟ = W P (since x ∈ W P ) m y W Q ⏟ = W Q (since y ∈ W Q ) = W P m W Q = W P w W Q W_P\left(m_1m_2\cdots\widehat{m_a}\cdots m_\ell\right)W_Q
=
\underbrace{W_Px}_{=W_P\text{ (since }x\in W_P\text{)}}
\ m\
\underbrace{yW_Q}_{=W_Q\text{ (since }y\in W_Q\text{)}}
=
W_PmW_Q
=
W_PwW_Q W P ( m 1 m 2 ⋯ m a ⋯ m ℓ ) W Q = = W P (since x ∈ W P ) W P x m = W Q (since y ∈ W Q ) y W Q = W P m W Q = W P w W Q (since m ∈ W P w W Q m\in W_PwW_Q m ∈ W P w W Q m 1 m 2 ⋯ m a ^ ⋯ m ℓ ∈ W P w W Q m_1m_2\cdots\widehat{m_a}\cdots m_\ell\in W_PwW_Q m 1 m 2 ⋯ m a ⋯ m ℓ ∈ W P w W Q m 1 m 2 ⋯ m a ^ ⋯ m ℓ m_1m_2\cdots\widehat{m_a}\cdots m_\ell m 1 m 2 ⋯ m a ⋯ m ℓ ≤ ℓ − 1 < ℓ = ℓ ( m ) \leq\ell-1<\ell=\ell(m) ≤ ℓ − 1 < ℓ = ℓ ( m ) W P w W Q W_PwW_Q W P w W Q m m m W P w W Q W_PwW_Q W P w W Q m 1 m 2 ⋯ m a ^ ⋯ m ℓ ∈ W P w W Q m_1m_2\cdots\widehat{m_a}\cdots m_\ell\in W_PwW_Q m 1 m 2 ⋯ m a ⋯ m ℓ ∈ W P w W Q
Thus we know that Cases 1, 2, 3 and 4 are all impossible. Hence, we must be in Case 5. In other words, one of the redundant factors has the form p a p_a p a a ∈ { 1 , 2 , … , i } a\in\{1,2,\ldots,i\} a ∈ { 1 , 2 , … , i } q b q_b q b b ∈ { 1 , 2 , … , j } b\in\{1,2,\ldots,j\} b ∈ { 1 , 2 , … , j } a a a b b b
p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p a ^ ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q b ^ ⋯ q j \begin{aligned}
p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j
&=p_1p_2\cdots\widehat{p_a}\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots\widehat{q_b}\cdots q_j
\end{aligned} p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p a ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q b ⋯ q j (since the removal of the redundant factors leaves the product p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j
x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p a ^ ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q b ^ ⋯ q j = p 1 p 2 ⋯ p a ^ ⋯ p i ⋅ m ⋅ q 1 q 2 ⋯ q b ^ ⋯ q j . \begin{aligned}
xmy &= p_1p_2\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots q_j \\
&= p_1p_2\cdots\widehat{p_a}\cdots p_i m_1m_2\cdots m_\ell q_1q_2\cdots\widehat{q_b}\cdots q_j \\
&= p_1p_2\cdots\widehat{p_a}\cdots p_i\cdot m\cdot q_1q_2\cdots\widehat{q_b}\cdots q_j.
\end{aligned} x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j = p 1 p 2 ⋯ p a ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q b ⋯ q j = p 1 p 2 ⋯ p a ⋯ p i ⋅ m ⋅ q 1 q 2 ⋯ q b ⋯ q j . If we set x ′ : = p 1 p 2 ⋯ p a ^ ⋯ p i x' := p_1p_2\cdots\widehat{p_a}\cdots p_i x ′ := p 1 p 2 ⋯ p a ⋯ p i y ′ : = q 1 q 2 ⋯ q b ^ ⋯ q j y' := q_1q_2\cdots\widehat{q_b}\cdots q_j y ′ := q 1 q 2 ⋯ q b ⋯ q j x m y = x ′ m y ′ xmy=x'my' x m y = x ′ m y ′ x ′ ∈ W P x'\in W_P x ′ ∈ W P p k p_k p k P P P y ′ ∈ W Q y'\in W_Q y ′ ∈ W Q x ′ ≤ x x'\leq x x ′ ≤ x Proposition 1.1 , since x ′ = p 1 p 2 ⋯ p a ^ ⋯ p i x'=p_1p_2\cdots\widehat{p_a}\cdots p_i x ′ = p 1 p 2 ⋯ p a ⋯ p i p 1 p 2 ⋯ p i p_1p_2\cdots p_i p 1 p 2 ⋯ p i x x x ℓ ( x ′ ) < ℓ ( x ) \ell(x')<\ell(x) ℓ ( x ′ ) < ℓ ( x ) x ′ = p 1 p 2 ⋯ p a ^ ⋯ p i x'=p_1p_2\cdots\widehat{p_a}\cdots p_i x ′ = p 1 p 2 ⋯ p a ⋯ p i ≤ i − 1 < i = ℓ ( x ) \leq i-1<i=\ell(x) ≤ i − 1 < i = ℓ ( x ) x ′ ≠ x x'\ne x x ′ = x x ′ < x x'<x x ′ < x x ′ ≤ x x'\leq x x ′ ≤ x y ′ < y y'<y y ′ < y x ′ ∈ W P x'\in W_P x ′ ∈ W P y ′ ∈ W Q y'\in W_Q y ′ ∈ W Q x ′ < x x'<x x ′ < x y ′ < y y'<y y ′ < y x m y = x ′ m y ′ xmy=x'my' x m y = x ′ m y ′
(c) Since m ∈ W P w W Q m \in W_PwW_Q m ∈ W P w W Q u ∈ W P w W Q u \in W_PwW_Q u ∈ W P w W Q m ≤ u m \leq u m ≤ u
Let u ∈ W P w W Q u \in W_PwW_Q u ∈ W P w W Q (b) shows that there exists a factorization u = x m y u = xmy u = x m y u u u x ∈ W P x \in W_P x ∈ W P y ∈ W Q y \in W_Q y ∈ W Q ℓ ( x ) \ell(x) ℓ ( x ) ℓ ( x ) \ell(x) ℓ ( x ) x ′ ∈ W P x' \in W_P x ′ ∈ W P y ′ ∈ W Q y' \in W_Q y ′ ∈ W Q u = x ′ m y ′ u = x'my' u = x ′ m y ′
ℓ ( x ′ ) ≥ ℓ ( x ) . (2) \ell(x') \geq\ell(x). \tag*{(2)} ℓ ( x ′ ) ≥ ℓ ( x ) . (2) If we had ℓ ( x m y ) ≠ ℓ ( x ) + ℓ ( m ) + ℓ ( y ) \ell(xmy) \ne\ell(x) + \ell(m) + \ell(y) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) (d) would yield that there exist x ′ ∈ W P x' \in W_P x ′ ∈ W P y ′ ∈ W Q y' \in W_Q y ′ ∈ W Q x ′ < x x' < x x ′ < x y ′ < y y' < y y ′ < y x m y = x ′ m y ′ xmy = x'my' x m y = x ′ m y ′ u = x m y = x ′ m y ′ u = xmy = x'my' u = x m y = x ′ m y ′ ℓ ( x ′ ) < ℓ ( x ) \ell(x') < \ell(x) ℓ ( x ′ ) < ℓ ( x ) x ′ < x x' < x x ′ < x ℓ ( x m y ) ≠ ℓ ( x ) + ℓ ( m ) + ℓ ( y ) \ell(xmy) \ne\ell(x) + \ell(m) + \ell(y) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) \ell(xmy) = \ell(x) + \ell(m) + \ell(y) ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y )
Let p 1 p 2 ⋯ p i p_1p_2 \cdots p_i p 1 p 2 ⋯ p i q 1 q 2 ⋯ q j q_1q_2 \cdots q_j q 1 q 2 ⋯ q j m 1 m 2 ⋯ m ℓ m_1m_2 \cdots m_\ell m 1 m 2 ⋯ m ℓ x x x y y y m m m p k p_k p k P P P x ∈ W P x \in W_P x ∈ W P q k q_k q k Q Q Q y ∈ W Q y \in W_Q y ∈ W Q ℓ ( x ) = i \ell(x) = i ℓ ( x ) = i ℓ ( y ) = j \ell(y) = j ℓ ( y ) = j ℓ ( m ) = ℓ \ell(m) = \ell ℓ ( m ) = ℓ
x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j xmy = p_1p_2 \cdots p_i m_1m_2 \cdots m_\ell q_1q_2 \cdots q_j x m y = p 1 p 2 ⋯ p i m 1 m 2 ⋯ m ℓ q 1 q 2 ⋯ q j is a reduced expression for x m y xmy x m y ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) = i + ℓ + j \ell(xmy) = \ell(x) + \ell(m) + \ell(y) = i + \ell+ j ℓ ( x m y ) = ℓ ( x ) + ℓ ( m ) + ℓ ( y ) = i + ℓ + j
But removing all the factors p 1 , p 2 , … , p i p_1,p_2,\ldots,p_i p 1 , p 2 , … , p i q 1 , q 2 , … , q j q_1,q_2,\ldots,q_j q 1 , q 2 , … , q j m 1 m 2 ⋯ m ℓ = m m_1m_2 \cdots m_\ell= m m 1 m 2 ⋯ m ℓ = m m m m m ≤ x m y = u m \leq xmy = u m ≤ x m y = u (c) .
(a) We have already proved Proposition 3.1 (c) under the condition that m m m W P w W Q W_PwW_Q W P w W Q W P w W Q W_PwW_Q W P w W Q W P w W Q W_PwW_Q W P w W Q W P w W Q W_PwW_Q W P w W Q X X X W W W W P w W Q W_PwW_Q W P w W Q (a) . □
Proof of Theorem 2.2
Now, we prove our first main result:
Proof of Theorem 2.2. Proposition 3.1 (a) shows that the double coset W I u W J W_IuW_J W I u W J u ′ u' u ′ W J v W K W_JvW_K W J v W K v ′ v' v ′ u ′ u' u ′ v ′ v' v ′ W I u W J = W I u ′ W J W_IuW_J = W_Iu'W_J W I u W J = W I u ′ W J u ′ ∈ W I u W J u' \in W_IuW_J u ′ ∈ W I u W J W J v W K = W J v ′ W K W_JvW_K = W_Jv'W_K W J v W K = W J v ′ W K v ′ ∈ W J v W K v' \in W_JvW_K v ′ ∈ W J v W K
W I u W J v W K = W I u ′ W J v W K = W I u ′ W J v ′ W K . W_IuW_JvW_K = W_Iu'W_JvW_K = W_Iu'W_Jv'W_K. W I u W J v W K = W I u ′ W J v W K = W I u ′ W J v ′ W K . Hence,
u ′ v ′ = 1 ⏟ ∈ W I u ′ 1 ⏟ ∈ W J v ′ 1 ⏟ ∈ W K ∈ W I u ′ W J v ′ W K = W I u W J v W K . u'v'=\underbrace{1}_{\in W_I}u'\underbrace{1}_{\in W_J}v'\underbrace{1}_{\in W_K}\in W_Iu'W_Jv'W_K=W_IuW_JvW_K. u ′ v ′ = ∈ W I 1 u ′ ∈ W J 1 v ′ ∈ W K 1 ∈ W I u ′ W J v ′ W K = W I u W J v W K . Proposition 3.1 (a) shows that the double coset W I u ′ v ′ W K W_Iu'v'W_K W I u ′ v ′ W K m m m m m m
m ∈ W I u ′ v ′ W K = W I u ′ 1 ⏟ ∈ W J v ′ W K ⊆ W I u ′ W J v ′ W K = W I u W J v W K . m\in W_Iu'v'W_K=W_Iu'\underbrace{1}_{\in W_J}v'W_K\subseteq W_Iu'W_Jv'W_K=W_IuW_JvW_K. m ∈ W I u ′ v ′ W K = W I u ′ ∈ W J 1 v ′ W K ⊆ W I u ′ W J v ′ W K = W I u W J v W K . Proposition 3.1 (c) shows that m m m W I u ′ v ′ W K W_Iu'v'W_K W I u ′ v ′ W K r ∈ W I u ′ v ′ W K r\in W_Iu'v'W_K r ∈ W I u ′ v ′ W K m ≤ r m\leq r m ≤ r
We shall now prove that every p ∈ W I u W J v W K p\in W_IuW_JvW_K p ∈ W I u W J v W K m ≤ p m\leq p m ≤ p m m m W I u W J v W K W_IuW_JvW_K W I u W J v W K m ∈ W I u W J v W K m\in W_IuW_JvW_K m ∈ W I u W J v W K
First, let us show the following:
Claim 1: For every x ∈ W I x\in W_I x ∈ W I y ∈ W J y\in W_J y ∈ W J z ∈ W K z\in W_K z ∈ W K m ≤ x u ′ y v ′ z m\leq xu'yv'z m ≤ x u ′ y v ′ z
Proof of Claim 1. We proceed by strong induction on ℓ ( y ) \ell(y) ℓ ( y )
Base case: It is easy to see that Claim 1 holds for ℓ ( y ) = 0 \ell(y)=0 ℓ ( y ) = 0 ℓ ( y ) = 0 \ell(y)=0 ℓ ( y ) = 0 y = 1 y=1 y = 1 x u ′ y v ′ z = x u ′ v ′ z ∈ W I u ′ v ′ W K xu'yv'z=xu'v'z\in W_Iu'v'W_K x u ′ y v ′ z = x u ′ v ′ z ∈ W I u ′ v ′ W K m ≤ x u ′ v ′ z m\leq xu'v'z m ≤ x u ′ v ′ z r ∈ W I u ′ v ′ W K r\in W_Iu'v'W_K r ∈ W I u ′ v ′ W K m ≤ r m\leq r m ≤ r
Induction step: Let N N N ℓ ( y ) < N \ell(y)<N ℓ ( y ) < N ℓ ( y ) = N \ell(y)=N ℓ ( y ) = N
So let x ∈ W I x\in W_I x ∈ W I y ∈ W J y\in W_J y ∈ W J z ∈ W K z\in W_K z ∈ W K ℓ ( y ) = N \ell(y)=N ℓ ( y ) = N m ≤ x u ′ y v ′ z m\leq xu'yv'z m ≤ x u ′ y v ′ z
Since y ∈ W J y\in W_J y ∈ W J ℓ ( y ) = N > 0 \ell(y)=N>0 ℓ ( y ) = N > 0 y = y ˉ j y=\bar{y}j y = y ˉ j j ∈ J j\in J j ∈ J y ˉ ∈ W J \bar{y}\in W_J y ˉ ∈ W J ℓ ( y ˉ ) = ℓ ( y ) − 1 \ell(\bar{y})=\ell(y)-1 ℓ ( y ˉ ) = ℓ ( y ) − 1 y y y j j j y ˉ \bar{y} y ˉ j j j y ˉ \bar{y} y ˉ j ∈ J ⊆ W J j\in J\subseteq W_J j ∈ J ⊆ W J ℓ ( j ) = 1 \ell(j)=1 ℓ ( j ) = 1 j ∈ J ⊆ S j\in J\subseteq S j ∈ J ⊆ S ℓ ( y ˉ ) + 1 = ℓ ( y ) \ell(\bar{y})+1=\ell(y) ℓ ( y ˉ ) + 1 = ℓ ( y ) ℓ ( y ˉ ) = ℓ ( y ) − 1 \ell(\bar{y})=\ell(y)-1 ℓ ( y ˉ ) = ℓ ( y ) − 1 ℓ ( y ˉ ) = ℓ ( y ) − 1 < ℓ ( y ) = N \ell(\bar{y})=\ell(y)-1<\ell(y)=N ℓ ( y ˉ ) = ℓ ( y ) − 1 < ℓ ( y ) = N y ˉ \bar{y} y ˉ y y y
m ≤ x u ′ y ˉ v ′ z . (3) m\leq xu'\bar{y}v'z. \tag*{(3)} m ≤ x u ′ y ˉ v ′ z . (3) We are in one of the following three cases:
Case 1: We have ℓ ( x u ′ y ˉ ) ≠ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) \ell(xu'\bar{y})\ne\ell(x)+\ell(u')+\ell(\bar{y}) ℓ ( x u ′ y ˉ ) = ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ )
Case 2: We have ℓ ( j v ′ z ) ≠ ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) \ell(jv'z)\ne\ell(j)+\ell(v')+\ell(z) ℓ ( j v ′ z ) = ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z )
Case 3: None of the above holds.
Let us first consider Case 1. In this case, we have ℓ ( x u ′ y ˉ ) ≠ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) \ell(xu'\bar{y})\ne\ell(x)+\ell(u')+\ell(\bar{y}) ℓ ( x u ′ y ˉ ) = ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) I I I J J J u u u u ′ u' u ′ P P P Q Q Q w w w m m m x ′ ∈ W I x'\in W_I x ′ ∈ W I y ′ ∈ W J y'\in W_J y ′ ∈ W J x ′ < x x'<x x ′ < x y ′ < y y'<y y ′ < y x u ′ y ˉ = x ′ u ′ y ′ xu'\bar{y}=x'u'y' x u ′ y ˉ = x ′ u ′ y ′ x ′ x' x ′ y ′ y' y ′ y ′ < y y'<y y ′ < y ℓ ( y ′ ) < ℓ ( y ) = N \ell(y') < \ell(y) = N ℓ ( y ′ ) < ℓ ( y ) = N x ′ x' x ′ y ′ y' y ′ x x x y y y m ≤ x ′ u ′ y ′ z m \leq x'u'y'z m ≤ x ′ u ′ y ′ z x u ′ y = x ′ u ′ y ′ xu'y = x'u'y' x u ′ y = x ′ u ′ y ′ m ≤ x u ′ y v ′ z m \leq xu'yv'z m ≤ x u ′ y v ′ z
Let us next consider Case 2. In this case, we have ℓ ( j v ′ z ) ≠ ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) \ell(jv'z) \ne\ell(j) + \ell(v') + \ell(z) ℓ ( j v ′ z ) = ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) J J J K K K v v v v ′ v' v ′ j j j z z z P P P Q Q Q w w w m m m x x x y y y x ′ ∈ W J x' \in W_J x ′ ∈ W J y ′ ∈ W K y' \in W_K y ′ ∈ W K x ′ < j x' < j x ′ < j y ′ < z y' < z y ′ < z j v ′ z = x ′ v ′ y ′ jv'z = x'v'y' j v ′ z = x ′ v ′ y ′ j ∈ W J j \in W_J j ∈ W J x ′ x' x ′ y ′ y' y ′ x ′ < j x' < j x ′ < j ℓ ( x ′ ) < ℓ ( j ) = 1 \ell(x') < \ell(j) = 1 ℓ ( x ′ ) < ℓ ( j ) = 1 ℓ ( x ′ ) = 0 \ell(x') = 0 ℓ ( x ′ ) = 0 x ′ = 1 x' = 1 x ′ = 1 j v ′ z = x ′ v ′ y ′ jv'z = x'v'y' j v ′ z = x ′ v ′ y ′ j v ′ z = v ′ y ′ jv'z = v'y' j v ′ z = v ′ y ′
But our induction hypothesis (applied to y ˉ \bar{y} y ˉ y ′ y' y ′ y y y z z z m ≤ x u ′ y ˉ v ′ y ′ m \leq xu'\bar{y}v'y' m ≤ x u ′ y ˉ v ′ y ′ y ˉ ∈ W J \bar{y} \in W_J y ˉ ∈ W J y ′ ∈ W K y' \in W_K y ′ ∈ W K ℓ ( y ˉ ) < N \ell(\bar{y}) < N ℓ ( y ˉ ) < N y ˉ v ′ z ⏟ = y ′ j v ′ z = y ˉ v ′ y ′ \underbrace{\bar{y}v'z}_{=y'jv'z} = \bar{y}v'y' = y ′ j v ′ z y ˉ v ′ z = y ˉ v ′ y ′ m ≤ x u ′ y ˉ v ′ z m \leq xu'\bar{y}v'z m ≤ x u ′ y ˉ v ′ z
Finally, let us consider Case 3. In this case, none of the assumptions of the previous two cases hold. Thus, we have
ℓ ( x u ′ y ˉ ) = ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) and ℓ ( j v ′ z ) = ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) . \ell(xu'\bar{y}) = \ell(x) + \ell(u') + \ell(\bar{y}) \qquad\text{and}\qquad\ell(jv'z) = \ell(j) + \ell(v') + \ell(z). ℓ ( x u ′ y ˉ ) = ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) and ℓ ( j v ′ z ) = ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) . Furthermore, the triangle inequality for lengths (i.e., the fact that ℓ ( a b ) ≤ ℓ ( a ) + ℓ ( b ) \ell(ab) \leq\ell(a) + \ell(b) ℓ ( ab ) ≤ ℓ ( a ) + ℓ ( b ) a , b ∈ W a,b \in W a , b ∈ W
ℓ ( x u ′ y ˉ ) ≤ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) and ℓ ( v ′ z ) ≤ ℓ ( v ′ ) + ℓ ( z ) . \ell(xu'\bar{y}) \leq\ell(x) + \ell(u') + \ell(\bar{y}) \qquad\text{and}\qquad\ell(v'z) \leq\ell(v') + \ell(z). ℓ ( x u ′ y ˉ ) ≤ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) and ℓ ( v ′ z ) ≤ ℓ ( v ′ ) + ℓ ( z ) . Hence,
ℓ ( x u ′ y ˉ ) ≤ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) < ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ) = ℓ ( x u ′ y ˉ j ⏟ = y ) . \ell(xu'\bar{y}) \leq\ell(x) + \ell(u') + \ell(\bar{y}) < \ell(x) + \ell(u') + \ell(y) = \ell\left(\underbrace{xu'\bar{y}j}_{=y}\right). ℓ ( x u ′ y ˉ ) ≤ ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ˉ ) < ℓ ( x ) + ℓ ( u ′ ) + ℓ ( y ) = ℓ = y x u ′ y ˉ j . Since j j j j ∈ J ⊆ S j \in J \subseteq S j ∈ J ⊆ S x u ′ y ˉ < x u ′ y ˉ j xu'\bar{y} < xu'\bar{y}j x u ′ y ˉ < x u ′ y ˉ j w = x u ′ y ˉ w = xu'\bar{y} w = x u ′ y ˉ t = j t = j t = j
Furthermore,
ℓ ( j v ′ z ) = ℓ ( j ) ⏟ = 1 > 0 + ℓ ( v ′ ) + ℓ ( z ) > ℓ ( v ′ ) + ℓ ( z ) ≥ ℓ ( v ′ z ) \ell(jv'z) = \underbrace{\ell(j)}_{=1>0} + \ell(v') + \ell(z) > \ell(v') + \ell(z) \geq\ell(v'z) ℓ ( j v ′ z ) = = 1 > 0 ℓ ( j ) + ℓ ( v ′ ) + ℓ ( z ) > ℓ ( v ′ ) + ℓ ( z ) ≥ ℓ ( v ′ z ) (since ℓ ( v ′ z ) ≤ ℓ ( v ′ ) + ℓ ( z ) \ell(v'z) \leq\ell(v') + \ell(z) ℓ ( v ′ z ) ≤ ℓ ( v ′ ) + ℓ ( z ) ℓ ( v ′ z ) < ℓ ( j v ′ z ) \ell(v'z) < \ell(jv'z) ℓ ( v ′ z ) < ℓ ( j v ′ z ) j j j v ′ z < j v ′ z v'z < jv'z v ′ z < j v ′ z w = v ′ z w = v'z w = v ′ z t = j t = j t = j
Now we know that j j j x u ′ y ˉ < x u ′ y ˉ j xu'\bar{y} < xu'\bar{y}j x u ′ y ˉ < x u ′ y ˉ j v ′ z < j v ′ z v'z < jv'z v ′ z < j v ′ z x u ′ y ˉ xu'\bar{y} x u ′ y ˉ v ′ z v'z v ′ z j j j x x x y y y t t t x u ′ y ˉ v ′ z < x u ′ y ˉ j v ′ z xu'\bar{y}v'z < xu'\bar{y}jv'z x u ′ y ˉ v ′ z < x u ′ y ˉ j v ′ z
m ≤ x u ′ y ˉ v ′ z < x u ′ y ˉ j ⏟ = y v ′ z = x u ′ y v ′ z . m \leq xu'\bar{y}v'z < xu'\underbrace{\bar{y}j}_{=y}v'z = xu'yv'z. m ≤ x u ′ y ˉ v ′ z < x u ′ = y y ˉ j v ′ z = x u ′ y v ′ z . Thus, the induction step is complete in Case 3. We have now completed the induction step in all three cases. Thus, the induction is complete, and Claim 1 is proved.
Now, every p ∈ W I u W J v W K p \in W_IuW_JvW_K p ∈ W I u W J v W K p = x u ′ y v ′ z p = xu' yv' z p = x u ′ y v ′ z x ∈ W I x \in W_I x ∈ W I y ∈ W J y \in W_J y ∈ W J z ∈ W K z \in W_K z ∈ W K p ∈ W I u W J v W K = W I u ′ W J v ′ W K p \in W_IuW_JvW_K = W_Iu'W_Jv'W_K p ∈ W I u W J v W K = W I u ′ W J v ′ W K m ≤ p m \le p m ≤ p m ≤ x u ′ y v ′ z m \le xu'yv'z m ≤ x u ′ y v ′ z m ∈ W I u W J v W K m \in W_IuW_JvW_K m ∈ W I u W J v W K m m m W I u W J v W K W_IuW_JvW_K W I u W J v W K
Reminders on the longest element
Next, we recall the definition of the so-called longest element (aka longest word ) in a finite Coxeter group:
Proposition 5.1. Let W W W w ∈ W w \in W w ∈ W ℓ ( w ) \ell(w) ℓ ( w )
Proof. This is (contained in) [12 §11.1] or [1 Proposition 2.3.1] or [15 Proposition 80] or [5 Lemma 4.6.1] or [14 Lemma 1.33] .
The unique element w w w longest element of W W W w 0 w_0 w 0
Proposition 5.2. Let W W W w 0 w_0 w 0 u ∈ W u \in W u ∈ W ℓ ( u w 0 ) = ℓ ( w 0 ) − ℓ ( u ) \ell(uw_0) = \ell(w_0) - \ell(u) ℓ ( u w 0 ) = ℓ ( w 0 ) − ℓ ( u )
Proof. This is [12 §11.2 (a)] or [15 Proposition 80 (c)] or [1 Proposition 2.3.2 (ii)] . (It also follows easily from [5 Lemma 4.6.1 (a)] .)
Classical coset lemmas
We shall next prove a sequence of lemmas – most of them likely well-known – that will be used in the proofs of our other two results. The first is a classic (see, e.g., [12 Lemma 9.7 (b)] ):
Lemma 6.1. Let I I I S S S w ∈ W w \in W w ∈ W a a a b b b w W I wW_I w W I W I w W_Iw W I w
(a) For each x ∈ W I x \in W_I x ∈ W I ℓ ( x b ) = ℓ ( x ) + ℓ ( b ) \ell(xb) = \ell(x) + \ell(b) ℓ ( x b ) = ℓ ( x ) + ℓ ( b )
(b) For each y ∈ W I y \in W_I y ∈ W I ℓ ( a y ) = ℓ ( a ) + ℓ ( y ) \ell(ay) = \ell(a) + \ell(y) ℓ ( a y ) = ℓ ( a ) + ℓ ( y )
Proof. (a) Let x ∈ W I x \in W_I x ∈ W I ℓ ( x b ) = ℓ ( x ) + ℓ ( b ) \ell(xb) = \ell(x) + \ell(b) ℓ ( x b ) = ℓ ( x ) + ℓ ( b ) ℓ ( x b ) ≠ ℓ ( x ) + ℓ ( b ) \ell(xb) \ne\ell(x) + \ell(b) ℓ ( x b ) = ℓ ( x ) + ℓ ( b )
ℓ ( x b 1 ) = ℓ ( x b ) ≠ ℓ ( x ) + ℓ ( b ) = ℓ ( x ) + ℓ ( b ) + ℓ ( 1 ) \ell(xb1) = \ell(xb) \ne\ell(x) + \ell(b) = \ell(x) + \ell(b) + \ell(1) ℓ ( x b 1 ) = ℓ ( x b ) = ℓ ( x ) + ℓ ( b ) = ℓ ( x ) + ℓ ( b ) + ℓ ( 1 ) (since ℓ ( 1 ) = 0 \ell(1) = 0 ℓ ( 1 ) = 0
Since b ∈ W I w b \in W_Iw b ∈ W I w W I b = W I w = W I w W ∅ W_Ib = W_Iw = W_IwW_{\varnothing} W I b = W I w = W I w W ∅ W ∅ = { 1 } W_{\varnothing} = \{1\} W ∅ = { 1 } P = I P = I P = I Q = ∅ Q = \varnothing Q = ∅ W I w W ∅ W_IwW_{\varnothing} W I w W ∅ b b b b b b W I w = W I w W ∅ W_Iw = W_IwW_{\varnothing} W I w = W I w W ∅ ℓ ( x b 1 ) ≠ ℓ ( x ) + ℓ ( b ) + ℓ ( 1 ) \ell(xb1) \ne\ell(x) + \ell(b) + \ell(1) ℓ ( x b 1 ) = ℓ ( x ) + ℓ ( b ) + ℓ ( 1 ) P = I P = I P = I Q = ∅ Q = \varnothing Q = ∅ m = b m = b m = b y = 1 y = 1 y = 1 x ′ ∈ W I x' \in W_I x ′ ∈ W I y ′ ∈ W ∅ y' \in W_{\varnothing} y ′ ∈ W ∅ x ′ < x x' < x x ′ < x y ′ < 1 y' < 1 y ′ < 1 x b 1 = x ′ b y ′ xb1 = x'by' x b 1 = x ′ b y ′ y ′ < 1 y' < 1 y ′ < 1 W W W 1 1 1 Lemma 6.1 (a) is proved.
(b) This is similar to part (a) , but with the left and right sides swapped (i.e., use w W I = W ∅ w W I wW_I = W_{\varnothing}wW_I w W I = W ∅ w W I a y ay a y 1 a y 1ay 1 a y
The following lemma says that any element in a finite parabolic subgroup can be connected to the longest element of that subgroup by a sequence of multiplications by simple reflections, each of which increases the length by 1.
Lemma 6.2. Let I ⊆ S I \subseteq S I ⊆ S W I W_I W I y ∈ W I y \in W_I y ∈ W I w I w_I w I W I W_I W I r ≥ 0 r \ge0 r ≥ 0 y 0 , y 1 , … , y r ∈ W I y_0, y_1, \ldots, y_r \in W_I y 0 , y 1 , … , y r ∈ W I i 0 , i 1 , … , i r − 1 ∈ I i_0, i_1, \ldots, i_{r-1} \in I i 0 , i 1 , … , i r − 1 ∈ I y 0 = y y_0 = y y 0 = y y r = w I y_r = w_I y r = w I
y j + 1 = y j i j and ℓ ( y j + 1 ) = ℓ ( y j ) + 1 for every j ∈ { 0 , 1 , … , r − 1 } . y_{j+1} = y_ji_j \quad\text{and} \quad\ell(y_{j+1}) = \ell(y_j) + 1 \qquad\text{for every } j \in\{0,1,\ldots,r-1\}. y j + 1 = y j i j and ℓ ( y j + 1 ) = ℓ ( y j ) + 1 for every j ∈ { 0 , 1 , … , r − 1 } . Proof. We have y − 1 w I ∈ W I y^{-1}w_I \in W_I y − 1 w I ∈ W I y ∈ W I y \in W_I y ∈ W I w I ∈ W I w_I \in W_I w I ∈ W I
y − 1 w I = i 0 i 1 ⋯ i r − 1 y^{-1}w_I = i_0i_1\cdots i_{r-1} y − 1 w I = i 0 i 1 ⋯ i r − 1 for y − 1 w I y^{-1}w_I y − 1 w I i 0 , i 1 , … , i r − 1 ∈ I i_0, i_1, \ldots, i_{r-1} \in I i 0 , i 1 , … , i r − 1 ∈ I ℓ ( y − 1 w I ) = r \ell(y^{-1}w_I) = r ℓ ( y − 1 w I ) = r
w I = y i 0 i 1 ⋯ i r − 1 . w_I = yi_0i_1\cdots i_{r-1}. w I = y i 0 i 1 ⋯ i r − 1 . By Proposition 5.2 (applied to W I W_I W I w I w_I w I y − 1 y^{-1} y − 1 W W W w 0 w_0 w 0 u u u
ℓ ( y − 1 w I ) = ℓ ( w I ) − ℓ ( y − 1 ) = ℓ ( w I ) − ℓ ( y ) \ell(y^{-1}w_I) = \ell(w_I) - \ell(y^{-1}) = \ell(w_I) - \ell(y) ℓ ( y − 1 w I ) = ℓ ( w I ) − ℓ ( y − 1 ) = ℓ ( w I ) − ℓ ( y ) (since ℓ ( y − 1 ) = ℓ ( y ) \ell(y^{-1}) = \ell(y) ℓ ( y − 1 ) = ℓ ( y )
and therefore
ℓ ( w I ) − ℓ ( y ) = ℓ ( y − 1 w I ) = r . \ell(w_I) - \ell(y) = \ell(y^{-1}w_I) = r. ℓ ( w I ) − ℓ ( y ) = ℓ ( y − 1 w I ) = r . Next, for all j ∈ { 0 , 1 , … , r } j \in\{0,1,\ldots,r\} j ∈ { 0 , 1 , … , r }
y j = y i 0 i 1 ⋯ i j − 1 ∈ W I . y_j = yi_0i_1\cdots i_{j-1} \in W_I. y j = y i 0 i 1 ⋯ i j − 1 ∈ W I . In particular, y 0 = y y_0 = y y 0 = y y r = y i 0 i 1 ⋯ i r − 1 = w I y_r = y i_0 i_1 \cdots i_{r-1} = w_I y r = y i 0 i 1 ⋯ i r − 1 = w I j ∈ { 0 , 1 , … , r − 1 } j \in\{0,1,\ldots,r-1\} j ∈ { 0 , 1 , … , r − 1 } y j + 1 = y j i j y_{j+1} = y_j i_j y j + 1 = y j i j ℓ ( y j + 1 ) = ℓ ( y j i j ) = ℓ ( y j ) ± 1 \ell(y_{j+1}) = \ell(y_j i_j) = \ell(y_j) \pm1 ℓ ( y j + 1 ) = ℓ ( y j i j ) = ℓ ( y j ) ± 1 i j ∈ I ⊆ S i_j \in I \subseteq S i j ∈ I ⊆ S
ℓ ( y j + 1 ) − ℓ ( y j ) = ± 1 ≤ 1. (4) \ell(y_{j+1}) - \ell(y_j) = \pm1 \leq1. \tag*{(4)} ℓ ( y j + 1 ) − ℓ ( y j ) = ± 1 ≤ 1. (4) But the sum of these inequalities (4) over all j ∈ { 0 , 1 , … , r − 1 } j \in\{0,1,\ldots,r-1\} j ∈ { 0 , 1 , … , r − 1 } ℓ ( y r ) − ℓ ( y 0 ) ≤ r \ell(y_r) - \ell(y_0) \leq r ℓ ( y r ) − ℓ ( y 0 ) ≤ r ℓ ( y r ) − ℓ ( y 0 ) = ℓ ( w I ) − ℓ ( y ) = r \ell(y_r) - \ell(y_0) = \ell(w_I) - \ell(y) = r ℓ ( y r ) − ℓ ( y 0 ) = ℓ ( w I ) − ℓ ( y ) = r (4) must have already been an equality. That is, ℓ ( y j + 1 ) = ℓ ( y j ) + 1 \ell(y_{j+1}) = \ell(y_j) + 1 ℓ ( y j + 1 ) = ℓ ( y j ) + 1 j j j i j i_j i j y j y_j y j
Lemma 6.3. Let I ⊆ S I \subseteq S I ⊆ S W I W_I W I a , b ∈ W a,b \in W a , b ∈ W w I w_I w I W I W_I W I a 0 a_0 a 0 b 0 b_0 b 0 a W I aW_I a W I W I b W_Ib W I b a W I b = a 0 W I b 0 aW_Ib = a_0W_Ib_0 a W I b = a 0 W I b 0 a 0 w I b 0 a_0w_Ib_0 a 0 w I b 0
Proof. First, note that a W I = a 0 W I aW_I = a_0W_I a W I = a 0 W I a 0 ∈ a W I a_0 \in aW_I a 0 ∈ a W I W I b = W I b 0 W_Ib = W_Ib_0 W I b = W I b 0 b 0 ∈ W I b b_0 \in W_Ib b 0 ∈ W I b
a W I b = a 0 W I b = a 0 W I b 0 . aW_Ib = a_0W_Ib = a_0W_Ib_0. a W I b = a 0 W I b = a 0 W I b 0 . It remains to show that a 0 w I b 0 a_0w_Ib_0 a 0 w I b 0 a 0 W I b 0 a_0W_Ib_0 a 0 W I b 0 a 0 w I b 0 ∈ a 0 W I b 0 a_0w_Ib_0 \in a_0W_Ib_0 a 0 w I b 0 ∈ a 0 W I b 0 p ∈ a 0 W I b 0 p \in a_0W_Ib_0 p ∈ a 0 W I b 0 p ≤ a 0 w I b 0 p \leq a_0w_Ib_0 p ≤ a 0 w I b 0
So let p ∈ a 0 W I b 0 p \in a_0W_Ib_0 p ∈ a 0 W I b 0 p = a 0 y b 0 p = a_0yb_0 p = a 0 y b 0 y ∈ W I y \in W_I y ∈ W I y y y r ≥ 0 r \geq0 r ≥ 0 y 0 , y 1 , … , y r ∈ W I y_0,y_1,\ldots,y_r \in W_I y 0 , y 1 , … , y r ∈ W I i 0 , i 1 , … , i r − 1 ∈ I i_0,i_1,\ldots,i_{r-1} \in I i 0 , i 1 , … , i r − 1 ∈ I Lemma 6.2 . Then, y 0 = y y_0 = y y 0 = y y r = w I y_r = w_I y r = w I
y j + 1 = y j i j and ℓ ( y j + 1 ) = ℓ ( y j ) + 1 for every j ∈ { 0 , 1 , … , r − 1 } . y_{j+1} = y_j i_j \quad\text{and}\quad\ell(y_{j+1}) = \ell(y_j) + 1 \quad\text{for every } j \in\{0,1,\ldots,r-1\}. y j + 1 = y j i j and ℓ ( y j + 1 ) = ℓ ( y j ) + 1 for every j ∈ { 0 , 1 , … , r − 1 } . Let j ∈ { 0 , 1 , … , r − 1 } j \in\{0,1,\ldots,r-1\} j ∈ { 0 , 1 , … , r − 1 } a 0 a_0 a 0 a W I aW_I a W I Lemma 6.1 (b) to a a a a 0 a_0 a 0 y j y_j y j w w w a a a y y y
ℓ ( a 0 y j ) = ℓ ( a 0 ) + ℓ ( y j ) . \ell(a_0y_j) = \ell(a_0) + \ell(y_j). ℓ ( a 0 y j ) = ℓ ( a 0 ) + ℓ ( y j ) . Applying the same lemma to y j + 1 y_{j+1} y j + 1 y j y_j y j
ℓ ( a 0 y j + 1 ) = ℓ ( a 0 ) + ℓ ( y j + 1 ) . \ell(a_0y_{j+1}) = \ell(a_0) + \ell(y_{j+1}). ℓ ( a 0 y j + 1 ) = ℓ ( a 0 ) + ℓ ( y j + 1 ) . In view of y j + 1 = y j i j y_{j+1} = y_j i_j y j + 1 = y j i j ℓ ( y j + 1 ) = ℓ ( y j ) + 1 \ell(y_{j+1}) = \ell(y_j) + 1 ℓ ( y j + 1 ) = ℓ ( y j ) + 1
ℓ ( a 0 y j i j ) = ℓ ( a 0 ) + ℓ ( y j ) + 1 > ℓ ( a 0 ) + ℓ ( y j ) = ℓ ( a 0 y j ) . \ell(a_0y_j i_j) = \ell(a_0) + \ell(y_j) + 1 > \ell(a_0) + \ell(y_j) = \ell(a_0y_j). ℓ ( a 0 y j i j ) = ℓ ( a 0 ) + ℓ ( y j ) + 1 > ℓ ( a 0 ) + ℓ ( y j ) = ℓ ( a 0 y j ) . That is, ℓ ( a 0 y j ) < ℓ ( a 0 y j i j ) \ell(a_0y_j) < \ell(a_0y_j i_j) ℓ ( a 0 y j ) < ℓ ( a 0 y j i j ) i j ∈ I ⊆ S i_j \in I \subseteq S i j ∈ I ⊆ S
a 0 y j < a 0 y j i j (5) a_0y_j < a_0y_j i_j \tag*{(5)} a 0 y j < a 0 y j i j (5) (by Proposition 1.3 (b), applied to a 0 y j a_0y_j a 0 y j w w w i j i_j i j t t t i j ∈ I ⊆ W I i_j \in I \subseteq W_I i j ∈ I ⊆ W I i j b 0 ∈ W I b 0 = W I b i_jb_0 \in W_Ib_0 = W_Ib i j b 0 ∈ W I b 0 = W I b b 0 ∈ W I b b_0 \in W_Ib b 0 ∈ W I b ℓ ( i j b 0 ) ≥ ℓ ( b 0 ) \ell(i_jb_0) \geq\ell(b_0) ℓ ( i j b 0 ) ≥ ℓ ( b 0 ) b 0 b_0 b 0 W I b W_Ib W I b i j i_j i j ℓ ( i j b 0 ) = ℓ ( b 0 ) ± 1 \ell(i_jb_0)=\ell(b_0)\pm1 ℓ ( i j b 0 ) = ℓ ( b 0 ) ± 1 ℓ ( i j b 0 ) ≥ ℓ ( b 0 ) \ell(i_jb_0)\ge\ell(b_0) ℓ ( i j b 0 ) ≥ ℓ ( b 0 ) ± \pm ± + + + ℓ ( i j b 0 ) = ℓ ( b 0 ) + 1 > ℓ ( b 0 ) \ell(i_jb_0)=\ell(b_0)+1>\ell(b_0) ℓ ( i j b 0 ) = ℓ ( b 0 ) + 1 > ℓ ( b 0 ) ℓ ( b 0 ) < ℓ ( i j b 0 ) \ell(b_0)<\ell(i_jb_0) ℓ ( b 0 ) < ℓ ( i j b 0 ) i j ∈ I ⊆ S i_j\in I\subseteq S i j ∈ I ⊆ S
b 0 < i j b 0 (6) b_0<i_jb_0 \tag*{(6)} b 0 < i j b 0 (6) (by Proposition 1.3 (a), applied to b 0 b_0 b 0 w w w i j i_j i j t t t
Now, we can use the inequalities (5) and (6) to apply Lemma 1.4 with a 0 y j a_0y_j a 0 y j x x x i j i_j i j t t t b 0 b_0 b 0 y y y i j ∈ I i_j\in I i j ∈ I
a 0 y j b 0 < a 0 y j i j b 0 = a 0 y j + 1 b 0 a_0y_jb_0<a_0y_ji_jb_0=a_0y_{j+1}b_0 a 0 y j b 0 < a 0 y j i j b 0 = a 0 y j + 1 b 0 (since y j i j = y j + 1 y_ji_j=y_{j+1} y j i j = y j + 1
Since this is true for all j ∈ { 0 , 1 , … , r − 1 } j\in\{0,1,\ldots,r-1\} j ∈ { 0 , 1 , … , r − 1 }
a 0 y 0 b 0 < a 0 y 1 b 0 < ⋯ < a 0 y r b 0 , a_0y_0b_0<a_0y_1b_0<\cdots<a_0y_rb_0, a 0 y 0 b 0 < a 0 y 1 b 0 < ⋯ < a 0 y r b 0 , hence a 0 y 0 b 0 ≤ a 0 y r b 0 a_0y_0b_0\le a_0y_rb_0 a 0 y 0 b 0 ≤ a 0 y r b 0 r = 0 r=0 r = 0 y 0 = y y_0=y y 0 = y y r = w I y_r=w_I y r = w I a 0 y b 0 ≤ a 0 w I b 0 a_0yb_0\le a_0w_Ib_0 a 0 y b 0 ≤ a 0 w I b 0 p = a 0 y b 0 ≤ a 0 w I b 0 p=a_0yb_0\le a_0w_Ib_0 p = a 0 y b 0 ≤ a 0 w I b 0
If w ∈ W w\in W w ∈ W s ∈ S s\in S s ∈ S w s < w ws<w w s < w w s > w ws>w w s > w ℓ ( w s ) \ell(ws) ℓ ( w s ) ℓ ( w ) − 1 \ell(w)-1 ℓ ( w ) − 1 ℓ ( w ) + 1 \ell(w)+1 ℓ ( w ) + 1 w s < w ws<w w s < w w s > w ws>w w s > w
Lemma 6.4. Consider v , w ∈ W v,w\in W v , w ∈ W v ≤ w v\le w v ≤ w s ∈ S s\in S s ∈ S
(a) If s w < w sw<w s w < w s v ≤ w sv\le w s v ≤ w
(b) If w s < w ws<w w s < w v s ≤ w vs\le w v s ≤ w
Proof. (a) This is well-known. Indeed, assume that s w < w sw<w s w < w [12 Corollary 2.5 (a)] (applied to y = v y=v y = v z = w z=w z = w v ≤ w v\le w v ≤ w s v ≤ w sv\le w s v ≤ w
(b) Assume that w s < w ws<w w s < w w s ws w s u u u w w w v v v
( w s ) − 1 < w − 1 . (ws)^{-1}<w^{-1}. ( w s ) − 1 < w − 1 . But s ∈ S s\in S s ∈ S s − 1 = s s^{-1}=s s − 1 = s ( w s ) − 1 = s − 1 w − 1 = s w − 1 (ws)^{-1}=s^{-1}w^{-1}=sw^{-1} ( w s ) − 1 = s − 1 w − 1 = s w − 1 s − 1 = s s^{-1}=s s − 1 = s
s w − 1 = ( w s ) − 1 < w − 1 . sw^{-1}=(ws)^{-1}<w^{-1}. s w − 1 = ( w s ) − 1 < w − 1 . Next, recall that v ≤ w v\le w v ≤ w v − 1 ≤ w − 1 v^{-1}\le w^{-1} v − 1 ≤ w − 1 s w − 1 < w − 1 sw^{-1}<w^{-1} s w − 1 < w − 1 v − 1 v^{-1} v − 1 w − 1 w^{-1} w − 1 v v v w w w
s v − 1 ≤ w − 1 . sv^{-1}\le w^{-1}. s v − 1 ≤ w − 1 . We can again apply Proposition 1.2 (c), with s v − 1 sv^{-1} s v − 1 u u u w − 1 w^{-1} w − 1 v v v ( s v − 1 ) − 1 ≤ ( w − 1 ) − 1 (sv^{-1})^{-1}\le(w^{-1})^{-1} ( s v − 1 ) − 1 ≤ ( w − 1 ) − 1 s − 1 = s s^{-1}=s s − 1 = s v s ≤ w vs\le w v s ≤ w
Lemma 6.5. Choose I ⊆ S I \subseteq S I ⊆ S W I W_I W I v , w ∈ W v,w \in W v , w ∈ W v ≤ w v \le w v ≤ w W I w = 1 W I w W_Iw = 1W_Iw W I w = 1 W I w w W I = w W I 1 wW_I = wW_I1 w W I = w W I 1 Lemma 6.3 .
(a) Let M M M W I w W_Iw W I w x ∈ W I x \in W_I x ∈ W I x v ≤ M xv \le M xv ≤ M
(b) Let N N N w W I wW_I w W I x ∈ W I x \in W_I x ∈ W I v x ≤ N vx \le N v x ≤ N
Proof. (a) We induct on the length of x x x x = 1 x = 1 x = 1 x v = v ≤ w ≤ M xv = v \le w \le M xv = v ≤ w ≤ M w ∈ W I w w \in W_Iw w ∈ W I w M M M W I w W_Iw W I w
Next, assume that x ≠ 1 x \ne1 x = 1 t ∈ I t \in I t ∈ I x ′ ∈ W I x' \in W_I x ′ ∈ W I x = t x ′ x = tx' x = t x ′ ℓ ( x ′ ) = ℓ ( x ) − 1 \ell(x') = \ell(x) - 1 ℓ ( x ′ ) = ℓ ( x ) − 1 t t t x x x x ′ x' x ′ x ′ v ≤ M x'v \le M x ′ v ≤ M M ∈ W I w M \in W_Iw M ∈ W I w t ∈ I ⊆ W I t \in I \subseteq W_I t ∈ I ⊆ W I t M ∈ W I W I w = W I w tM \in W_IW_Iw = W_Iw tM ∈ W I W I w = W I w t M ≤ M tM \le M tM ≤ M M M M W I w W_Iw W I w t M < M tM < M tM < M t ≠ 1 t \ne1 t = 1 t M ≠ M tM \ne M tM = M Lemma 6.4 (a) (with x ′ v x'v x ′ v v v v M M M w w w t t t s s s t x ′ v ≤ M tx'v \le M t x ′ v ≤ M t x ′ = x tx' = x t x ′ = x x v ≤ M xv \le M xv ≤ M
(b) This is analogous to part (a); just read all products right-to-left (in particular, W I w W_Iw W I w w W I wW_I w W I Lemma 6.4 (b) instead of Lemma 6.4 (a). (Of course, t t t x x x
Lemma 6.6. Choose P , Q ⊆ S P,Q \subseteq S P , Q ⊆ S W P W_P W P W Q W_Q W Q r , s ∈ W r,s \in W r , s ∈ W r ≤ s r \le s r ≤ s
(a) The double coset W P s W Q W_PsW_Q W P s W Q
(b) Let M M M x ∈ W P x \in W_P x ∈ W P z ∈ W Q z \in W_Q z ∈ W Q x r z ≤ M xrz \le M x r z ≤ M
Proof. The coset s W Q sW_Q s W Q s W Q 1 sW_Q1 s W Q 1 Lemma 6.3 (applied to I = Q I = Q I = Q a = s a = s a = s b = 1 b = 1 b = 1 N N N W P N W_PN W P N 1 W P N 1W_PN 1 W P N M M M M ∈ W P N ⊆ W P s W Q M \in W_PN \subseteq W_PsW_Q M ∈ W P N ⊆ W P s W Q N ∈ s W Q N \in sW_Q N ∈ s W Q M M M W P s W Q W_PsW_Q W P s W Q
Let x ∈ W P x \in W_P x ∈ W P z ∈ W Q z \in W_Q z ∈ W Q r ≤ s r \le s r ≤ s Lemma 6.5 (b) with I = Q I = Q I = Q v = r v = r v = r w = s w = s w = s x = z x = z x = z r z ≤ N rz \le N r z ≤ N Lemma 6.5 (a) with I = P I = P I = P v = r z v = rz v = r z w = N w = N w = N
x r z ≤ M . (7) xrz \le M. \tag*{(7)} x r z ≤ M . (7) Note that the definitions of N N N M M M r r r r = s r = s r = s s ≤ s s \le s s ≤ s x s z ≤ M xsz \le M x sz ≤ M
So we have shown that x s z ≤ M xsz \le M x sz ≤ M x ∈ W P x \in W_P x ∈ W P z ∈ W Q z \in W_Q z ∈ W Q W P s W Q W_PsW_Q W P s W Q ≤ M \le M ≤ M M ∈ W P s W Q M \in W_PsW_Q M ∈ W P s W Q M M M W P s W Q W_{P} s W_{Q} W P s W Q (7) .
Proof of Theorem 2.1
Next we take aim at Theorem 2.1. First we establish a useful lemma that allows us to “disassemble” a Bruhat inequality:
Lemma 7.1. Let p , m , q , w ∈ W p,m,q,w \in W p , m , q , w ∈ W w ≤ p m q w \leq pmq w ≤ p m q p ′ , m ′ , q ′ ∈ W p',m',q' \in W p ′ , m ′ , q ′ ∈ W p ′ ≤ p p' \leq p p ′ ≤ p m ′ ≤ m m' \leq m m ′ ≤ m q ′ ≤ q q' \leq q q ′ ≤ q w = p ′ m ′ q ′ w = p'm'q' w = p ′ m ′ q ′
Proof. Let p 1 ⋯ p i p_1 \cdots p_i p 1 ⋯ p i m 1 ⋯ m j m_1 \cdots m_j m 1 ⋯ m j q 1 ⋯ q k q_1 \cdots q_k q 1 ⋯ q k p p p m m m q q q
p m q = p 1 ⋯ p i m 1 ⋯ m j q 1 ⋯ q k (8) pmq = p_1 \cdots p_i m_1 \cdots m_j q_1 \cdots q_k \tag*{(8)} p m q = p 1 ⋯ p i m 1 ⋯ m j q 1 ⋯ q k (8) is not necessarily a reduced expression. If it is not, then, by the deletion property (Theorem 3.2 ), we can delete two factors from it without changing the product. By applying this repeatedly, we eventually arrive at a reduced expression of p m q pmq p m q
p m q = p u 1 ⋯ p u x m v 1 ⋯ m v y q w 1 ⋯ q w z (9) pmq = p_{u_1} \cdots p_{u_x}m_{v_1} \cdots m_{v_y}q_{w_1} \cdots q_{w_z} \tag*{(9)} p m q = p u 1 ⋯ p u x m v 1 ⋯ m v y q w 1 ⋯ q w z (9) for some 1 ≤ u 1 < ⋯ < u x ≤ i 1 \leq u_1 < \cdots< u_x \leq i 1 ≤ u 1 < ⋯ < u x ≤ i 1 ≤ v 1 < ⋯ < v y ≤ j 1 \leq v_1 < \cdots< v_y \leq j 1 ≤ v 1 < ⋯ < v y ≤ j 1 ≤ w 1 < ⋯ < w z ≤ k 1 \leq w_1 < \cdots< w_z \leq k 1 ≤ w 1 < ⋯ < w z ≤ k p ∗ = p u 1 ⋯ p u x p^* = p_{u_1} \cdots p_{u_x} p ∗ = p u 1 ⋯ p u x m ∗ = m v 1 ⋯ m v y m^* = m_{v_1} \cdots m_{v_y} m ∗ = m v 1 ⋯ m v y q ∗ = q w 1 ⋯ q w z q^* = q_{w_1} \cdots q_{w_z} q ∗ = q w 1 ⋯ q w z p m q = p ∗ m ∗ q ∗ pmq = p^*m^*q^* p m q = p ∗ m ∗ q ∗ p ∗ ≤ p p^* \leq p p ∗ ≤ p p ∗ = p u 1 ⋯ p u x p^* = p_{u_1} \cdots p_{u_x} p ∗ = p u 1 ⋯ p u x p 1 ⋯ p i p_1 \cdots p_i p 1 ⋯ p i p p p m ∗ ≤ m m^* \leq m m ∗ ≤ m q ∗ ≤ q q^* \leq q q ∗ ≤ q
Since w ≤ p m q w \leq pmq w ≤ p m q w w w p u 1 ⋯ p u x m v 1 ⋯ m v y q w 1 ⋯ q w z p_{u_1} \cdots p_{u_x}m_{v_1} \cdots m_{v_y}q_{w_1} \cdots q_{w_z} p u 1 ⋯ p u x m v 1 ⋯ m v y q w 1 ⋯ q w z p m q pmq p m q Proposition 1.1 ). That is, we can write w w w
w = p u 1 ′ ⋯ p u a ′ m v 1 ′ ⋯ m v b ′ q w 1 ′ ⋯ q w c ′ (10) w = p_{u'_1} \cdots p_{u'_a}m_{v'_1} \cdots m_{v'_b}q_{w'_1} \cdots q_{w'_c} \tag*{(10)} w = p u 1 ′ ⋯ p u a ′ m v 1 ′ ⋯ m v b ′ q w 1 ′ ⋯ q w c ′ (10) where p u 1 ′ ⋯ p u a ′ p_{u'_1} \cdots p_{u'_a} p u 1 ′ ⋯ p u a ′ p u 1 ⋯ p u x p_{u_1} \cdots p_{u_x} p u 1 ⋯ p u x m v 1 ′ ⋯ m v b ′ m_{v'_1} \cdots m_{v'_b} m v 1 ′ ⋯ m v b ′ m v 1 ⋯ m v y m_{v_1} \cdots m_{v_y} m v 1 ⋯ m v y q w 1 ′ ⋯ q w c ′ q_{w'_1} \cdots q_{w'_c} q w 1 ′ ⋯ q w c ′ q w 1 ⋯ q w z q_{w_1} \cdots q_{w_z} q w 1 ⋯ q w z
Set p ′ : = p u 1 ′ ⋯ p u a ′ p' := p_{u'_1} \cdots p_{u'_a} p ′ := p u 1 ′ ⋯ p u a ′ m ′ : = m v 1 ′ ⋯ m v b ′ m' := m_{v'_1} \cdots m_{v'_b} m ′ := m v 1 ′ ⋯ m v b ′ q ′ : = q w 1 ′ ⋯ q w c ′ q' := q_{w'_1} \cdots q_{w'_c} q ′ := q w 1 ′ ⋯ q w c ′ p ′ = p u 1 ′ ⋯ p u a ′ p' = p_{u'_1} \cdots p_{u'_a} p ′ = p u 1 ′ ⋯ p u a ′ p u 1 ⋯ p u x p_{u_1} \cdots p_{u_x} p u 1 ⋯ p u x p 1 ⋯ p i p_1 \cdots p_i p 1 ⋯ p i p p p p ′ p' p ′ p p p Proposition 1.1 , we have p ′ ≤ p p' \leq p p ′ ≤ p m ′ ≤ m m' \leq m m ′ ≤ m q ′ ≤ q q' \leq q q ′ ≤ q w = p ′ m ′ q ′ w = p'm'q' w = p ′ m ′ q ′ p ′ , m ′ , q ′ p',m',q' p ′ , m ′ , q ′ p ′ , m ′ , q ′ p',m',q' p ′ , m ′ , q ′
Now we are ready to prove Theorem 2.1:
Proof of Theorem 2.1. Proposition 3.1 (c) yields that the double coset W I w W J W_IwW_J W I w W J m m m
By Lemma 6.6 (a) (applied to P = I P=I P = I Q = J Q=J Q = J s = w s=w s = w r = w r=w r = w W I w W J W_IwW_J W I w W J M M M
We shall now show that W I w W J W_IwW_J W I w W J [ m , M ] [m,M] [ m , M ]
Note that W I M W J = W I w W J W_IMW_J=W_IwW_J W I M W J = W I w W J M ∈ W I w W J M\in W_IwW_J M ∈ W I w W J W I m W J = W I w W J W_ImW_J=W_IwW_J W I m W J = W I w W J
Each x ∈ W I w W J x\in W_IwW_J x ∈ W I w W J m ≤ x m\leq x m ≤ x m m m W I w W J W_IwW_J W I w W J x ≤ M x\leq M x ≤ M M M M W I w W J W_IwW_J W I w W J x ∈ [ m , M ] x\in[m,M] x ∈ [ m , M ]
W I w W J ⊆ [ m , M ] . (11) W_IwW_J\subseteq[m,M]. \tag*{(11)} W I w W J ⊆ [ m , M ] . (11) We shall now prove the reverse inclusion.
Let u ∈ [ m , M ] u\in[m,M] u ∈ [ m , M ] u ≤ M u\leq M u ≤ M M ∈ W I w W J = W I m W J M\in W_IwW_J=W_ImW_J M ∈ W I w W J = W I m W J M = p m q M=pmq M = p m q p ∈ W I p\in W_I p ∈ W I q ∈ W J q\in W_J q ∈ W J p p p q q q u ≤ M = p m q u\leq M=pmq u ≤ M = p m q u u u w w w p ′ , m ′ , q ′ ∈ W p',m',q'\in W p ′ , m ′ , q ′ ∈ W p ′ ≤ p p'\leq p p ′ ≤ p m ′ ≤ m m'\leq m m ′ ≤ m q ′ ≤ q q'\leq q q ′ ≤ q u = p ′ m ′ q ′ u=p'm'q' u = p ′ m ′ q ′ p ′ , m ′ , q ′ p',m',q' p ′ , m ′ , q ′
Since p ∈ W I p\in W_I p ∈ W I p p p I I I p ′ ≤ p p'\leq p p ′ ≤ p p ′ p' p ′ p ′ p' p ′ I I I p ′ ∈ W I p'\in W_I p ′ ∈ W I q ′ ≤ q q'\leq q q ′ ≤ q q ∈ W J q\in W_J q ∈ W J q ′ ∈ W J q'\in W_J q ′ ∈ W J
Next, from u ∈ [ m , M ] u\in[m,M] u ∈ [ m , M ] m ≤ u = p ′ m ′ q ′ m\leq u=p'm'q' m ≤ u = p ′ m ′ q ′ p ′ p' p ′ m ′ m' m ′ q ′ q' q ′ m m m p p p m m m q q q w w w p ′ ′ , m ′ ′ , q ′ ′ ∈ W p'',m'',q''\in W p ′′ , m ′′ , q ′′ ∈ W p ′ ′ ≤ p ′ p''\leq p' p ′′ ≤ p ′ m ′ ′ ≤ m ′ m''\leq m' m ′′ ≤ m ′ q ′ ′ ≤ q ′ q''\leq q' q ′′ ≤ q ′ m = p ′ ′ m ′ ′ q ′ ′ m=p''m''q'' m = p ′′ m ′′ q ′′ p ′ ′ , m ′ ′ , q ′ ′ p'',m'',q'' p ′′ , m ′′ , q ′′
Previously we obtained p ′ ∈ W I p'\in W_I p ′ ∈ W I p ∈ W I p\in W_I p ∈ W I p ′ ≤ p p'\leq p p ′ ≤ p p ′ ′ ∈ W I p''\in W_I p ′′ ∈ W I p ′ ∈ W I p'\in W_I p ′ ∈ W I p ′ ′ ≤ p ′ p''\leq p' p ′′ ≤ p ′ q ′ ′ ∈ W J q''\in W_J q ′′ ∈ W J m = p ′ ′ m ′ ′ q ′ ′ ∈ W I m ′ ′ W J m=p''m''q''\in W_Im''W_J m = p ′′ m ′′ q ′′ ∈ W I m ′′ W J W I m W J = W I m ′ ′ W J W_ImW_J=W_Im''W_J W I m W J = W I m ′′ W J m ′ ′ ∈ W I m W J = W I w W J m''\in W_ImW_J=W_IwW_J m ′′ ∈ W I m W J = W I w W J m m m W I w W J W_IwW_J W I w W J m ≤ m ′ ′ m\leq m'' m ≤ m ′′
m ≤ m ′ ′ ≤ m ′ ≤ m . m\leq m''\leq m'\leq m. m ≤ m ′′ ≤ m ′ ≤ m . Thus, all ≤ \leq ≤ m ′ = m m'=m m ′ = m
u = p ′ m ′ q ′ ∈ W I m W J = W I w W J . u=p'm'q'\in W_ImW_J=W_IwW_J. u = p ′ m ′ q ′ ∈ W I m W J = W I w W J . So we have proved that each u ∈ [ m , M ] u\in[m,M] u ∈ [ m , M ] u ∈ W I w W J u\in W_IwW_J u ∈ W I w W J [ m , M ] ⊆ W I w W J [m,M]\subseteq W_IwW_J [ m , M ] ⊆ W I w W J W I w W J = [ m , M ] W_IwW_J=[m,M] W I w W J = [ m , M ] W I w W J W_IwW_J W I w W J
Proof of Theorem 2.3
At last, we prove Theorem 2.3 .
Proof of Theorem 2.3. First, since W J W_J W J Lemma 6.3 , the set u W J v uW_Jv u W J v n n n
Next, consider the double coset W I n W K W_InW_K W I n W K Lemma 6.6 (a) (applied to P = I P=I P = I Q = K Q=K Q = K r = n r=n r = n s = n s=n s = n W I n W K W_InW_K W I n W K M M M M M M W I u W J v W K W_IuW_JvW_K W I u W J v W K
First note that M ∈ W I n W K ⊆ W I u W J v W K M \in W_InW_K \subseteq W_IuW_JvW_K M ∈ W I n W K ⊆ W I u W J v W K n ∈ u W J v n \in uW_Jv n ∈ u W J v
Next, let p ∈ W I u W J v W K p \in W_IuW_JvW_K p ∈ W I u W J v W K p p p p = x u y v z p=xuyvz p = xu y v z x ∈ W I x \in W_I x ∈ W I y ∈ W J y \in W_J y ∈ W J z ∈ W K z \in W_K z ∈ W K u y v ∈ u W J v uyv \in uW_Jv u y v ∈ u W J v u y v ≤ n uyv \leq n u y v ≤ n n n n u W J v uW_Jv u W J v Lemma 6.6 (b) with r = u y v r=uyv r = u y v s = n s=n s = n P = I P=I P = I Q = K Q=K Q = K
p = x u y v z ≤ M . p=xuyvz\leq M. p = xu y v z ≤ M . Since this is true for all p ∈ W I u W J v W K p \in W_IuW_JvW_K p ∈ W I u W J v W K M M M W I u W J v W K W_IuW_JvW_K W I u W J v W K M ∈ W I u W J v W K M \in W_IuW_JvW_K M ∈ W I u W J v W K
Acknowledgments
We thank Hankyung Ko for conversations on double cosets at ICERM (2025), and Elizabeth Milicevic and Mark Skandera for further helpful discussions. Various versions of GPT were used to generate parts of the proofs of Theorems 2.1 and 2.3. None of the writing is AI-made, but everything was proofread by GPT (and by us as well).
