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The parabolic intersection W_I cap dW_Jd inverse for d in ^IW^J

Statement

Let $(S, m)$ , $W$ , $ℓ$ be as in Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, let $I, J \subseteq S$ , and let $d \in^{I} W^{J}$ . Put

$K : = I \cap d J d^{- 1} = {s \in I : d^{- 1} s d \in J},$

where $d J d^{- 1} = {d j d^{- 1} : j \in J} \subseteq T$ . Let $V_{I}$ , $Φ_{I}$ be as in Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives (2) and let $α \mapsto t_{α}$ be the root-reflection dictionary of The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1).

(1) The intersection. $W_{I} \cap d W_{J} d^{- 1} = W_{K}$ . Conjugating by $d^{- 1}$ gives

$d^{- 1} W_{K} d = W_{d^{- 1} K d}, d^{- 1} K d = {d^{- 1} s d : s \in K} \subseteq J,$

and equivalently $W_{J} \cap d^{- 1} W_{I} d = W_{d^{- 1} K d}$ .

(2) The descent form. If $y \in W_{I} \cap d W_{J} d^{- 1}$ and $y = s_{1} \dots s_{p}$ is a reduced expression of $y$ with letters $s_{1}, \dots, s_{p} \in I$ , then

$d^{- 1} s_{i} d \in J for every i = 1, \dots, p .$

(3) The conjugated positive root is simple. Let $s \in I$ . Then $z : = d^{- 1} s d$ is a reflection of $W$ (that is, $z \in T$ ), its root is $ρ (d)^{- 1} e_{s} \in Φ_{+}$ , and $z \in J$ if and only if this root is one of the simple roots $e_{j}$ , $j \in J$ . Moreover, if $z \in W_{J}$ then already $z \in J$ : a reflection of the form $d^{- 1} s d$ with $s \in I$ that lies in $W_{J}$ is necessarily a simple reflection of the parabolic root system $Φ_{J} = Φ \cap V_{J}$ of Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives (2), never a non-simple positive combination such as $e_{j} + e_{j^{'}}$ .

Facts & Assumptions

Given: a finite Coxeter matrix $(S, m)$ with presented group $W$ and length $ℓ$ , subsets $I, J \subseteq S$ , an element $d \in^{I} W^{J}$ and the subset $K = {s \in I : d^{- 1} s d \in J}$ .

[F1]

$W_{I} = ⟨ s : s \in I ⟩$ , $^{I} W = {w : ℓ (s w) > ℓ (w) for all s \in I}$ , $W^{J} = {w : ℓ (w s) > ℓ (w) for all s \in J}$ and $^{I} W^{J} =^{I} W \cap W^{J}$ ; in particular $d \in^{I} W^{J}$ satisfies $ℓ (s d) > ℓ (d)$ for $s \in I$ and $ℓ (d s) > ℓ (d)$ for $s \in J$ (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups).

[F2]

$W_{J} = ⟨ J ⟩ = {w : S (w) \subseteq J}$ , $W_{J} \cap S = J$ , $(W_{J}, J)$ is a Coxeter system with intrinsic length $ℓ ∣_{W_{J}}$ , every left coset $a W_{J}$ has a unique minimal element $d$ , characterized by $ℓ (d s) > ℓ (d)$ for $s \in J$ and satisfying $ℓ (d u) = ℓ (d) + ℓ (u)$ for all $u \in W_{J}$ , and dually $ℓ (u d) = ℓ (u) + ℓ (d)$ for the minimal representative $d \in^{J} W$ of a right coset and all $u \in W_{J}$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification).

[F3]

For $d \in^{I} W^{J}$ , every $x \in W_{I} d W_{J}$ admits some factorization $x = u d v$ with $u \in W_{I}$ , $v \in W_{J}$ and $ℓ (x) = ℓ (u) + ℓ (d) + ℓ (v)$ ; moreover $ℓ (d) \leq ℓ (x)$ for all $x \in W_{I} d W_{J}$ (Descent reduction, minimum-length elements, and the additive factorization in a double coset).

[F4]

Strong exchange: if $w = w_{1} \dots w_{n}$ is a reduced expression and $t \in T$ satisfies $ℓ (t w) < ℓ (w)$ , then there is a unique index $i$ with $t w = w_{1} \dots \hat{w_{i}} \dots w_{n}$ and $t = w_{1} \dots w_{i - 1} w_{i} w_{i - 1} \dots w_{1}$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange).

[F5]

The root-reflection dictionary: for $α \in Φ$ and $w \in W$ , $s \in S$ with $α = ρ (w) e_{s}$ the element $t_{α} : = w s w^{- 1}$ is well defined, $t_{ρ (w) α} = w t_{α} w^{- 1}$ , the map $Φ_{+} \to T$ , $α \mapsto t_{α}$ , is a bijection and $t_{α} = t_{β}$ if and only if $α = \pm β$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange).

[F6]

For all $w \in W$ and $s \in S$ : $ℓ (w s) > ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{+}$ and $ℓ (w s) < ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{-}$ (The root-length criterion and faithfulness of the canonical reflection representation).

[F7]

$T = {w s w^{- 1} : w \in W, s \in S}$ , so every conjugate $d^{- 1} s d$ of a simple reflection is a reflection (The canonical reflection homomorphism, roots, reflections, and the positive cone).

[F8]

$Φ_{J} = Φ \cap V_{J}$ , and the reflections lying in $W_{J}$ are exactly the $t_{α}$ with $α \in Φ_{J}^{+}$ ; in particular the simple roots $e_{j}$ , $j \in J$ , correspond to the simple reflections of $W_{J}$ , and a reflection of $W_{J}$ whose positive root is not any $e_{j}$ ( $j \in J$ ) cannot have length one: by [F2] a length-one element lies in $J$ , and [F5] then identifies its positive root with $e_{j}$ (Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives).

[F9]

For $w = s_{1} \dots s_{k}$ one has $ℓ (w) \leq k$ , so $ℓ (u v) \leq ℓ (u) + ℓ (v)$ , and $ℓ (w^{- 1}) = ℓ (w)$ with $(a b)^{- 1} = b^{- 1} a^{- 1}$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Group and abelian group).

Proof

technique · direct; strong exchange at the first letter of a reduced expression, then iteration along the expression

1.1F1F7

The easy inclusion and the conjugation identities. If $s \in K$ , then $s \in I \subseteq W_{I}$ and $d^{- 1} s d \in J$ , so $s = d (d^{- 1} s d) d^{- 1} \in d W_{J} d^{- 1}$ ; hence $W_{K} \subseteq W_{I} \cap d W_{J} d^{- 1}$ . Conjugation by $d^{- 1}$ is an isomorphism of groups carrying the generating set $K$ to $d^{- 1} K d$ , so $d^{- 1} W_{K} d = W_{d^{- 1} K d}$ , and $d^{- 1} K d \subseteq J$ holds by the definition of $K$ . Conjugating the coming equality $W_{I} \cap d W_{J} d^{- 1} = W_{K}$ by $d^{- 1}$ will give $W_{J} \cap d^{- 1} W_{I} d = W_{d^{- 1} K d}$ .

1.2F1F2F3F4F9

The key step: the first letter conjugated lies in $W_{J}$ . Let $y \in W_{I} \cap d W_{J} d^{- 1}$ and let $y = s_{1} \dots s_{p}$ be a reduced expression with $p \geq 1$ and all $s_{i} \in I$ by [F2]; put $\tilde{y} : = d^{- 1} y d \in W_{J}$ and write $\tilde{y} = {\tilde{y}}_{1} \dots {\tilde{y}}_{p}$ for a reduced expression of $\tilde{y}$ . Then $ℓ (y) = ℓ (\tilde{y}) = p$ : indeed $y d = d \tilde{y}$ , while $ℓ (y d) = ℓ (y) + ℓ (d)$ because $y \in W_{I}$ , $d \in^{I} W$ , and $ℓ (d \tilde{y}) = ℓ (d) + ℓ (\tilde{y})$ because $d \in W^{J}$ , $\tilde{y} \in W_{J}$ , both by [F2]. Consequently the word $s_{1} \dots s_{p}$ followed by a reduced word $a_{1} \dots a_{q}$ of $d$ , and the word $a_{1} \dots a_{q}$ followed by ${\tilde{y}}_{1} \dots {\tilde{y}}_{p}$ , are two reduced expressions of the same element $y d = d \tilde{y}$ of length $ℓ (d) + p$ , where $q = ℓ (d)$ . The element $s_{1}$ is a left descent of $y d$ : $s_{1} y = s_{2} \dots s_{p}$ , so $ℓ (s_{1} \cdot y d) = ℓ (s_{1} y) + ℓ (d) = (p - 1) + ℓ (d) < ℓ (y d)$ , using the additivity of [F2] and [F9] for $ℓ (s_{1} y) = p - 1$ . Applying strong exchange [F4] to the reduced expression $a_{1} \dots a_{q} {\tilde{y}}_{1} \dots {\tilde{y}}_{p}$ of $y d$ and the reflection $s_{1}$ gives a unique index $i$ with $s_{1} (y d)$ equal to that word with its $i$ -th letter deleted, and $s_{1} = a_{1} \dots a_{i - 1} a_{i} a_{i - 1} \dots a_{1}$ if $i \leq q$ , while $s_{1} = X {\tilde{y}}_{j} X^{- 1}$ with $X : = d {\tilde{y}}_{1} \dots {\tilde{y}}_{j - 1}$ if $i = q + j$ . The first case is impossible: then the deleted word exhibits $d_{i} : = a_{1} \dots a_{i - 1} a_{i + 1} \dots a_{q}$ with $d_{i} \tilde{y} = (s_{2} \dots s_{p}) d$ , hence $d_{i} = (s_{2} \dots s_{p}) d {\tilde{y}}^{- 1} \in W_{I} d W_{J}$ , while $ℓ (d_{i}) \leq q - 1 < q = ℓ (d)$ by [F9], contradicting the minimality of $d$ in $W_{I} d W_{J}$ given by [F3]. Hence $i = q + j$ for some $j$ , and substituting $X = d {\tilde{y}}_{1} \dots {\tilde{y}}_{j - 1}$ gives $d^{- 1} s_{1} d = {\tilde{y}}_{1} \dots {\tilde{y}}_{j - 1} {\tilde{y}}_{j} {\tilde{y}}_{j - 1} \dots {\tilde{y}}_{1} \in W_{J}$ , a conjugate in $W_{J}$ of the letter ${\tilde{y}}_{j} \in J$ .

1.3F1F5F6F7F9

The conjugated root is positive. Let $s \in I$ and $z : = d^{- 1} s d$ . Then $z \in T$ by [F7], and $z = t_{ρ (d)^{- 1} e_{s}}$ by the dictionary [F5], since $d^{- 1} s d = d^{- 1} t_{e_{s}} d = t_{ρ (d^{- 1}) e_{s}}$ . The root $ρ (d)^{- 1} e_{s}$ lies in $Φ_{+}$ : because $d \in^{I} W$ we have $ℓ (s d) > ℓ (d)$ by [F1], and $ℓ (s d) = ℓ (d^{- 1} s)$ and $ℓ (d) = ℓ (d^{- 1})$ by [F9], so the root-length criterion [F6] applied to $w : = d^{- 1}$ , $s$ gives $ρ (d^{- 1}) e_{s} = ρ (d)^{- 1} e_{s} \in Φ_{+}$ .

2.1F1F2F9step 1.2

Length one forces simplicity. Let $s \in I$ and suppose $z : = d^{- 1} s d \in W_{J}$ . Repeating the length computation of step 1.2 with $(y, \tilde{y})$ replaced by $(s, z)$ is legitimate because $s \in W_{I}$ , $z \in W_{J}$ and $s d = d z$ : it gives $ℓ (z) = ℓ (s) = 1$ . An element of $W_{J}$ of length one is a product of one generator, hence lies in $W_{J} \cap S = J$ by [F2]. Applying this to the first letter of step 1.2, $s_{1} \in K$ ; indeed $d^{- 1} s_{1} d \in W_{J}$ there, so $d^{- 1} s_{1} d \in J$ and $s_{1} \in I$ .

2.2F5step 1.3

The root of a conjugate that lies in $J$ . Let $s \in I$ and let $φ : = ρ (d)^{- 1} e_{s} \in Φ_{+}$ be the root of $z = d^{- 1} s d$ from step 1.3, so that $z = t_{φ}$ . If $z \in J$ , then $z = j = t_{e_{j}}$ for an element $j \in J$ , so $t_{φ} = t_{e_{j}}$ ; by [F5] $φ = \pm e_{j}$ , and since both $φ$ and $e_{j}$ lie in $Φ_{+}$ while $Φ_{+}$ and $Φ_{-} = - Φ_{+}$ are disjoint, $φ = e_{j}$ . Conversely, if $φ = e_{j}$ with $j \in J$ , then $z = t_{e_{j}} = j \in J$ .

3.1F1F2F9step 1.1step 2.1

The intersection. Let $y \in W_{I} \cap d W_{J} d^{- 1}$ with reduced expression $y = s_{1} \dots s_{p}$ , letters in $I$ , and write $y = d \tilde{y} d^{- 1}$ with $\tilde{y} \in W_{J}$ . If $p = 0$ , then $y = 1 \in W_{K}$ and there is nothing to prove, so assume $p \geq 1$ . By step 2.1 the first letter $s_{1}$ lies in $K$ , i.e. $d^{- 1} s_{1} d \in J$ ; then $y^{'} : = s_{1} y = s_{2} \dots s_{p}$ satisfies $y^{'} \in W_{I}$ and $y^{'} = d (d^{- 1} s_{1} d) \tilde{y} d^{- 1} \in d W_{J} d^{- 1}$ , and it has length $p - 1$ , because $ℓ (y^{'}) \leq p - 1$ by [F9] and $p = ℓ (y) = ℓ (s_{1} y^{'}) \leq 1 + ℓ (y^{'})$ . Iterating this argument through the suffixes shows $d^{- 1} s_{i} d \in J$ , hence $s_{i} \in K$ , for every $i = 1, \dots, p$ . Thus $y \in W_{K}$ , which proves (2) and, together with the inclusion of step 1.1, gives $W_{I} \cap d W_{J} d^{- 1} = W_{K}$ , that is (1) and its conjugation identities.

4.1F2F5F8step 1.3step 2.1step 2.2∎

Conclusion of (3). Let $s \in I$ . By step 1.3 the element $z = d^{- 1} s d$ lies in $T$ with root $φ = ρ (d)^{- 1} e_{s} \in Φ_{+}$ ; by step 2.2 one has $z \in J$ if and only if $φ = e_{j}$ for some $j \in J$ . If instead $z \in W_{J}$ , then by step 2.1 (applied to this $s$ ) $ℓ (z) = 1$ and $z \in W_{J} \cap S = J$ by [F2], so here too $φ = e_{j}$ for some $j \in J$ : the conjugated positive root is then a simple root of $Φ_{J} = Φ \cap V_{J}$ and never a non-simple positive combination such as $e_{j} + e_{j^{'}}$ , because [F8] shows that every reflection with a non-simple positive root has length $> 1$ , whereas $ℓ (z) = 1$ . The illustrative sum $e_{j} + e_{j^{'}}$ need not itself be a root; when it is a root for distinct $j, j^{'}$ , it is non-simple. This proves (3) for every $s \in I$ .

Remarks

The key step is Lusztig's argument for the intersection of parabolic subgroups: strong exchange at the first letter of a reduced expression of $y \in W_{I} \cap d W_{J} d^{- 1}$ either deletes a letter of the middle representative $d$ (impossible by minimality) or exhibits $d^{- 1} s_{1} d$ as a conjugate inside $W_{J}$ of a letter of $J$ .
The warning in (3) is not vacuous: in a parabolic subsystem of type $A_{2}$ the sum $e_{j} + e_{j^{'}}$ is a positive root whose reflection $t_{e_{j} + e_{j^{'}}}$ lies in $W_{J}$ with length $3$ , so "lying in $W_{J}$ " alone would not make the conjugated root simple; it is the length-one conclusion $ℓ (d^{- 1} s d) = ℓ (s) = 1$ that forces $z \in J$ and hence $φ = e_{j}$ .

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Sources

George Lusztig, Hecke Algebras with Unequal Parameters (revised arXiv edition of the CRM monograph, arXiv:math/0208154v2) (standard reference, not scraped)
Sara Billey, Matjaz Konvalinka, T. Kyle Petersen, William Slofstra and Bridget Tenner, Parabolic double cosets in Coxeter groups (arXiv:1612.00736v2) (standard reference, not scraped)
Dongwen Qi, A Note on Parabolic Subgroups of a Coxeter Group (arXiv:math/0512408) (standard reference, not scraped)