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Parabolic root subsystems and unique coset minima

Statement

Let $(W, S)$ be a Coxeter system with finite $S$ , presented by a Coxeter matrix, and let $ℓ$ be word length in $S$ . Let $ρ$ be its canonical real reflection representation on $V = R^{S}$ , with basis $(e_{s})_{s \in S}$ , root system $Φ = {ρ (w) e_{s} : w \in W, s \in S}$ and positive roots $Φ_{+}$ , the roots with nonnegative basis coordinates. For $I, K \subseteq S$ , write $W_{I} = ⟨ I ⟩$ , $V_{I} = span {e_{i} : i \in I}$ , $Φ_{I} = {ρ (u) e_{i} : u \in W_{I}, i \in I}$ and $Φ_{I}^{+} = Φ_{I} \cap Φ_{+}$ .

Then $Φ_{I} = Φ \cap V_{I}$ and $W_{I} \cap W_{K} = W_{I \cap K}$ . Every set $x W_{I} = {x u : u \in W_{I}}$ has a unique element $d$ of minimum length, and $x = d u$ with $u \in W_{I}$ and $ℓ (x) = ℓ (d) + ℓ (u)$ . For any $y \in W$ , the following are equivalent: $y$ is the minimum of $y W_{I}$ ; $ℓ (y s) > ℓ (y)$ for every $s \in I$ ; and $ρ (y) Φ_{I}^{+} \subseteq Φ_{+}$ . The inverted counterpart holds for $W_{I} x$ : its unique minimum $d$ satisfies $ℓ (s d) > ℓ (d)$ for every $s \in I$ and $ρ (d^{- 1}) Φ_{I}^{+} \subseteq Φ_{+}$ , and the corresponding factorization is length additive.

Statement references

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

The canonical reflection homomorphism, roots, reflections, and the positive cone

Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups

Root sign coherence and the action of simple reflections on positive roots

Facts & Assumptions

[F1]

The standard parabolic intersection and root-subsystem identities are $W_{I} \cap W_{K} = W_{I \cap K}$ and $Φ_{I} = Φ \cap V_{I}$ (Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives).

[F2]

Each $x W_{I}$ has a unique minimum $d$ , characterized by $ℓ (d s) > ℓ (d)$ for all $s \in I$ ; for every $u \in W_{I}$ , $ℓ (d u) = ℓ (d) + ℓ (u)$ . The analogous assertions hold for $W_{I} x$ with multiplication on the left (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification; Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups).

[F3]

For $y \in W$ and $s \in S$ , $ℓ (y s) > ℓ (y)$ if and only if $ρ (y) e_{s} \in Φ_{+}$ . Each $e_{s}$ is positive and roots have exactly one sign (The root-length criterion and faithfulness of the canonical reflection representation; Root sign coherence and the action of simple reflections on positive roots).

[F4]

For a reduced expression $w = s_{1} \dots s_{n}$ , the positive roots sent negative by $ρ (w^{- 1})$ are exactly $ρ (s_{1} \dots s_{j - 1}) e_{s_{j}}$ for $1 \leq j \leq n$ . Each root $α$ has an associated reflection $t_{α}$ satisfying $ρ (t_{α}) α = - α$ ; if $α \in Φ_{I}$ , then $t_{α} \in W_{I}$ (The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange; Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives).

[F5]

Generators are involutions, $ℓ (w^{- 1}) = ℓ (w)$ , and $ρ$ is a homomorphism (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; The canonical reflection homomorphism, roots, reflections, and the positive cone).

Proof

Given: the finite-rank Coxeter system, representation, roots and subsets in the statement.

Given: $x, y \in W$ and $I, K \subseteq S$ , with the notation and hypotheses of the statement.

Goal: prove the subsystem and intersection identities, the coset factorizations, and the three equivalent minimum criteria.

1.1F1F2

The two displayed subsystem and intersection identities are exactly [F1]. By [F2], $x W_{I}$ has a unique minimum $d$ , and writing $x = d u$ with $u \in W_{I}$ gives $ℓ (x) = ℓ (d) + ℓ (u)$ . The first two criteria for a general $y$ are equivalent by the same fact.

1.2F2F4F5

Suppose $y$ is the minimum of $y W_{I}$ . For $α \in Φ_{I}^{+}$ , put $t = t_{α} \in W_{I}$ . Choose a reduced expression $t = s_{1} \dots s_{n}$ and a reduced expression for $y$ . By [F2], $ℓ (y t) = ℓ (y) + ℓ (t)$ , so concatenating these expressions gives a reduced expression for $y t$ . Since $ρ (t^{- 1}) α = - α$ , [F4] expresses $α$ as $ρ (s_{1} \dots s_{j - 1}) e_{s_{j}}$ for some $j$ . The prefix formula in [F4] applied to the concatenated expression shows that $ρ (y) α$ is positive. Thus $ρ (y) Φ_{I}^{+} \subseteq Φ_{+}$ .

1.3F2F3

Conversely, if $ρ (y) Φ_{I}^{+} \subseteq Φ_{+}$ , then for every $s \in I$ the root $e_{s}$ lies in $Φ_{I}^{+}$ , so $ρ (y) e_{s}$ is positive. By [F3], $ℓ (y s) > ℓ (y)$ for every $s \in I$ , and [F2] makes $y$ the unique minimum of $y W_{I}$ . This completes the three-way equivalence.

2.1F5step 1.1step 1.2step 1.3∎

Inversion maps $W_{I} x$ bijectively to $x^{- 1} W_{I}$ and preserves length. Applying the established right-multiplication result to the latter set and inverting gives the unique minimum and length-additive factorization for $W_{I} x$ . Its descent criterion is $ℓ (s d) > ℓ (d)$ , and its root criterion is $ρ (d^{- 1}) Φ_{I}^{+} \subseteq Φ_{+}$ . Together with steps 1.1–1.3, this proves all assertions.

Depends on

Used by · 10

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