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Concatenation of length-additive standard expressions

PDF p. 18: Remark 4.14

Statement

Let $y : I \to K$ and $z : K \to L$ be composable morphisms in the Brink–Howlett groupoid, and put $x = y z$ . If $ℓ (x) = ℓ (y) + ℓ (z)$ , then concatenating any chosen standard expression $y$ for $y$ with any chosen standard expression $z$ for $z$ gives a standard expression $y z$ for $x$ . This includes empty expressions for identity morphisms.

Statement references

Standard expressions and their lengths

Proof

Given: Composable morphisms $y : I \to K$ and $z : K \to L$ , their product $x = y z$ , the equality $ℓ (x) = ℓ (y) + ℓ (z)$ , and chosen standard expressions $y$ and $z$ .

Goal: Show that their concatenation is a standard expression for $x$ .

[F1] By Standard expressions and their lengths, a standard expression is a finite composable sequence of the specified wall-crossing morphisms whose product is the morphism and whose factor lengths sum to its Coxeter length. The empty sequence represents an identity and has length sum zero; composable labels multiply in sequence order.

1.1F1given

Write $y = (u_{1}, \dots, u_{m})$ and $z = (v_{1}, \dots, v_{n})$ . By [F1], their sequences consist of the required wall-crossing morphisms, with products $y$ and $z$ and respective length sums $ℓ (y)$ and $ℓ (z)$ . Their endpoint $K$ agrees, so their concatenation is composable from $I$ to $L$ and has product $y z = x$ . This remains true when either sequence is empty.

2.1F1step 1.1given∎

The sum of the Coxeter lengths of the concatenated factors is $ℓ (y) + ℓ (z) = ℓ (x)$ , including when a sequence is empty. Hence [F1] identifies the concatenation $y z$ as a standard expression for $x$ , as claimed.

Depends on

Standard expressions and their lengths

Used by · 1

Atomic Matsumoto theorem

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 18, Remark 4.14