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Coxeter Presentations, Exchange, and Reduced Word Theorems — Examples

1 · Prerequisites

2 · Summary

Work out rank-one, finite dihedral and type-A reduced words, a nonreduced word deleted by exchange, and minimal representatives for S2⊂S3. No geometric braid-group construction is a prerequisite.

This companion is a dependency leaf: its five examples use only the theory of coxeter-presentations-exchange-and-reduced-word-theorems and that page's established prerequisite closure, and no other page or item depends on them. The examples involving the symmetric group display permutations on the letters ${1, 2, 3}$ , identified with the library's ${0, 1, 2}$ by the order-preserving letter shift $j \mapsto j - 1$ declared in Type-A reduced words and inversion numbers in $S_{3}$ ; the shift preserves inversion numbers and lengths.

Reduced words in rank one computes the rank-one group $W = {1, s}$ with its two reduced words, its reflection and root sets and its faithful signed action. Reduced words and lengths in a finite dihedral group exhibits the $2 m$ normal forms $(s t)^{k}$ , $(s t)^{k} s$ of a finite dihedral group, the reducedness of alternating words of length at most $m$ , the length formulae $ℓ ((s t)^{k}) = \min (2 k, 2 (m - k))$ and $ℓ ((s t)^{k} s) = \min (2 k + 1, 2 (m - k) - 1)$ , and the two braid-related reduced expressions of the longest element for even $m$ . A nonreduced word deleted by its repeated prefix reflection, and an exchange step runs the exchange step in $I_{2} (3)$ , finds the repeated prefix reflection $r_{1} = r_{4}$ of the word $(s, t, s, t)$ that licenses the deletion of its first and last letter, and exhibits the failed deletion of the unequal-reflection pair at positions $1$ and $3$ . Type-A reduced words and inversion numbers in $S_{3}$ tabulates the six elements of $S_{3}$ with $ℓ = inv$ , the reduced words of lengths $1$ and $2$ , the two braid-related length-three expressions of the longest element, and the nonreduced word $(s_{1}, s_{2}, s_{1}, s_{2})$ with its two-letter deletion. Minimal coset representatives of $S_{2}$ in $S_{3}$ enumerates the three left and three right cosets of $W_{J} = {1, s_{1}}$ in $S_{3}$ , with their unique minimal representatives, the descent characterisations and the length-additivity identities.

The results tested here are proved on the theory page: the exchange and deletion statements of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action, the rank-two computation and ambient reducedness of The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness, and the coset theorem with the type-A identification of Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification. The computations are evidence within their finite scope and do not replace those proofs.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Reduced words in rank one

Example

Let $S = {s}$ and $m (s, s) = 1$ , so that the relator set of the presentation of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups is $R = {s^{2}}$ and $W = F ({s}) / ⟨ ⁣ ⟨ s^{2} ⟩ ⁣ ⟩$ . Then:

Every element of $W$ is $1$ or $s$ , with $s^{2} = 1$ and $s \neq 1$ ; hence $W ≅ Z / 2$ .
$ℓ (1) = 0$ and $ℓ (s) = 1$ .
A word $(s, \dots, s)$ of length $k$ has value $s^{k}$ , which is $1$ for even $k$ and $s$ for odd $k$ . Hence the only reduced words are the empty word $()$ for $1$ and the word $(s)$ for $s$ , and every word of length $k \geq 2$ is nonreduced and reduces to a reduced word by repeated deletion of two letters.
The reflection set is $T = {s}$ , the root set is $Φ = {α_{s}, - α_{s}}$ (because $σ_{s} (α_{s}) = - α_{s}$ ), and the signed action on ${\pm 1} \times {s}$ is $(ε, s) \cdot s = (- ε, s)$ , which is faithful.

Facts & Assumptions

Given: The Coxeter matrix on $S = {s}$ with $m (s, s) = 1$ ; the presented group $W$ with its length $ℓ$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the field $K$ of characteristic $0$ , the space $E$ with basis $(α_{s})$ and the involution $σ_{s}$ of The geometric representation on the simple-root basis over a common splitting field, and the root set; the reflection set $T$ with the right action of $W$ on ${\pm 1} \times T$ of The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness; and the deletion and faithfulness statements of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action.

[F1]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups: the relator set is " $R : = {s^{2} : s \in S} \cup {(s t)^{m (s, t)} : s, t \in S, m (s, t) < \infty}$ " with " $N : = ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (S)}$ " its normal closure, and "Define $W : = F (S) / N,$ and write $s$ (as well as $s N$ ) for the image of $s \in S$ in $W$ "; the length is " $ℓ (w) : = \min {k \in N : there exist s_{1}, \dots, s_{k} \in S with w = s_{1} \dots s_{k}}$ ", and "the empty word $()$ is a word in $S$ of length $0$ , and $ℓ (1) = 0$ ; it is the reduced expression of $1$ ".

[F2]

The geometric representation on the simple-root basis over a common splitting field, and the root set: "The prime subfield of $K$ is $Q$ " and " $char K = 0$ "; the maps satisfy " $σ_{s} (α_{s}) = - α_{s}, σ_{s} (α_{t}) = α_{t} + c_{s t} α_{s} (t \neq s)$ ", and the root set is " $Φ : = {σ_{s_{1}} σ_{s_{2}} \dots σ_{s_{k}} (α_{t}) : k \geq 0, s_{1}, \dots, s_{k}, t \in S} \subseteq E$ ".

[F3]

The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness: " $σ_{s}^{2} = {i d}_{E}$ for every $s \in S$ ; each $σ_{s}$ is invertible, and $ℓ (s) = 1$ ."; the reflection set is " $T : = {w s w^{- 1} : w \in W, s \in S} \subseteq W$ "; and "Then $U_{s}^{2} = i d$ for every $s$ , and the assignment $(ε, r) \cdot s : = U_{s} (ε, r)$ extends to a well-defined right action of $W$ on ${\pm 1} \times T$ ".

[F4]

Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action: "Hence repeated deletion of two letters transforms every word into a reduced expression for the same element, and a word is reduced if and only if it cannot be shortened by deleting two letters."; and "The right action of $W$ on ${\pm 1} \times T$ is faithful".

Verification

technique · direct computation in the rank-one presentation, with the general deletion and faithfulness statements applied to it

1.1F1F3

The group and its two elements. The relation $s^{2} = 1$ holds in $W$ because $s^{2} \in R \subseteq N$ [F1]. Every element of $W$ is the image of a product of the generator $s$ and its inverse, and $s^{- 1} = s$ in $W$ , so every element is a power $s^{k}$ with $k \geq 0$ ; since $s^{k}$ is $1$ for even $k$ and $s$ for odd $k$ , one has $W = {1, s}$ . By [F1] $ℓ (1) = 0$ and by [F3] $ℓ (s) = 1$ , so $s \neq 1$ , and the bijection $W \to Z / 2$ with $1 \mapsto 0$ , $s \mapsto 1$ is a homomorphism; hence $W ≅ Z / 2$ .

2.1F1F3F4step 1.1

Word values and reduced words. A word $(s, \dots, s)$ of length $k$ has value $s^{k}$ , which is $1$ for even $k$ and $s$ for odd $k$ by step 1.1; in particular $()$ has value $1$ and $(s)$ has value $s$ with lengths $0 = ℓ (1)$ and $1 = ℓ (s)$ [F1, F3], so both are reduced. If $k \geq 2$ , the word of length $k$ has value of length at most $1 < k$ , so it is not reduced; by [F4] it admits a deletion of two letters with the same value, and iterating this deletion, the length drops by two each time until the word has length $ℓ (1)$ or $ℓ (s)$ , namely until it is $()$ or $(s)$ . Hence the only reduced words are $()$ and $(s)$ .

3.1F2F3F4∎

Reflections, roots and the signed action. Since $W$ is abelian and $T = {w s w^{- 1} : w \in W, s \in S}$ , the reflection set is $T = {s}$ [F3, step 1.1]. By [F3] the group generated by $σ_{s}$ is ${1, σ_{s}}$ , so the root set of [F2] is $Φ = {α_{s}, σ_{s} (α_{s})} = {α_{s}, - α_{s}}$ , and these two vectors are distinct: $α_{s} = - α_{s}$ would give $2 α_{s} = 0$ , while the basis vector $α_{s}$ is nonzero and $char K = 0$ [F2]. For $r = s$ the formula of [F3] gives $(ε, s) \cdot s = U_{s} (ε, s) = (ε (- 1)^{δ (s, s)}, s s s) = (- ε, s)$ , and this action is faithful by [F4].

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Reduced words and lengths in a finite dihedral group

Example

Let $m \geq 2$ , $S = {s, t}$ and $m (s, t) = m$ , and put $W = ⟨ s, t ∣ s^{2} = t^{2} = (s t)^{m} = 1 ⟩$ . The exact order of $s t$ is $m$ by The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (4); the normal-form computation below verifies that $W$ is dihedral of order $2 m$ .

The $2 m$ elements $(s t)^{k}$ and $(s t)^{k} s$ with $0 \leq k < m$ are pairwise distinct and exhaust $W$ .
For $1 \leq q \leq m$ the two alternating words of length $q$ are reduced. Their values are distinct when $q < m$ and equal when $q = m$ ; explicitly, each has length $q$ , so values belonging to different lengths are distinct.
For $0 \leq k \leq m$ one has $ℓ ((s t)^{k}) = \min (2 k, 2 (m - k)),$ and for $0 \leq k \leq m - 1$ one has $(s t)^{k} s = (t s)^{m - k} s = (t s)^{m - k - 1} t$ and $ℓ ((s t)^{k} s) = \min (2 k + 1, 2 (m - k) - 1);$ in particular $ℓ (s) = 1$ and $ℓ ((s t)^{m - 1} s) = 1$ .
The element $(s t)^{k}$ with $2 k = m$ (only for even $m$ ) is the unique longest element $w_{0}$ of $W$ , of length $m$ ; its two reduced expressions are the two alternating words of length $m$ , and they are related by the braid move $s t s t \dots \mapsto t s t s \dots$ .

Facts & Assumptions

Given: A group $W$ presented by the Coxeter matrix on $S = {s, t}$ with $m (s, t) = m \geq 2$ , with the length function $ℓ$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the exact order of $s t$ and the ambient reducedness of alternating words from The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness; the deletion statement of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action; and the vocabulary of powers and orders.

[F1]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups: $W = F (S) / N$ is presented by the relators $s^{2}$ , $t^{2}$ and $(s t)^{m}$ (the set $R = {s^{2} : s \in S} \cup {(s t)^{m (s, t)} : s \neq t, m (s, t) < \infty}$ ); the length $ℓ (w)$ is the least $k$ such that $w = s_{1} \dots s_{k}$ for some $s_{1}, \dots, s_{k} \in S$ , and $ℓ (1) = 0$ .

[F2]

The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (1),(4): " $σ_{s}^{2} = {i d}_{E}$ for every $s \in S$ ; each $σ_{s}$ is invertible, and $ℓ (s) = 1$ "; and "Consequently, for any distinct $s, t \in S$ , one has $s \neq t$ in $W$ and $s t$ has order exactly $m (s, t)$ in $W$ (infinite when $m (s, t) = \infty$ ).".

[F3]

The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (7): "Let $s \neq t$ , $m : = m (s, t)$ , and for $q \geq 1$ let $w_{q}$ be the value of the alternating word of length $q$ beginning with $s$ . If $m < \infty$ and $q \leq m$ , or if $m = \infty$ and $q \geq 1$ , then $ℓ (w_{q}) = q$ , every word in $S$ representing $w_{q}$ has length at least $q$ , and $w_{1}, \dots, w_{q}$ are pairwise distinct."

[F4]

Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action (3): "If the word $(s_{1}, \dots, s_{k})$ in $S$ is not reduced, then there are $i < j$ with $s_{1} \dots \hat{s_{i}} \dots \hat{s_{j}} \dots s_{k} = s_{1} \dots s_{k}$ . Hence repeated deletion of two letters transforms every word into a reduced expression for the same element".

[F5]

Powers $g^{n}$ : natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$ , The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity: $g^{k}$ for $k \geq 0$ is the $k$ -fold product of $g$ with $g^{0} = 1$ , $g^{- k} = (g^{- 1})^{k}$ , and the order of $g$ is the least $k \geq 1$ with $g^{k} = 1$ when such $k$ exist, with $g^{ord (g)} = 1$ .

Verification

technique · direct computation with the normal forms of the dihedral group, using the exact order of $st$ and the ambient reducedness of alternating words

1.1F1F5

Exhaustion. Every element of $W$ is the value of a word in $s, t$ whose inverse letters are again letters ( $s^{- 1} = s$ , $t^{- 1} = t$ in $W$ because $s^{2}, t^{2} \in R$ [F1]), so it suffices to treat words in $s, t$ . Cancelling all consecutive equal letters using $s^{2} = t^{2} = 1$ turns any such word into an alternating word $(u_{1}, \dots, u_{q})$ with $u_{i + 1} \neq u_{i}$ . If $q$ is even, the value of the alternating word beginning with $s$ is $(s t)^{q / 2}$ and that of the one beginning with $t$ is $(t s)^{q / 2} = (s t)^{- q / 2}$ [F5]; if $q$ is odd, the corresponding values are $(s t)^{(q - 1) / 2} s$ and $(t s)^{(q - 1) / 2} t = (s t)^{- (q - 1) / 2} t = (s t)^{- (q + 1) / 2} s$ , because $t = (s t)^{- 1} s$ . Since $(s t)^{m} = 1$ [F1], an integer power $(s t)^{j}$ equals $(s t)^{k}$ for the unique $k \in {0, \dots, m - 1}$ with $j \equiv k (m o d m)$ , and $(s t)^{j} s = (s t)^{k} s$ ; hence every element of $W$ is one of the $2 m$ listed elements $(s t)^{k}, (s t)^{k} s$ with $0 \leq k < m$ .

1.2F1F2F3

Reducedness of alternating words. Let $1 \leq q \leq m$ and let $w_{q}$ be the value of the alternating word of length $q$ beginning with $s$ ; by [F3], $ℓ (w_{q}) = q$ and every word in $S$ representing $w_{q}$ has length at least $q$ , so that alternating word is reduced, and $w_{1}, \dots, w_{q}$ are pairwise distinct. The same length and reducedness statements hold for alternating words beginning with $t$ : interchanging the roles of $s$ and $t$ preserves every hypothesis, since $m (t, s) = m (s, t) = m$ and the presentation is symmetric in $s, t$ [F1]. To compare the two words of the same length, put $r = s t$ , so $t = r^{- 1} s$ . For $q = 2 h$ their values are $r^{h}$ and $r^{- h}$ ; for $q = 2 h + 1$ they are $r^{h} s$ and $r^{- h - 1} s$ . In either case equality is equivalent to $r^{q} = 1$ , which for $1 \leq q \leq m$ holds exactly at $q = m$ by [F2].

2.1F2F5step 1.1

Distinctness. If $(s t)^{k} = (s t)^{l}$ with $0 \leq k < l < m$ , then $(s t)^{l - k} = 1$ with $0 < l - k < m$ , contradicting the exact order $m$ of $s t$ [F2, F5]. If $(s t)^{k} s = (s t)^{l} s$ with $0 \leq k < l < m$ , then multiplying on the right by $s$ gives $(s t)^{k} = (s t)^{l}$ , the previous case. If finally $(s t)^{k} = (s t)^{l} s$ , then $(s t)^{k - l} = s$ , so $s \in ⟨ s t ⟩$ and also $t = s \cdot s t \in ⟨ s t ⟩$ , whence $W = ⟨ s t ⟩$ is cyclic and therefore abelian; then $s t = t s$ , so $(s t)^{2} = s t s t = s (t s) t = s (s t) t = (s s) (t t) = 1$ , and the order of $s t$ divides $2$ . Since that order is $m \geq 2$ by [F2], this forces $m = 2$ , in which case $⟨ s t ⟩ = {1, s t}$ while $s \in {1, s t}$ gives $s = 1$ or $t = 1$ ; both are impossible because $ℓ (s) = ℓ (t) = 1$ [F2]. Hence no rotation equals a reflection and the $2 m$ elements of step 1.1 are pairwise distinct, so they exhaust $W$ and $∣ W ∣ = 2 m$ . The identity $s (s t) s = (s t)^{- 1}$ , together with these distinct normal forms, identifies $W$ as the dihedral group.

2.2F1F5step 1.2

The rotation lengths. For $0 \leq k \leq m$ the element $(s t)^{k}$ also equals $(t s)^{m - k}$ : indeed $(t s) = (s t)^{- 1}$ [F5], so $(t s)^{m - k} = (s t)^{- (m - k)} = (s t)^{k - m} = (s t)^{k}$ , using $(s t)^{m} = 1$ [F1]. The two expressions $(s t)^{k}$ and $(t s)^{m - k}$ are alternating words of lengths $2 k$ and $2 (m - k)$ , whose minimum $q_{0} : = \min (2 k, 2 (m - k))$ satisfies $q_{0} \leq m$ because the two lengths sum to $2 m$ . The shorter of the two words has length $q_{0}$ ; if $q_{0} = 0$ it is the empty word with value $1$ , so $ℓ ((s t)^{k}) = 0 = q_{0}$ , and if $q_{0} \geq 1$ it is an alternating word of length $q_{0} \leq m$ whose value is $(s t)^{k}$ , so step 1.2 gives $ℓ ((s t)^{k}) = q_{0}$ and shows that no word represents $(s t)^{k}$ with fewer than $q_{0}$ letters. Hence $ℓ ((s t)^{k}) = \min (2 k, 2 (m - k))$ .

3.1F5step 1.2step 2.2

The reflection lengths. For $0 \leq k \leq m - 1$ one has $(s t)^{k} s = (t s)^{m - k} s = (t s)^{m - k - 1} (t s) s = (t s)^{m - k - 1} t$ , because $(t s)^{m - k} = (s t)^{k - m} = (s t)^{k}$ as in step 2.2; these are alternating words of lengths $2 k + 1$ and $2 (m - k) - 1$ , whose minimum $q_{1} : = \min (2 k + 1, 2 (m - k) - 1)$ is at most $m$ : indeed $q_{1} \leq 2 k + 1$ and $q_{1} \leq 2 (m - k) - 1$ , so $2 q_{1} \leq 2 m$ , that is $q_{1} \leq m$ . The shorter word is nonempty because $q_{1} \geq 1$ for $0 \leq k \leq m - 1$ , and step 1.2 applied to it (with the roles of $s$ and $t$ interchanged if it begins with $t$ ) gives that it is reduced, that its value $(s t)^{k} s$ has length exactly $q_{1}$ , and that no word for $(s t)^{k} s$ is shorter. Hence $ℓ ((s t)^{k} s) = \min (2 k + 1, 2 (m - k) - 1)$ ; the endpoint $k = 0$ gives $ℓ (s) = \min (1, 2 m - 1) = 1$ and the endpoint $k = m - 1$ gives $ℓ ((s t)^{m - 1} s) = \min (2 m - 1, 1) = 1$ .

4.1F4F5step 1.1step 2.1step 1.2step 2.2step 3.1∎

The longest element. Suppose $m$ is even and put $w_{0} : = (s t)^{m / 2}$ . By step 2.2, $ℓ (w_{0}) = \min (m, m) = m$ . Every rotation $(s t)^{k}$ with $k \neq m / 2$ has $ℓ = \min (2 k, 2 m - 2 k) < m$ : their minimum is at most $m$ , and equality would require both terms to equal $m$ because their sum is $2 m$ , forcing $k = m / 2$ . Every reflection $(s t)^{k} s$ with $0 \leq k \leq m - 1$ has $ℓ = \min (2 k + 1, 2 m - 2 k - 1) \leq m$ by step 3.1, and both entries of that minimum are odd while $m$ is even, so $ℓ \leq m - 1 < m$ . Together with steps 1.1 and 2.1 this shows that $w_{0}$ is the unique element of length $m$ , hence the unique longest element. A reduced word for $w_{0}$ of length $m$ cannot contain two consecutive equal letters, since deleting that pair would exhibit a shorter word for $w_{0}$ [F4] in contradiction to $ℓ (w_{0}) = m$ ; hence it is alternating. The two alternating words of length $m$ are $(s t)^{m / 2}$ and $(t s)^{m / 2}$ , both of which have value $w_{0}$ because $(t s)^{m / 2} = (s t)^{- m / 2} = (s t)^{m / 2}$ [F5], and they are reduced by step 1.2; so they are exactly the two reduced expressions of $w_{0}$ . They differ by the single replacement of the alternating block of length $m$ by the other alternating word of the same length, the braid move.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

A nonreduced word deleted by its repeated prefix reflection, and an exchange step

Example

Let $S = {s, t}$ with $m (s, t) = 3$ , so that $W = I_{2} (3)$ is the dihedral group of order $6$ in which $s t$ has order $3$ (The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (4), Reduced words and lengths in a finite dihedral group (1)).

Exchange. The word $(s, t)$ is a reduced expression of $s t$ with $ℓ (s t) = 2$ . Since $s \cdot s t = t$ has length $1 < 2$ , the exchange theorem (Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action (2)) predicts $s \cdot s t = t = s_{1} \dots \hat{s_{1}} \dots s_{2}$ , the deletion of the first letter; and indeed $s s t = t$ .
A nonreduced word. In the word $(s, t, s, t)$ the prefix reflections are $r_{1} = s$ , $r_{2} = s t s$ , $r_{3} = t$ , $r_{4} = s$ ; thus $r_{1} = r_{4}$ . Deleting the first and the last letter gives the word $(t, s)$ with value $t s$ , and indeed $s t s t = (s t)^{2} = (s t)^{- 1} = t s$ , so the deleted word is an expression of the same element: $s t s t = t s$ .
Consequences. The word $(s, t, s, t)$ is nonreduced: $ℓ (s t s t) = ℓ (t s) = 2 < 4$ , in agreement with the length formula $ℓ ((s t)^{2}) = \min (4, 2) = 2$ of Reduced words and lengths in a finite dihedral group (3). The reflection set of the element is $Φ (s t s t) = {s t s, t}$ , of cardinality $2 = ℓ (s t s t)$ , and $s \notin Φ (s t s t)$ because $s = r_{1} = r_{4}$ occurs an even number of times; deleting a different pair, e.g. the letters $s$ at positions $1$ and $3$ , does not preserve the value: $t \cdot t = 1 \neq t s$ .

Facts & Assumptions

Given: The Coxeter matrix on $S = {s, t}$ with $m (s, t) = 3$ ; the presented group $W$ with its length $ℓ$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the reflection set $T$ , the prefix reflections, the sign-change sets $Φ (w)$ and the exact order of $s t$ from The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness; the exchange and deletion statements of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action; and the length table for $I_{2} (3)$ of Reduced words and lengths in a finite dihedral group.

[F1]

The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (1),(4),(6): " $σ_{s}^{2} = {i d}_{E}$ for every $s \in S$ ; each $σ_{s}$ is invertible, and $ℓ (s) = 1$ "; "Consequently, for any distinct $s, t \in S$ , one has $s \neq t$ in $W$ and $s t$ has order exactly $m (s, t)$ in $W$ (infinite when $m (s, t) = \infty$ ).", so here $s t$ has order $3$ ; "(a) If $r_{i} = r_{j}$ for some $i < j$ , then $s_{1} \dots \hat{s_{i}} \dots \hat{s_{j}} \dots s_{k} = w$ : the two letters $s_{i}, s_{j}$ can be deleted"; "(b) $(- 1)^{n (r)} = : η (r, w)$ depends only on $w$ and $r$ "; and for a reduced word "(c) ... the set $Φ (w) : = {r_{1}, \dots, r_{k}} = {r \in T : η (r, w) = - 1}$ is independent of the reduced expression chosen, with $# Φ (w) = ℓ (w)$ ", where $r_{i} = s_{1} \dots s_{i - 1} s_{i} s_{i - 1} \dots s_{1}$ are the prefix reflections and $n (r) = # {i : r_{i} = r}$ .

[F2]

Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action (2): "Let $w = s_{1} \dots s_{k}$ be a reduced expression and let $s \in S$ satisfy $ℓ (s w) = k - 1$ . Then $s w = s_{1} \dots \hat{s_{i}} \dots s_{k}$ for some $i \in {1, \dots, k}$ "; and (3): "If the word $(s_{1}, \dots, s_{k})$ in $S$ is not reduced, then there are $i < j$ with $s_{1} \dots \hat{s_{i}} \dots \hat{s_{j}} \dots s_{k} = s_{1} \dots s_{k}$ ."

[F3]

Reduced words and lengths in a finite dihedral group (3): for $0 \leq k \leq m$ one has " $ℓ ((s t)^{k}) = \min (2 k, 2 (m - k))$ ", so with $m = 3$ and $k = 2$ this reads $ℓ ((s t)^{2}) = \min (4, 2) = 2$ .

[F4]

Powers $g^{n}$ : natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$ : $g^{k}$ is the $k$ -fold product with $g^{0} = 1$ and $g^{- k} = (g^{- 1})^{k}$ , so $g \cdot g^{- 1} = 1$ and $(s t)^{- 1} = t^{- 1} s^{- 1} = t s$ in $W$ , where $s^{- 1} = s$ and $t^{- 1} = t$ because $s^{2}, t^{2}$ are relators of the presentation of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups.

[F5]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups: the relators are $s^{2}$ , $t^{2}$ and $(s t)^{3}$ , and $ℓ (w)$ is the least length of a word in $S$ representing $w$ .

Verification

technique · direct computation in $I_2(3)$, with the prefix reflections read off from the definition and the general exchange and deletion statements applied to this word

1.1F1F2F3

The exchange step. The word $(s, t)$ has value $s t$ and length $2$ , and $ℓ (s t) = 2$ by [F3] (the case $m = 3$ , $k = 1$ gives $\min (2, 4) = 2$ ), so $(s, t)$ is a reduced expression of $s t$ . In $W$ one has $s^{2} = 1$ , so $s \cdot s t = s^{2} t = t$ ; and $ℓ (t) = 1$ by [F1]. Hence $ℓ (s \cdot s t) = 1 = 2 - 1$ and [F2] (2) applies to the reduced word $(s_{1}, s_{2}) = (s, t)$ with the letter $s$ : there is $i \in {1, 2}$ with $s \cdot s t = s_{1} \dots \hat{s_{i}} \dots s_{2}$ . Since $s \cdot s t = t$ and the two deletion words are $\hat{s_{1}} s_{2} = t$ and $s_{1} \hat{s_{2}} = s$ , only $i = 1$ gives the value $t$ , so $s \cdot s t = \hat{s_{1}} s_{2} = t$ : the predicted deletion is the deletion of the first letter.

1.2F1F4F5

The prefix reflections and the repeated one. Compute the prefix reflections of $(s, t, s, t)$ from $r_{i} = w_{i - 1} s_{i} w_{i - 1}^{- 1}$ with $w_{0} = 1$ , $w_{1} = s$ , $w_{2} = s t$ , $w_{3} = s t s$ : $r_{1} = s$ ; $r_{2} = s t s = s t s$ ; $r_{3} = (s t) s (s t)^{- 1} = s t s \cdot t s = s t s t s = (s t)^{2} s = t s \cdot s = t$ , where $(s t)^{- 1} = t s$ and $(s t)^{2} = (s t)^{- 1}$ because $s t$ has order $3$ [F1, F4]; and $r_{4} = (s t s) t (s t s) = s t s t s t s = (s t)^{3} s = s$ , using $(s t)^{3} = 1$ and $s^{2} = t^{2} = 1$ [F5]. Hence $r_{1} = r_{4} = s$ while $r_{2} = s t s$ and $r_{3} = t$ .

1.3F1F4

The deletion and the value identity. Since $r_{1} = r_{4}$ , [F1] (6)(a) with $i = 1$ , $j = 4$ gives $s t s t = \hat{s_{1}} t s \hat{s_{4}} = t s$ . Independently, $(s t)^{2} = (s t)^{- 1}$ because $s t$ has order $3$ [F1], and $(s t)^{- 1} = t^{- 1} s^{- 1} = t s$ [F4]; hence $s t s t = (s t)^{2} = t s$ , confirming that deleting the first and the last letter of $(s, t, s, t)$ preserves the value.

2.1F1F3F5step 1.2

The element and its reflection set. By step 1.3 the word $(s, t, s, t)$ represents $s t s t = t s$ and has length $4$ , while $ℓ (s t s t) = ℓ (t s) = 2$ by [F3] with $m = 3$ , $k = 2$ and $k = 1$ ; hence the word is not reduced. By [F1] (6)(b) the function $η (r, s t s t) = (- 1)^{n (r)}$ is expression-independent, so it may be computed from the word $(s, t, s, t)$ with the prefix reflections of step 1.2: of these $r_{1} = r_{4} = s$ occurs twice and $r_{2} = s t s$ , $r_{3} = t$ occur once each, so $Φ (s t s t) = {r \in T : η (r, s t s t) = - 1} = {s t s, t}$ has cardinality $2 = ℓ (s t s t)$ , in agreement with the cardinality forced for any reduced expression by [F1] (6)(c); in particular $s \notin Φ (s t s t)$ , and the two-letter deletion licensed by [F1] (6)(a) is the one deleting the equal reflections $r_{1} = r_{4}$ , not an arbitrary pair of equal letters. The last point: deleting the letters at positions $1$ and $3$ of $(s, t, s, t)$ leaves the word $(t, t)$ with value $t^{2} = 1$ [F5], and $1 \neq t s$ since $ℓ (1) = 0 \neq 2 = ℓ (t s)$ ; so that deletion does not preserve the value.

3.1step 1.1step 1.2step 1.3step 2.1∎

Collected. The example exhibits the exchange step of [F2] (2) in complete detail (step 1.1), a nonreduced word whose two equal prefix reflections $r_{1} = r_{4}$ license the deletion of its first and last letter (steps 1.2, 1.3), and the resulting expression-independent sign-change set $Φ (s t s t) = {s t s, t}$ together with a failed deletion of an unequal-reflection pair (step 2.1).

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Type-A reduced words and inversion numbers in $S_{3}$

Example

Let $n = 3$ , so $S = {s_{1}, s_{2}}$ with $m (s_{1}, s_{2}) = 3$ , and let $W ≅ S_{3}$ be the identification of Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4) sending $s_{1} \mapsto (1 2)$ and $s_{2} \mapsto (2 3)$ . As on the published examples pages, permutations are displayed on the letters ${1, 2, 3}$ , identified with the library's ${0, 1, 2}$ by the order-preserving letter shift $j \mapsto j - 1$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation); the shift preserves the order, hence preserves inversion numbers and lengths (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations). Then $ℓ (w) = inv (φ (w))$ for all $w \in W$ , and the six elements of $W$ and their data are:

$w$	permutation	length $ℓ (w)$	inversion number
$1$	$i d$	$0$	$0$
$s_{1}$	$(1 2)$	$1$	$1$
$s_{2}$	$(2 3)$	$1$	$1$
$s_{1} s_{2}$	$(1 2 3)$	$2$	$2$
$s_{2} s_{1}$	$(1 3 2)$	$2$	$2$
$s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$	$(1 3)$	$3$	$3$

Consequently: the words $(s_{1}, s_{2})$ , $(s_{2}, s_{1})$ and $(s_{1}, s_{2}, s_{1})$ are reduced, the two reduced expressions $s_{1} s_{2} s_{1}$ and $s_{2} s_{1} s_{2}$ of the longest element are related by the braid move, and the word $(s_{1}, s_{2}, s_{1}, s_{2})$ of length $4$ is nonreduced: it represents $s_{2} s_{1}$ (inversion number $2$ ) and deleting its first and last letters gives the reduced word $(s_{2}, s_{1})$ .

Facts & Assumptions

Given: The type-A Coxeter matrix on $S = {s_{1}, s_{2}}$ with $m (s_{1}, s_{2}) = 3$ ; the presented group $W$ with its length $ℓ$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the isomorphism $W \to S_{3}$ with $ℓ = inv$ and the relators of the presentation from Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4); the rank-two length formula of the dihedral specialisation Reduced words and lengths in a finite dihedral group (3); and the deletion statement of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action.

[F1]

Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4): for the type-A matrix, " $s_{i} \mapsto (i i + 1)$ extends to an isomorphism $W \to S_{n}$ , and for every $w \in W$ , $ℓ (w) = inv (φ (w))$ "; and "a word in the $s_{i}$ is reduced if and only if its length equals the inversion number of its value".

[F2]

The finite symmetric group $S_{n}$ , one-line notation, and cycle notation: $S_{n} = Sym ({0, 1, \dots, n - 1})$ with composition $(σ τ) (i) = σ (τ (i))$ , so the right factor acts first, and "An element of $S_{n}$ is named by either of the two notations below", one-line notation $[σ (0), \dots, σ (n - 1)]$ and cycle notation.

[F3]

Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations: "An inversion of $σ$ is a pair $(i, j)$ with $i < j < n$ and $σ (i) > σ (j)$ ", and $inv (σ) : = ∣ Inv (σ) ∣$ .

[F4]

Reduced words and lengths in a finite dihedral group (3): with $m = 3$ the formulae " $ℓ ((s t)^{k}) = \min (2 k, 2 (m - k))$ " and " $ℓ ((s t)^{k} s) = \min (2 k + 1, 2 (m - k) - 1)$ " give, for the rotation values $k = 0, 1, 2, 3$ and the reflection values $k = 0, 1, 2$ , the lengths $0, 2, 2, 0$ and $1, 3, 1$ ; the maximum is $3$ , attained only by $(s t)^{1} s$ , and the same values hold after interchanging the roles of $s$ and $t$ . In particular $ℓ (s_{1}) = ℓ (s_{2}) = 1$ , $ℓ (s_{1} s_{2}) = ℓ (s_{2} s_{1}) = 2$ and $ℓ (s_{1} s_{2} s_{1}) = 3$ , and the element $s_{1} s_{2} s_{1}$ , which also equals the alternating word $s_{2} s_{1} s_{2}$ of length $3$ , is the unique longest element of $W$ .

[F5]

Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action (3): "Hence repeated deletion of two letters transforms every word into a reduced expression for the same element, and a word is reduced if and only if it cannot be shortened by deleting two letters."

Verification

technique · direct multiplication in $S_3$, with the lengths read off from the inversion-number identification of the type-A theorem

1.1F2

The six products. Multiplying in $S_{3}$ with the right factor acting first [F2]: $s_{1} s_{2} = (1 2) (2 3) = (1 2 3)$ , since the right factor $(2 3)$ sends $2 \mapsto 3$ and $3 \mapsto 2$ and the left factor $(1 2)$ then sends $1 \mapsto 2$ , giving $1 \mapsto 2 \mapsto 3 \mapsto 1$ ; and symmetrically $s_{2} s_{1} = (2 3) (1 2) = (1 3 2)$ . Further $s_{1} s_{2} s_{1} = (1 2) (2 3) (1 2) = (1 3)$ and $s_{2} s_{1} s_{2} = (2 3) (1 2) (2 3) = (1 3)$ , so the two length-three words have the same value. The six values $i d$ , $(1 2)$ , $(2 3)$ , $(1 2 3)$ , $(1 3 2)$ , $(1 3)$ are exactly the six elements of $S_{3}$ , so the images of the six words of the table are correct.

2.1F1F3step 1.1

The inversion numbers. Read the inversions of each value off its one-line form on the letters $1 < 2 < 3$ [F3]: $i d = [1, 2, 3]$ and $(1 2) = [2, 1, 3]$ , $(2 3) = [1, 3, 2]$ have inversion numbers $0, 1, 1$ ; the $3$ -cycles $(1 2 3) = [2, 3, 1]$ and $(1 3 2) = [3, 1, 2]$ have inversion numbers $2$ and $2$ ; and $(1 3) = [3, 2, 1]$ has all three pairs inverted, so its inversion number is $3$ . Hence the inversion-number column of the table is $(0, 1, 1, 2, 2, 3)$ , and the length column is the same by [F1].

3.1F1F4step 1.1step 2.1

Reducedness of the short words and the braid move. By step 2.1, $ℓ (s_{1} s_{2}) = 2$ and $ℓ (s_{2} s_{1}) = 2$ equal the lengths of the words $(s_{1}, s_{2})$ and $(s_{2}, s_{1})$ , so both are reduced; likewise $ℓ (s_{1} s_{2} s_{1}) = ℓ (s_{2} s_{1} s_{2}) = 3$ equals the length of the words $(s_{1}, s_{2}, s_{1})$ and $(s_{2}, s_{1}, s_{2})$ , so both are reduced expressions of the common element $s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2} = (1 3)$ of step 1.1. By [F4] the maximum of $ℓ$ on $W$ is $3$ , attained only by the two equal alternating words of length $3$ , so this common element is the unique longest element of $W$ and its two reduced expressions are the two alternating words; the replacement of $(s_{1}, s_{2}, s_{1})$ by $(s_{2}, s_{1}, s_{2})$ is the single braid move exchanging the two alternating words of length $m (s_{1}, s_{2}) = 3$ .

4.1F5step 2.1step 3.1

The nonreduced word and its deletion. For the word $(s_{1}, s_{2}, s_{1}, s_{2})$ one computes in $W$ , using $s_{2}^{2} = 1$ , that $s_{1} s_{2} s_{1} s_{2} = (s_{1} s_{2} s_{1}) s_{2} = (s_{2} s_{1} s_{2}) s_{2} = s_{2} s_{1}$ , by the relation $s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ of the type-A presentation; the value $s_{2} s_{1}$ has inversion number $2 < 4$ by step 2.1, so the word is not reduced. Its first and last letters are $s_{1}$ and $s_{2}$ , and deleting them leaves the word $(s_{2}, s_{1})$ , which represents $s_{2} s_{1}$ and is reduced by step 3.1; this is the two-letter deletion asserted in [F5], here deleting the two letters at positions $1$ and $4$ .

5.1step 1.1step 2.1step 3.1step 4.1∎

Collected. The table and the length identification (steps 1.1, 2.1) verify $ℓ = inv$ on all six elements of the type-A group and exhibit the two reduced expressions of the longest element (step 3.1) together with a nonreduced word whose first and last letters may be deleted (step 4.1).

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Minimal coset representatives of $S_{2}$ in $S_{3}$

Example

Let $W ≅ S_{3}$ be the type-A group of Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4) with $S = {s_{1}, s_{2}}$ , and let $J = {s_{1}}$ , so that $W_{J} = {1, s_{1}} ≅ S_{2}$ . Then $W$ has three right cosets $W_{J} a = {u a : u \in W_{J}}$ ,

${1, s_{1}}, {s_{2}, s_{1} s_{2}}, {s_{2} s_{1}, s_{1} s_{2} s_{1}},$

with unique minimal elements $1$ , $s_{2}$ and $s_{2} s_{1}$ of lengths $0$ , $1$ and $2$ . Each minimal element $d$ satisfies $ℓ (s_{1} d) > ℓ (d)$ (namely $ℓ (s_{1}) = 1$ , $ℓ (s_{1} s_{2}) = 2$ , $ℓ (s_{1} s_{2} s_{1}) = 3$ ), and length additivity holds: $ℓ (s_{1} \cdot s_{2}) = 1 + 1 = 2$ and $ℓ (s_{1} \cdot s_{2} s_{1}) = 1 + 2 = 3$ . The corresponding left cosets $a W_{J} = {a u : u \in W_{J}}$ have the minimal representatives $1$ , $s_{2}$ , $s_{1} s_{2}$ , and for these $ℓ (d s_{1}) = ℓ (d) + 1$ holds.

Facts & Assumptions

Given: The type-A Coxeter matrix on $S = {s_{1}, s_{2}}$ with $m (s_{1}, s_{2}) = 3$ ; the presented group $W$ with its length $ℓ$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the parabolic subgroup $W_{J} = ⟨ {s : s \in J} ⟩$ and the coset theorem of Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification; the element list and length table of Type-A reduced words and inversion numbers in $S_{3}$ ; and the coset vocabulary of Left and right cosets $g H$ and $H g$ of a subgroup.

[F1]

Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3): "Every right coset $W_{J} a : = {u a : u \in W_{J}}$ ( $a \in W$ , Left and right cosets $g H$ and $H g$ of a subgroup) has a unique element $d$ of minimal length; it is characterized by $ℓ (s d) > ℓ (d)$ for all $s \in J$ , and it satisfies $ℓ (u d) = ℓ (u) + ℓ (d) for all u \in W_{J} .$ "; and "every left coset $a W_{J} : = {a u : u \in W_{J}}$ has a unique minimal element $d$ , characterized by $ℓ (d s) > ℓ (d)$ for all $s \in J$ and satisfying $ℓ (d u) = ℓ (d) + ℓ (u)$ for all $u \in W_{J}$ ".

[F2]

Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1),(2): " $W_{J} = ⟨ J ⟩ = {w \in W : S (w) \subseteq J}$ ", and the canonical map is an isomorphism onto $W_{J}$ , so "its intrinsic length function $ℓ_{J}$ agrees with the ambient length $ℓ$ on $W_{J}$ , and $W_{J} \cap S = J$ ".

[F3]

Type-A reduced words and inversion numbers in $S_{3}$ : the six elements of $W$ are $1, s_{1}, s_{2}, s_{1} s_{2}, s_{2} s_{1}, s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ with lengths $0, 1, 1, 2, 2, 3$ and $ℓ (w) = inv (φ (w))$ ; and $s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ is the longest element of $W$ .

[F4]

Left and right cosets $g H$ and $H g$ of a subgroup: "For $g \in G$ , the left coset and right coset of $H$ represented by $g$ are $g H : = {g h : h \in H}, H g : = {h g : h \in H} .$ ", so the sets $W_{J} a = {u a : u \in W_{J}}$ and $a W_{J} = {a u : u \in W_{J}}$ are the two coset families. Thus $W_{J} a$ is a right coset and $a W_{J}$ a left coset, consistently with [F1] and the displayed sets.

[F5]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups: $W_{J} : = ⟨ {s : s \in J} ⟩ \leq W$ for $J \subseteq S$ , and $ℓ (w)$ is the least length of a word in $S$ representing $w$ ; in particular $ℓ (1) = 0$ .

[F6]

The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness (1),(4): " $ℓ (s) = 1$ " for the generators, and " $σ_{s}^{2} = {i d}_{E}$ for every $s \in S$ "; in particular $ℓ (s_{1}) = 1$ implies $s_{1} \neq 1$ .

Verification

technique · direct enumeration of the six elements of $W$, with the coset, minimality and additivity statements of the parabolic theorem applied to $J=\{s_1\}$

1.1F2F5F6

The parabolic subgroup. $W_{J} = ⟨ J ⟩ = ⟨ s_{1} ⟩ = {1, s_{1}}$ : the definition of $W_{J}$ as the subgroup generated by $J$ is [F5] and its identity with $⟨ s_{1} ⟩$ is immediate for $J = {s_{1}}$ [F2]; the relator $s_{1}^{2}$ of the presentation gives $s_{1}^{2} = 1$ [F5], so $⟨ s_{1} ⟩ \subseteq {1, s_{1}}$ , and $s_{1} \neq 1$ because $ℓ (s_{1}) = 1$ [F6], so $⟨ s_{1} ⟩$ has the two distinct elements $1, s_{1}$ ; it is a group of order $2$ , hence isomorphic to $S_{2}$ .

1.2F1F3F4

The left cosets. The left cosets $a W_{J}$ are ${1, s_{1}}$ , $s_{2} W_{J} = {s_{2}, s_{2} s_{1}}$ and $s_{1} s_{2} W_{J} = {s_{1} s_{2}, s_{1} s_{2} s_{1}}$ , disjoint and exhausting $W$ by the six distinct elements listed in [F3]; their minimal elements are $1$ (length $0$ ), $s_{2}$ (length $1$ , versus $ℓ (s_{2} s_{1}) = 2$ ) and $s_{1} s_{2}$ (length $2$ , versus $ℓ (s_{1} s_{2} s_{1}) = 3$ ), unique by the left-coset half of [F1]. For these three elements $d$ the right-handed additivity of [F1] gives $ℓ (d s_{1}) = ℓ (d) + ℓ (s_{1}) = ℓ (d) + 1$ , namely $ℓ (s_{1}) = 1$ , $ℓ (s_{2} s_{1}) = 2 = 1 + 1$ and $ℓ (s_{1} s_{2} s_{1}) = 3 = 2 + 1$ , as asserted; equivalently the left coset representatives satisfy the characterisation $ℓ (d s_{1}) > ℓ (d)$ of [F1].

2.1F1F3F4step 1.1

The right cosets and their minimal elements. For $W_{J} = {1, s_{1}}$ the sets $W_{J} a = {u a : u \in W_{J}}$ of [F4], with $a \in W$ , can be enumerated using the six-element list of [F3] they are $W_{J} \cdot 1 = {1, s_{1}}$ , $W_{J} s_{2} = {s_{2}, s_{1} s_{2}}$ and $W_{J} s_{2} s_{1} = {s_{2} s_{1}, s_{1} s_{2} s_{1}}$ , which are disjoint and exhaust $W$ . Reading the lengths off [F3], the minimum of $ℓ$ is $0$ on ${1, s_{1}}$ (attained at $1$ ), $1$ on ${s_{2}, s_{1} s_{2}}$ (attained at $s_{2}$ ) and $2$ on ${s_{2} s_{1}, s_{1} s_{2} s_{1}}$ (attained at $s_{2} s_{1}$ ); by [F1] each coset has a unique element of minimal length, so the minimal representatives are $1$ , $s_{2}$ and $s_{2} s_{1}$ with lengths $0$ , $1$ and $2$ .

3.1F1F3step 2.1

The descent characterisation and length additivity. For the three minimal representatives $d = 1$ , $d = s_{2}$ and $d = s_{2} s_{1}$ , the table of [F3] gives $ℓ (s_{1} \cdot 1) = ℓ (s_{1}) = 1 > 0 = ℓ (1)$ , $ℓ (s_{1} s_{2}) = 2 > 1 = ℓ (s_{2})$ and $ℓ (s_{1} s_{2} s_{1}) = 3 > 2 = ℓ (s_{2} s_{1})$ , so each of them satisfies the characterisation $ℓ (s_{1} d) > ℓ (d)$ of [F1]. The additivity identity of [F1] for $u = s_{1} \in W_{J}$ reads $ℓ (s_{1} d) = ℓ (s_{1}) + ℓ (d)$ for the minimal $d$ ; at $d = s_{2}$ this is $ℓ (s_{1} s_{2}) = 1 + 1 = 2$ and at $d = s_{2} s_{1}$ it is $ℓ (s_{1} s_{2} s_{1}) = 1 + 2 = 3$ , the two values asserted in the statement.

4.1step 1.1step 2.1step 3.1step 1.2∎

Collected. The example enumerates the three right cosets and the three left cosets of $W_{J} = {1, s_{1}}$ in $W ≅ S_{3}$ , identifies their unique minimal representatives with the lengths predicted by the parabolic theorem, verifies the descent characterisations on both sides and the length-additivity identities $ℓ (u d) = ℓ (u) + ℓ (d)$ and $ℓ (d u) = ℓ (d) + ℓ (u)$ for $u = s_{1}$ (steps 2.1, 3.1, 1.2).

Coxeter Presentations, Exchange, and Reduced Word Theorems — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Reduced words in rank one

Example

Facts & Assumptions

Verification

Reduced words and lengths in a finite dihedral group

Example

Facts & Assumptions

Verification

A nonreduced word deleted by its repeated prefix reflection, and an exchange step

Example

Facts & Assumptions

Verification

Type-A reduced words and inversion numbers in S3S3​

Example

Facts & Assumptions

Verification

Minimal coset representatives of S2S2​ in S3S3​

Example

Facts & Assumptions

Verification

Sources

Type-A reduced words and inversion numbers in $S_{3}$

Minimal coset representatives of $S_{2}$ in $S_{3}$