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✓ 4 results · all verified · 1 also independently AI-judged

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Generic Coxeter Hecke Algebras and the Standard Basis — Examples

1 · Prerequisites

2 · Summary

This companion works out the smallest computations of generic-coxeter-hecke-algebras-and-the-standard-basis and the one obstruction that the parameter rule removes. The rank-one example records the complete multiplication in both normalizations: the normalized table generated by $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ and the multiplicative table generated by $(S_{s} - Q) (S_{s} + 1) = 0$ with $S_{s} = v T_{s}$ and $Q = v^{2}$ , together with the conversion $S_{s} = v T_{s}$ and the domain statement for $R = Z [v^{\pm 1}]$ .

For $S_{3}$ the example gives the complete $6 \times 6$ left-multiplication table in both normalizations, computed from the length-multiplication rules: the normalized table in $T_{w}$ and the multiplicative table in $S_{w} = v^{ℓ (w)} T_{w}$ , with the explicit entry $S_{w_{0}} S_{w_{0}}$ and the sanity checks of the braid identity, associativity and the specialization at $v = 1$ .

The third example isolates what unequal parameters can and cannot do in dihedral type. For odd $m$ the two standard reflections are conjugate, and the two length operators fail to commute at the longest alternating element unless the two parameter differences agree: the difference is $(u_{t} - u_{s})$ times an explicit basis vector, and in the universal Laurent ring equality holds exactly for $v_{t} = v_{s}$ or $v_{t} = - v_{s}^{- 1}$ . For even $m$ the two generators lie in different odd components, no length configuration obstructs commutation, and the standard basis exists for every pair of independent unit parameters. The last example specializes the generic algebra: at $u_{s} = 1$ the quotient is the group ring $A [W]$ with $T_{s} \mapsto s$ and $T_{w} \mapsto w$ , on matching bases; for general units the quadratic relation reads $T_{s}^{2} = (u_{s} - u_{s}^{- 1}) T_{s} + 1$ , so the substitution $T_{s} \mapsto s$ is an algebra map exactly when $u_{s}^{2} = 1$ , and both $v \mapsto 1$ and $v \mapsto - 1$ give the group ring.

The examples use the A-page items of generic-coxeter-hecke-algebras-and-the-standard-basis and its prerequisite closure; the specialization also uses the published group-ring construction and basis theorem from the-group-algebra-and-representations. This companion is a dependency leaf: it supplies no theorem to another page, and the two application pages named in the specialization example are reading pointers only, not dependencies.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-08Open item page →

Unequal parameters in the dihedral cases: the odd-edge obstruction and the even-edge freedom

Example

Let $S = {s, t}$ with $m : = m (s, t)$ finite and $W$ the dihedral group of order $2 m$ (Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups, part 3). For the operator test, let $R$ be any commutative ring and let $v_{s}, v_{t} \in R^{\times}$ be arbitrary units, without imposing the odd-component rule. Put $E : = R^{(W)}$ with basis $(e_{w})$ and define $P_{s}, Q_{t}$ by the length formulas of The commuting left and right length operators and their Hecke relations, using $u_{x} : = v_{x} - v_{x}^{- 1}$ for $x \in {s, t}$ . These formulas define linear maps even when the parameters fail the rule; the commutation test below determines the obstruction. For the generic Hecke algebra itself, the coefficient ring and class-constant parameters are those of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy.

$m$ odd forces equal differences. Suppose $m$ is odd, so that $s$ and $t$ are conjugate and Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy gives them one parameter. If instead one tries independent parameters $u_{s}, u_{t}$ , the commutation check of The commuting left and right length operators and their Hecke relations fails: at the weight $w = s t s$ (the case $m = 3$ ) one has $s w = w t = t s$ and $P_{s} Q_{t} (e_{w}) - Q_{t} P_{s} (e_{w}) = (u_{t} - u_{s}) e_{t s},$ and for general odd $m$ the same computation at the longest element $w = s t s t \dots$ of length $m$ gives $(u_{t} - u_{s}) e_{w t}$ with $w t$ the alternating element of length $m - 1$ ; the difference vanishes if and only if $u_{s} = u_{t}$ . Since the quadratic relation depends on the parameter only through the difference $u_{x}$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, convention (ii)), the condition $u_{s} = u_{t}$ means precisely that the quadratic relations of $T_{s}$ and $T_{t}$ coincide; for unit-valued parameters in an integral domain its solutions are $v_{t} = v_{s}$ and $v_{t} = - v_{s}^{- 1}$ ; over an arbitrary commutative ring the exact condition is $(v_{t} - v_{s}) (v_{t} + v_{s}^{- 1}) = 0$ . The class-constant assignment of the A page satisfies this condition on an odd edge; it is a canonical choice of unit parameters, rather than the only choice with these quadratic relations.
$m$ even allows unequal parameters. Suppose $m$ is even, e.g. $m = 4$ or $6$ . Then $s$ and $t$ lie in different components of the odd-edge graph and are not conjugate; the assignment $v_{s} = u$ , $v_{t} = w$ with independent units $u, w$ is a legitimate instance of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, and for every $z \in W$ the two expansions of $P_{s} Q_{t} (e_{z})$ and $Q_{t} P_{s} (e_{z})$ agree term by term with no relation imposed between $u$ and $w$ . Consequently $P_{s} Q_{t} = Q_{t} P_{s}$ , and the standard basis theorem The standard basis of the generic Hecke algebra and base change applies with distinct parameters $u \neq w$ .
Conclusion. Independent parameter differences are allowed across generators that are not conjugate. Within an odd component the differences must agree; distinct unit parameters can still give the same difference, as in $v_{t} = - v_{s}^{- 1}$ over an integral domain. This is the local (rank-two) content of the parameter rule of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy.

Facts & Assumptions

Given: $S = {s, t}$ with $m = m (s, t)$ finite, the dihedral group $W$ of order $2 m$ , the free module $E$ with basis $(e_{w})_{w \in W}$ and the operators $P_{s}$ , $Q_{t}$ built from the parameters $u_{s}$ , $u_{t}$ .

[F1]

Simple generators are conjugate in $W$ if and only if they are joined by a chain of odd edges; in the generic presentation the units are assigned equally on each conjugacy class, while the quadratic relation depends on $v_{s}$ only through $u_{s} = v_{s} - v_{s}^{- 1}$ . (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

[F2]

The operators $P_{s}, Q_{t}$ are defined by the two length clauses, so on each basis vector $e_{z}$ they act by the clause determined by $ℓ (s z)$ and $ℓ (z t)$ ; whenever $ℓ (s z t) = ℓ (z)$ and $ℓ (s z) = ℓ (z t)$ one has $s z = z t$ (the two-length lemma). (The commuting left and right length operators and their Hecke relations, Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action)

[F3]

The group on $s, t$ with finite $m (s, t) = m$ is dihedral of order $2 m$ (Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups, part 3). Alternating dihedral words are reduced in the ambient group: for $q \leq m$ the alternating word of length $q$ has length exactly $q$ in $W$ . (The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness)

[F4]

${T_{w} : w \in W}$ is an $R$ -basis of the Hecke algebra whenever the parameters are constant on odd-edge components; in particular the standard basis theorem applies to the two-parameter algebra of the even case $m$ , where $s$ and $t$ lie in different odd components. (The standard basis of the generic Hecke algebra and base change)

[F5]

The integers are a commutative ring (The integers form a commutative ring) with no zero divisors (The integers have no zero divisors; multiplicative cancellation), and $0 \neq 1$ by injectivity of the natural-number embedding (The naturals embed in the integers), hence an integral domain. The Laurent polynomial ring in finitely many variables over an integral domain is an integral domain; in an integral domain a product is zero only if one factor is zero. The independent formal variables $v_{s}, v_{t}$ in $Z [v_{s}^{\pm 1}, v_{t}^{\pm 1}]$ do not satisfy either factor equation; the two alternatives describe unit-valued specializations into integral domains. (Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields)

Verification

technique · direct

1.1F2algebra

Fix $x \in W$ and expand $P_{s} Q_{t} (e_{x})$ and $Q_{t} P_{s} (e_{x})$ from the two length clauses. The intermediate length $ℓ (s x t)$ differs from both $ℓ (s x)$ and $ℓ (x t)$ by $\pm 1$ , so only the following length configurations occur: (i) $ℓ (s x) = ℓ (x t) = ℓ (x) + 1$ , $ℓ (s x t) = ℓ (x) + 2$ ; (ii) $ℓ (s x) = ℓ (x t) = ℓ (x) - 1$ , $ℓ (s x t) = ℓ (x) - 2$ ; (iii) $ℓ (s x) = ℓ (x) - 1$ , $ℓ (x t) = ℓ (x) + 1$ , $ℓ (s x t) = ℓ (x)$ ; (iv) $ℓ (s x) = ℓ (x) + 1$ , $ℓ (x t) = ℓ (x) - 1$ , $ℓ (s x t) = ℓ (x)$ ; (v) $ℓ (s x) = ℓ (x t) = ℓ (x) - 1$ , $ℓ (s x t) = ℓ (x)$ ; (vi) $ℓ (s x) = ℓ (x t) = ℓ (x) + 1$ , $ℓ (s x t) = ℓ (x)$ . In (i)-(iv) the two expansions agree identically for all values of $u_{s}, u_{t}$ ; in (v) they are $P_{s} Q_{t} (e_{x}) = e_{s x t} + u_{t} e_{s x} + u_{s} u_{t} e_{x}$ and $Q_{t} P_{s} (e_{x}) = e_{s x t} + u_{s} e_{x t} + u_{s} u_{t} e_{x}$ , and in (vi) they are $P_{s} Q_{t} (e_{x}) = e_{s x t} + u_{s} e_{x t}$ and $Q_{t} P_{s} (e_{x}) = e_{s x t} + u_{t} e_{s x}$ ; moreover in (v) and (vi) the hypotheses $ℓ (s x t) = ℓ (x)$ and $ℓ (s x) = ℓ (x t)$ hold, so $s x = x t$ ([F2]).

2.1F1F3F5step 1.1algebra

Let $m = 2 k + 1$ be odd and let $ξ : = (s t)^{k} s$ , the alternating word of length $m$ ; by [F3] $ℓ (ξ) = m$ , and $s ξ = s (s t)^{k} s = (t s)^{k} = (s t)^{- k} = (s t)^{k + 1}$ while $ξ t = (s t)^{k} s t = (s t)^{k + 1}$ , the middle identities using $(s t)^{- 1} = t s$ and the relator $(s t)^{m} = 1$ together with $m - k = k + 1$ (the values here are elements of $W$ , not literal words). Put $z : = s ξ = ξ t$ ; then $z$ is the alternating element of length $m - 1$ , so $ℓ (s ξ) = ℓ (ξ t) = ℓ (z) = m - 1 = ℓ (ξ) - 1$ by [F3], while $s ξ t = z t = (s t)^{k + 1} t = (s t)^{k} s t t = (s t)^{k} s = ξ$ gives $ℓ (s ξ t) = ℓ (ξ)$ . This is configuration (v) of 1.1 at the weight $ξ$ , so $P_{s} Q_{t} (e_{ξ}) - Q_{t} P_{s} (e_{ξ}) = (u_{t} - u_{s}) e_{z}$ with $z = ξ t$ the alternating element of length $m - 1$ ; for $m = 3$ this is the displayed identity with $z = t s$ . The difference vanishes in $E$ if and only if $u_{t} - u_{s} = 0$ in the coefficient ring, because $(e_{w})$ is a basis. Consequently independent parameters with $u_{s} \neq u_{t}$ violate the commutation of the length operators, while $u_{s} = u_{t}$ holds exactly when the quadratic relations of $T_{s}$ and $T_{t}$ coincide ([F1]); the identity $v_{t} (u_{t} - u_{s}) = (v_{t} - v_{s}) (v_{t} + v_{s}^{- 1})$ shows that $u_{s} = u_{t}$ is equivalent to $(v_{t} - v_{s}) (v_{t} + v_{s}^{- 1}) = 0$ , since $v_{t}$ is a unit. For unit-valued specializations into an integral domain this holds exactly when $v_{t} = v_{s}$ or $v_{t} = - v_{s}^{- 1}$ by [F5]; over a ring with zero divisors only the product-zero condition is asserted.

2.2F1F2F4step 1.1

Let $m$ be even. If some $x$ satisfied $ℓ (s x t) = ℓ (x)$ and $ℓ (s x) = ℓ (x t)$ , then $s x = x t$ by [F2], so $s = x t x^{- 1}$ is conjugate to $t$ ; but for $m$ even the odd-edge graph on $S = {s, t}$ has no edge, so $s$ and $t$ are not conjugate by [F1]. Hence no $x$ realizes configurations (v) or (vi) of 1.1, and only the configurations (i)-(iv) occur, in which the expansions agree identically for arbitrary $u_{s}, u_{t}$ ; therefore $P_{s} Q_{t} = Q_{t} P_{s}$ in the two-parameter algebra, and by [F4] the standard basis theorem applies there with independent parameters $u \neq w$ .

3.1F1step 2.1step 2.2algebra∎

By 2.1 the parameter difference is forced within the odd component ${s, t}$ : commutation of the length operators requires $u_{s} = u_{t}$ , equivalently the two quadratic relations coincide, and over an integral domain these are the assignments $v_{t} = v_{s}$ or $v_{t} = - v_{s}^{- 1}$ . The first is class-constant, as in [F1]; the second has the same quadratic relation and hence the same length operators as the first. By 2.2, when $s$ and $t$ lie in different odd components the parameter difference is free and the standard basis exists for any independent units. This is precisely the rank-two content of the parameter rule [F1], and no choice is used: all expansions are computed from the explicit length clauses, and the two equations solved above are quadratic identities in units.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-08Open item page →

Rank-one Hecke multiplication in both normalizations

Example

Let $S = {s}$ with $m (s, s) = 1$ , so that $W = {1, s} ≅ Z / 2$ with $ℓ (1) = 0$ and $ℓ (s) = 1$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). The odd-edge graph has one vertex and one component, so $R = Z [v^{\pm 1}]$ with $v : = v_{s}$ is the coefficient ring of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, and $H$ is the quotient of the free associative $R$ -algebra $R ⟨ T_{s} ⟩$ by the single relation $(T_{s} - v) (T_{s} + v^{- 1}) = 0$ .

Normalized table. ${1, T_{s}}$ is an $R$ -basis of $H$ (The standard basis of the generic Hecke algebra and base change), and $1 \cdot 1 = 1, 1 \cdot T_{s} = T_{s} = T_{s} \cdot 1, T_{s}^{2} = (v - v^{- 1}) T_{s} + 1,$ equivalently $(T_{s} - v) (T_{s} + v^{- 1}) = 0$ . Moreover $T_{s}^{- 1} = T_{s} - (v - v^{- 1})$ , so $T_{s}^{- 1} T_{s} = T_{s} T_{s}^{- 1} = 1$ (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization).
Multiplicative table. Put $Q : = v^{2}$ and $S_{s} : = v T_{s}$ . Then ${1, S_{s}}$ is again an $R$ -basis, $S_{s}$ is a unit, and $1 \cdot 1 = 1, 1 \cdot S_{s} = S_{s} = S_{s} \cdot 1, S_{s}^{2} = (Q - 1) S_{s} + Q,$ equivalently $(S_{s} - Q) (S_{s} + 1) = 0$ ; indeed $S_{s}^{- 1} = Q^{- 1} (S_{s} + 1 - Q)$ .
Conversion. The two tables are interconverted by $S_{s} = v T_{s}$ , $T_{s} = v^{- 1} S_{s}$ : substituting in $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ gives $v^{- 2} S_{s}^{2} = (1 - v^{- 2}) S_{s} + 1$ , i.e. $S_{s}^{2} = (Q - 1) S_{s} + Q$ ; conversely $T_{s}^{2} = v^{- 2} S_{s}^{2} = v^{- 2} ((Q - 1) S_{s} + Q) = (v - v^{- 1}) T_{s} + 1$ . In particular $R$ is a domain, both bases persist after every base change by The standard basis of the generic Hecke algebra and base change, part 4, and no choice or finiteness hypothesis beyond $∣ S ∣ = 1$ is used.

Facts & Assumptions

Given: The one-element generator set $S = {s}$ , the group $W = {1, s}$ , the Laurent ring $R = Z [v^{\pm 1}]$ and the algebra $H = R ⟨ T_{s} ⟩ / ((T_{s} - v) (T_{s} + v^{- 1}))$ .

[F1]

$R = Λ_{Z, 1}$ is a commutative ring in which $v$ is a unit, $H$ is presented by the generator $T_{s}$ and the single quadratic relation $(T_{s} - v) (T_{s} + v^{- 1}) = 0$ , and each $v_{s}$ is a unit; there are no braid relations because $∣ S ∣ = 1$ . (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

[F2]

For $S = {s}$ the presentation of $W$ has the single generator $s$ and the relation $s^{2} = 1$ . Its universal property extends any assignment of $s$ to an involution in a group, and $ℓ$ is the least word length. (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups)

[F3]

${T_{w} : w \in W}$ is an $R$ -basis of $H$ , and after base change along any ring homomorphism the specialized family is an $R^{'}$ -basis; in particular $T_{1} = 1$ and ${1, T_{s}}$ is a basis. (The standard basis of the generic Hecke algebra and base change)

[F4]

$T_{s}$ is a unit with $T_{s}^{- 1} = T_{s} - (v - v^{- 1})$ ; with $Q_{s} = v_{s}^{2}$ and $S_{s} = v_{s} T_{s}$ , the element $S_{s}$ is a unit, $T_{s} = v_{s}^{- 1} S_{s}$ , and $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ . (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization)

[F5]

Verification

technique · direct

1.1F1F2F3F4

By [F2] every word in the single generator reduces using $s^{2} = 1$ to $1$ or $s$ . The assignment $s \mapsto - 1$ extends by the universal property to a homomorphism $W \to {\pm 1}$ , so $s \neq 1$ ; thus $W = {1, s}$ and the reduced expressions are the empty word and the one-letter word $s$ , with lengths $0$ and $1$ ; by [F3] the family ${T_{1}, T_{s}}$ is an $R$ -basis of $H$ and $T_{1} = 1$ is the empty product. The unit axioms give $1 \cdot T_{s} = T_{s} = T_{s} \cdot 1$ , and expanding $(T_{s} - v) (T_{s} + v^{- 1}) = 0$ from [F1] gives $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ ; the inverse formula $T_{s}^{- 1} = T_{s} - (v - v^{- 1})$ and $T_{s}^{- 1} T_{s} = T_{s} T_{s}^{- 1} = 1$ are [F4]. This is the normalized table of part 1.

1.2F1F3F4algebra

Put $Q : = v^{2}$ and $S_{s} : = v T_{s}$ . Since $v$ is a unit of $R$ ([F1]) and multiplication by it is an invertible $R$ -linear map, ${1, S_{s}}$ is again an $R$ -basis of $H$ ([F3]); $S_{s}$ is a unit with $S_{s}^{- 1} = v^{- 1} T_{s}^{- 1}$ as a product of units ([F4]). Multiplying $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ by $v^{2}$ gives $S_{s}^{2} = v (v - v^{- 1}) S_{s} + v^{2} = (Q - 1) S_{s} + Q$ , because $v (v - v^{- 1}) = v^{2} - 1 = Q - 1$ ; equivalently $(S_{s} - Q) (S_{s} + 1) = S_{s}^{2} + (1 - Q) S_{s} - Q = 0$ . Finally $S_{s} (S_{s} + 1 - Q) = S_{s}^{2} + S_{s} - Q S_{s} = ((Q - 1) S_{s} + Q) + S_{s} - Q S_{s} = Q$ , so $S_{s}^{- 1} = Q^{- 1} (S_{s} + 1 - Q)$ . This is the multiplicative table of part 2.

2.1F3F4F5step 1.1step 1.2∎

The two tables are interconverted by $S_{s} = v T_{s}$ and $T_{s} = v^{- 1} S_{s}$ ([F4]). Substituting $T_{s} = v^{- 1} S_{s}$ into $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ gives $v^{- 2} S_{s}^{2} = (v - v^{- 1}) v^{- 1} S_{s} + 1$ , and multiplying by the unit $v^{2}$ gives $S_{s}^{2} = v (v - v^{- 1}) S_{s} + v^{2} = (Q - 1) S_{s} + Q$ ; conversely, substituting $S_{s} = v T_{s}$ into $S_{s}^{2} = (Q - 1) S_{s} + Q$ and multiplying by $v^{- 2}$ returns $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ . The ring $R = Z [v^{\pm 1}]$ is a domain by [F5], both bases persist under every base change by [F3], and no choice is used: all identities are explicit polynomial identities in $v$ involving no selection. This completes the conversion of part 3 and with 1.1 and 1.2 all three parts of the example.

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

The complete S3 multiplication table in both normalizations

Example

Let $S = {s, t}$ with $m (s, t) = 3$ , so that $W = ⟨ s, t ∣ s^{2} = t^{2} = (s t)^{3} = 1 ⟩ ≅ S_{3}$ with elements $1, s, t, s t, t s, w_{0} : = s t s = t s t$ of lengths $0, 1, 1, 2, 2, 3$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; the identification with $S_{3}$ is the type- $A$ clause of Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification). The odd-edge graph is connected, so $R = Z [v^{\pm 1}]$ , $v_{s} = v_{t} = v$ , and $H$ is free with basis ${T_{w}}$ (The standard basis of the generic Hecke algebra and base change).

Normalized table. With $u : = v - v^{- 1}$ and the columns indexed by $1, s, t, s t, t s, w_{0}$ , the complete $6 \times 6$ left-multiplication table of $H$ is $\begin{array}{ccccccc} T_{x} \ T_{y} & T_{1} & T_{s} & T_{t} & T_{s t} & T_{t s} & T_{w_{0}} \\ T_{1} & T_{1} & T_{s} & T_{t} & T_{s t} & T_{t s} & T_{w_{0}} \\ T_{s} & T_{s} & u T_{s} + T_{1} & T_{s t} & T_{t} + u T_{s t} & T_{w_{0}} & T_{t s} + u T_{w_{0}} \\ T_{t} & T_{t} & T_{t s} & u T_{t} + T_{1} & T_{w_{0}} & T_{s} + u T_{t s} & T_{s t} + u T_{w_{0}} \\ T_{s t} & T_{s t} & T_{w_{0}} & T_{s} + u T_{s t} & T_{t s} + u T_{w_{0}} & T_{1} + u T_{s} + u T_{w_{0}} & T_{t} + u T_{s t} + u T_{t s} + u^{2} T_{w_{0}} \\ T_{t s} & T_{t s} & T_{t} + u T_{t s} & T_{w_{0}} & T_{1} + u T_{t} + u T_{w_{0}} & T_{s t} + u T_{w_{0}} & T_{s} + u T_{t s} + u T_{s t} + u^{2} T_{w_{0}} \\ T_{w_{0}} & T_{w_{0}} & T_{s t} + u T_{w_{0}} & T_{t s} + u T_{w_{0}} & T_{s} + u T_{s t} + u T_{t s} + u^{2} T_{w_{0}} & T_{t} + u T_{t s} + u T_{s t} + u^{2} T_{w_{0}} & T_{1} + u (T_{s} + T_{t}) + u^{2} (T_{s t} + T_{t s}) + (u^{3} + u) T_{w_{0}} \end{array}$ In particular $T_{s} T_{t} = T_{s t}$ , $T_{t} T_{s} = T_{t s}$ , the braid identity $T_{s} T_{t} T_{s} = T_{w_{0}} = T_{t} T_{s} T_{t}$ holds, and $T_{s}^{2} = u T_{s} + T_{1}$ , $T_{t}^{2} = u T_{t} + T_{1}$ (Reduced-word independence of T_w and the length-multiplication rules; the table is a finite computation from those rules).
Multiplicative table. Put $Q : = v^{2}$ , $S_{s} : = v T_{s}$ , $S_{t} : = v T_{t}$ and $S_{w} : = S_{s_{1}} \dots S_{s_{k}} = v^{ℓ (w)} T_{w}$ for a reduced expression $w = s_{1} \dots s_{k}$ (well defined, since the $S$ 's satisfy the braid relations). Then $S_{s} S_{w} = {\begin{cases} S_{s w}, & ℓ (s w) = ℓ (w) + 1, \\ Q S_{s w} + (Q - 1) S_{w}, & ℓ (s w) = ℓ (w) - 1, \end{cases}$ and the same rule holds with $S_{t}$ . With $a : = Q - 1$ , the complete multiplicative table is $\begin{array}{ccccccc} S_{x} \ S_{y} & S_{1} & S_{s} & S_{t} & S_{s t} & S_{t s} & S_{w_{0}} \\ S_{1} & S_{1} & S_{s} & S_{t} & S_{s t} & S_{t s} & S_{w_{0}} \\ S_{s} & S_{s} & Q S_{1} + a S_{s} & S_{s t} & Q S_{t} + a S_{s t} & S_{w_{0}} & Q S_{t s} + a S_{w_{0}} \\ S_{t} & S_{t} & S_{t s} & Q S_{1} + a S_{t} & S_{w_{0}} & Q S_{s} + a S_{t s} & Q S_{s t} + a S_{w_{0}} \\ S_{s t} & S_{s t} & S_{w_{0}} & Q S_{s} + a S_{s t} & Q S_{t s} + a S_{w_{0}} & Q^{2} S_{1} + Q a S_{s} + a S_{w_{0}} & Q^{2} S_{t} + Q a (S_{s t} + S_{t s}) + a^{2} S_{w_{0}} \\ S_{t s} & S_{t s} & Q S_{t} + a S_{t s} & S_{w_{0}} & Q^{2} S_{1} + Q a S_{t} + a S_{w_{0}} & Q S_{s t} + a S_{w_{0}} & Q^{2} S_{s} + Q a (S_{t s} + S_{s t}) + a^{2} S_{w_{0}} \\ S_{w_{0}} & S_{w_{0}} & Q S_{s t} + a S_{w_{0}} & Q S_{t s} + a S_{w_{0}} & Q^{2} S_{s} + Q a (S_{s t} + S_{t s}) + a^{2} S_{w_{0}} & Q^{2} S_{t} + Q a (S_{t s} + S_{s t}) + a^{2} S_{w_{0}} & Q^{3} S_{1} + Q^{2} a (S_{s} + S_{t}) + Q a^{2} (S_{s t} + S_{t s}) + (a^{3} + Q a) S_{w_{0}} \end{array}$ The rows for $S_{s t}$ , $S_{t s}$ and $S_{w_{0}}$ are obtained by composing the generator rows; in expanded $Q$ -notation, $S_{w_{0}} S_{w_{0}} = Q^{3} S_{1} + (Q^{3} - Q^{2}) (S_{s} + S_{t}) + (Q^{3} - 2 Q^{2} + Q) (S_{s t} + S_{t s}) + (Q^{3} - 2 Q^{2} + 2 Q - 1) S_{w_{0}}$ (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization, part 4). The conversion between the two tables is $T_{w} = v^{- ℓ (w)} S_{w}$ ; it maps the normalized table above to the multiplicative one.
Sanity checks. Both tables satisfy the braid identity and are associative, because they are the tables of the associative algebras $H$ and its rescaling; the substitution $v = 1$ (so $Q = 1$ ) turns them into the multiplication table of the group ring $Z [S_{3}]$ (Specialization of the generic Hecke algebra to the group ring).

Facts & Assumptions

Given: The generators $s, t$ with $m (s, t) = 3$ , the group $W ≅ S_{3}$ with its six elements and lengths, the ring $R = Z [v^{\pm 1}]$ and the algebra $H$ with standard basis ${T_{w}}$ .

[F1]

$R = Λ_{Z, 1}$ , $H$ is the quotient of the free associative $R$ -algebra on $T_{s}, T_{t}$ by the quadratic relations and the single braid relation $T_{s} T_{t} T_{s} = T_{t} T_{s} T_{t}$ , and $v$ is a unit. (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

[F2]

For every reduced expression $w = s_{1} \dots s_{k}$ the element $T_{w} = T_{s_{1}} \dots T_{s_{k}}$ is well defined, and $T_{s} T_{w} = T_{s w}$ if $ℓ (s w) = ℓ (w) + 1$ while $T_{s} T_{w} = T_{s w} + (v - v^{- 1}) T_{w}$ if $ℓ (s w) = ℓ (w) - 1$ ; the analogous right-multiplication rule holds. (Reduced-word independence of T_w and the length-multiplication rules)

[F3]

${T_{w} : w \in W}$ is an $R$ -basis of $H$ . (The standard basis of the generic Hecke algebra and base change)

[F4]

$T_{s}$ is a unit with $T_{s}^{- 1} = T_{s} - (v - v^{- 1})$ ; for $Q_{s} = v^{2}$ and $S_{s} = v T_{s}$ one has $T_{s} = v^{- 1} S_{s}$ , $S_{s}$ is a unit, and $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ . (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization)

[F5]

$W ≅ S_{3}$ via $s \mapsto (12)$ , $t \mapsto (23)$ ; its six elements $1, s, t, s t, t s, w_{0} = s t s = t s t$ have lengths $0, 1, 1, 2, 2, 3$ , and $ℓ$ agrees with the inversion number of the corresponding permutation. (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification)

Verification

technique · direct

1.1F1F3F5

The six elements of $W$ and their lengths are those recorded in [F5], and the odd-edge graph on $S = {s, t}$ (the single edge ${s, t}$ , since $m (s, t) = 3$ is odd) is connected, so $c = 1$ , $R = Z [v^{\pm 1}]$ and $v_{s} = v_{t} = v$ by the parameter rule of [F1]. By [F3] the family ${T_{w} : w \in W}$ is an $R$ -basis of $H$ , so all tables below record well-defined elements.

1.2F2F3algebra

Expansion from the rules of [F2] gives the normalized table of part 1. For example $T_{s} T_{s t}$ : $s \cdot (s t) = t$ has length $1 < 2$ , so the entry is $T_{t} + u T_{s t}$ ; $T_{s} T_{t s}$ : $s \cdot (t s) = w_{0}$ has length $3 > 2$ , so the entry is $T_{w_{0}}$ ; $T_{w_{0}} T_{s}$ : $w_{0} s = s t$ has length $2 < 3$ , so the entry is $T_{s t} + u T_{w_{0}}$ ; $T_{s t} T_{w_{0}}$ : $(s t) w_{0} = t$ has length $1 < 2$ , so $T_{s t} T_{w_{0}} = T_{s} (T_{t} T_{w_{0}}) = T_{s} (T_{s t} + u T_{w_{0}}) = T_{s} T_{s t} + u T_{s} T_{w_{0}} = (T_{t} + u T_{s t}) + u (T_{t s} + u T_{w_{0}}) = T_{t} + u T_{s t} + u T_{t s} + u^{2} T_{w_{0}}$ ; and $T_{w_{0}} T_{w_{0}} = (T_{w_{0}} T_{s t}) T_{s} = (T_{s} + u T_{s t} + u T_{t s} + u^{2} T_{w_{0}}) T_{s} = T_{1} + u (T_{s} + T_{t}) + u^{2} (T_{s t} + T_{t s}) + (u^{3} + u) T_{w_{0}}$ , using $T_{s t} T_{s} = T_{w_{0}}$ , $T_{t s} T_{s} = T_{t} + u T_{t s}$ , $T_{w_{0}} T_{s} = T_{s t} + u T_{w_{0}}$ and $T_{s}^{2} = T_{1} + u T_{s}$ . The remaining entries are computed in the same way by left multiplication by a reduced expression of the row element; the displayed table records all $36$ products.

2.1F2F4step 1.1step 1.2algebra

Multiplying the rules of 1.2 by $v^{ℓ (w) + 1}$ gives the multiplicative rule $S_{s} S_{w} = S_{s w}$ when $ℓ (s w) = ℓ (w) + 1$ and $S_{s} S_{w} = Q S_{s w} + (Q - 1) S_{w}$ when $ℓ (s w) = ℓ (w) - 1$ , and similarly for $S_{t}$ : for instance $S_{s} S_{s} = v^{2} T_{s}^{2} = v^{2} (u T_{s} + T_{1}) = v^{2} u v^{- 1} S_{s} + Q S_{1} = (Q - 1) S_{s} + Q S_{1}$ because $v u = v^{2} - 1 = Q - 1$ ; and $S_{s} S_{w_{0}} = v^{4} T_{s} T_{w_{0}} = v^{4} (T_{t s} + u T_{w_{0}}) = Q S_{t s} + (Q - 1) S_{w_{0}}$ . The $S_{s}$ - and $S_{t}$ -rows are the translations of the corresponding rows of the normalized table; since $S_{s t} = S_{s} S_{t}$ , $S_{t s} = S_{t} S_{s}$ and $S_{w_{0}} = S_{s} S_{t s}$ (each $S_{w}$ is the product of the $S$ 's along a reduced expression), associativity of $H$ composes the generator rows into the other displayed rows, with $a = Q - 1$ ; and converting the entry $T_{w_{0}} T_{w_{0}} = T_{1} + u (T_{s} + T_{t}) + u^{2} (T_{s t} + T_{t s}) + (u^{3} + u) T_{w_{0}}$ with $S_{w} = v^{ℓ (w)} T_{w}$ gives the stated $S_{w_{0}} S_{w_{0}}$ .

3.1F1F4step 1.2step 2.1∎

The two tables are converted into one another by $T_{w} = v^{- ℓ (w)} S_{w}$ , which is $T_{s} = v^{- 1} S_{s}$ on generators and extends to products; it maps each entry of the normalized table to the corresponding entry of the multiplicative table by the computation of 2.1. Both tables satisfy the braid identity $T_{s} T_{t} T_{s} = T_{w_{0}} = T_{t} T_{s} T_{t}$ because it is the defining braid relation of $H$ ([F1]), and both are associative because $H$ is a quotient of an associative algebra; the substitution $v = 1$ (so $u = 0$ and $Q = 1$ ) turns the normalized table into the group-ring table of $Z [S_{3}]$ , as recorded in Specialization of the generic Hecke algebra to the group ring, and the multiplicative table into the same table since $S_{w} = T_{w}$ and $Q = 1$ there. All computations are over $Z [v^{\pm 1}]$ and use no choice.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-08Open item page →

Specialization of the generic Hecke algebra to the group ring

Example

Let $(S, m)$ , $W$ , $R$ , $v_{s}$ , $H$ , $T_{s}$ be as in Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy and let $A$ be a commutative ring with units $u_{s} \in A^{\times}$ constant on the classes $[s]$ , inducing $φ : R \to A$ with $v_{s} \mapsto u_{s}$ (Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields, part 2). Write $H_{A} : = A \otimes_{R} H$ for the specialization.

The specialization at $u_{s} = 1$ . If $u_{s} = 1$ for all $s$ , then there is an isomorphism of $A$ -algebras $H_{A} ⟶ A [W], T_{s} ⟼ s, T_{w} ⟼ w,$ where $A [W]$ is the group ring (The group ring $R [G]$ of finitely supported formal $R$ -linear combinations of group elements); it is an isomorphism of free $A$ -modules on the specialized standard basis ${1 \otimes T_{w}}$ and the group basis ${w}$ (The standard basis of the generic Hecke algebra and base change, part 4).
Why the relation specializes. In $A [W]$ one has $s^{2} = 1$ , so the specialized quadratic relation $(s - 1) (s + 1) = s^{2} - 1 = 0$ holds, and the braid relations are the defining relations of $W$ ; conversely, sending $T_{s} \mapsto s$ kills the specialized relations, so it factors through $H_{A}$ by the universal property. This is the case $Q = 1$ of the normalization $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ of The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization, part 4.
Other specializations. For arbitrary units $u_{s}$ the specialization is still free with basis $(1 \otimes T_{w})$ (The standard basis of the generic Hecke algebra and base change, part 4), but the quadratic relation reads $T_{s}^{2} = (u_{s} - u_{s}^{- 1}) T_{s} + 1$ , so $T_{s} \mapsto s$ is an algebra map only when $u_{s}^{2} = 1$ in $A$ , i.e. $u_{s} - u_{s}^{- 1} = 0$ (over a field this means $u_{s} = \pm 1$ ); in particular $v \mapsto 1$ and $v \mapsto - 1$ both give the group ring, since $u - u^{- 1} = 0$ in either case. Applications: the specialization $v = 1$ is the bridge from this generic algebra to the type- $A$ principal-series page principal-series-representations-of-gl-n-over-a-finite-field and to the Hecke-Markov trace page hecke-markov-traces-and-polynomial-link-invariants, both of which work in the multiplicative normalization of The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization.

Facts & Assumptions

Given: A finite Coxeter matrix $(S, m)$ , the group $W$ , the parameters $R$ , $v_{s}$ , the algebra $H$ with standard basis ${T_{w}}$ , a commutative ring $A$ with units $u_{s}$ and the induced homomorphism $φ : R \to A$ .

[F1]

$H$ is the quotient of the free associative $R$ -algebra on $(T_{s})_{s \in S}$ by the relations (Q) and (B), and for every unital associative $R$ -algebra $B$ and every family $(t_{s})$ in $B$ satisfying (Q) and (B) there is a unique unital $R$ -algebra homomorphism $H \to B$ with $T_{s} \mapsto t_{s}$ ; the images of the $T_{s}$ , together with the coefficient image of $R$ , generate $H$ as a ring. (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

[F2]

${T_{w} : w \in W}$ is an $R$ -basis of $H$ , and for every ring homomorphism $R \to R^{'}$ the scalar extension $R^{'} \otimes_{R} H$ is free with basis $(1 \otimes T_{w})_{w \in W}$ ; there is no flatness or torsion hypothesis. (The standard basis of the generic Hecke algebra and base change)

[F3]

Each $T_{s}$ is a unit, $T_{s} = v_{s}^{- 1} S_{s}$ with $S_{s} = v_{s} T_{s}$ , and $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ with $Q_{s} = v_{s}^{2}$ . (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization)

[F4]

The scalar extension of a presented algebra is presented by the images of the relations: $R^{'} \otimes_{R} (R ⟨ X ⟩ / I) ≅ (R^{'} \otimes_{R} R ⟨ X ⟩) / im (R^{'} \otimes_{R} I)$ , with no flatness or freeness of $R^{'}$ over $R$ ; the scalar extension of a free module with basis $(a_{i})$ is free with basis $(1 \otimes a_{i})$ . (Presentation base change and transport of explicit bases to commutative specializations)

[F5]

Applying the Laurent universal property over $Z$ , a choice of units $u_{1}, \dots, u_{c}$ in the commutative $Z$ -algebra $A$ extends uniquely to a $Z$ -algebra homomorphism $φ : R = Λ_{Z, c} \to A$ with $v_{i} \mapsto u_{i}$ ; equivalently, $φ (v_{s}) = u_{s}$ when the units $u_{s}$ are constant on the classes $[s]$ . (Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields, part 2)

[F6]

The group ring $A [W]$ carries a unique multiplication with $[w] [w^{'}] = [w w^{'}]$ , making it a unital $A$ -algebra with $A$ -basis ${[w] : w \in W}$ whose identity is $[1]$ and whose every basis element $[w]$ is a unit with inverse $[w^{- 1}]$ . (The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ , The group ring $R [G]$ of finitely supported formal $R$ -linear combinations of group elements)

[F7]

$W$ is presented by the generators $s \in S$ and the relators $s^{2}$ and $(s t)^{m (s, t)}$ ( $s \neq t$ , $m (s, t) < \infty$ ); a map from $S$ into a group that kills every relator extends uniquely to a group homomorphism $W \to G$ . (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups)

[F8]

For a commutative ring $A$ and a set $X$ , the free associative $A$ -algebra $A ⟨ X ⟩$ has the universal property that every map $X \to B$ into a unital $A$ -algebra $B$ extends uniquely to a unital $A$ -algebra homomorphism $A ⟨ X ⟩ \to B$ ; and a unital $A$ -algebra homomorphism out of $A ⟨ X ⟩$ that kills a set $E \subseteq A ⟨ X ⟩$ factors uniquely through the quotient $A ⟨ X ⟩ / (E)$ , the presented $A$ -algebra with generators $X$ and relations $E$ . (The free associative R-algebra on a set and descent of relations)

Verification

technique · direct

1.1F1F4F6F7F8

By [F4] the specialization $H_{A} = A \otimes_{R} H$ is the quotient of the free associative $A$ -algebra on $(T_{s})_{s \in S}$ by the images under $φ$ of the relations (Q) and (B) of [F1]. When $u_{s} = 1$ for all $s$ , those images are $T_{s}^{2} - 1$ and the braid relations. The assignment $T_{s} \mapsto s$ extends uniquely to a unital $A$ -algebra homomorphism from the free algebra to $A [W]$ ([F8], first assertion); it kills $T_{s}^{2} - 1$ because $s^{2} = 1$ in $W$ , and it kills each braid relation because the defining relator $(s t)^{m (s, t)} = 1$ holds in $W$ ([F7]). By the quotient universal property ([F8], second assertion) it therefore factors uniquely through $H_{A}$ , giving a unital $A$ -algebra homomorphism $Φ : H_{A} \to A [W]$ with $Φ (T_{s}) = s$ .

1.2F3F4F6F7

Conversely, in $H_{A}$ the images of the relations give $T_{s}^{2} = 1$ (the image of (Q) at $u_{s} = 1$ ) and the braid relations, and each $T_{s}$ is a unit ([F3]). For $s \neq t$ with $m : = m (s, t) < \infty$ put $a : = T_{s}$ , $b : = T_{t}$ and $x : = a b$ ; then $a^{2} = b^{2} = 1$ , so $x^{- 1} = b a$ and $x^{- k} = (b a)^{k}$ for every $k \geq 0$ . The braid relation says that the two alternating products of $m$ factors coincide: if $m = 2 k$ it reads $x^{k} = x^{- k}$ , whence $x^{2 k} = 1$ , while if $m = 2 k + 1$ it reads $x^{k} a = x^{- k} b$ , and multiplying on the right by $b = b^{- 1}$ gives $x^{k + 1} = x^{- k}$ , whence $x^{2 k + 1} = 1$ . Thus $(T_{s} T_{t})^{m (s, t)} = x^{m} = 1$ ; with $T_{s}^{2} = 1$ this shows that the assignment $s \mapsto T_{s}$ kills every defining relator of $W$ ([F7]), so it extends to a group homomorphism $φ : W \to H_{A}^{\times}$ with $φ (w) = T_{w}$ for every $w \in W$ (the product along a reduced expression). Since $([w])_{w \in W}$ is an $A$ -basis of $A [W]$ and $[w] [w^{'}] = [w w^{'}]$ ([F6]), the formula $Ψ (\sum_{w} a_{w} [w]) : = \sum_{w} a_{w} φ (w)$ defines a unital $A$ -algebra homomorphism $Ψ : A [W] \to H_{A}$ with $Ψ (w) = T_{w}$ : it is $A$ -linear by construction, multiplicativity reduces on basis elements to $φ (w w^{'}) = φ (w) φ (w^{'})$ , and $Ψ ([1]) = φ (1) = 1$ .

2.1F1F2F6step 1.1step 1.2

The two maps are mutually inverse: $Φ (Ψ (s)) = Φ (T_{s}) = s$ for every $s \in S$ and $Ψ (Φ (T_{s})) = Ψ (s) = T_{s}$ , and both composites fix $A$ because the maps are $A$ -linear; they fix every $T_{w}$ since each is a product of the $T_{s}$ , and every group basis element $w$ since each is a product of the generators $s$ . The $A$ -bases ([F2], [F6]) therefore make the composites the respective identities. Hence $Φ$ is an isomorphism of $A$ -algebras. On bases, $Φ (T_{w}) = s_{1} \dots s_{k} = w$ for a reduced expression $w = s_{1} \dots s_{k}$ by multiplicativity, so $Φ$ carries the $A$ -basis $(1 \otimes T_{w})$ of $H_{A}$ ([F2]) to the $A$ -basis $(w)$ of $A [W]$ ([F6]) and is an isomorphism of free $A$ -modules.

2.2F2F4F6step 1.1

For arbitrary units $u_{s}$ , the image of (Q) under $φ$ reads $T_{s}^{2} - (u_{s} - u_{s}^{- 1}) T_{s} - 1 = 0$ in $H_{A}$ ([F4]), and $H_{A}$ is free with basis $(1 \otimes T_{w})$ ([F2]). If an $A$ -algebra homomorphism $H_{A} \to A [W]$ with $T_{s} \mapsto s$ existed, applying it to that relation would give $0 = s^{2} - (u_{s} - u_{s}^{- 1}) s - 1 = - (u_{s} - u_{s}^{- 1}) s$ in $A [W]$ ; since the elements $w$ form an $A$ -basis of $A [W]$ ([F6]), this forces $u_{s} - u_{s}^{- 1} = 0$ , i.e. $u_{s}^{2} = 1$ , and over a field $u_{s} = \pm 1$ . Conversely, if every $u_{s}^{2} = 1$ , all specialized quadratics are $T_{s}^{2} - 1$ , so the constructions of 1.1–2.1 apply and give the same basis-preserving isomorphism $H_{A} ≅ A [W]$ . Consequently, when $R = Z [v^{\pm 1}]$ is the one-component coefficient ring, the specializations $v \mapsto 1$ and $v \mapsto - 1$ both satisfy $u - u^{- 1} = 0$ ; for $u = \pm 1$ the image of (Q) is $T_{s}^{2} - 1$ in either case, so the construction of 1.1-2.1 applies verbatim and both give the group ring, whereas any specialization to a unit with $u_{s}^{2} \neq 1$ admits no such map $T_{s} \mapsto s$ : the standard basis $(1 \otimes T_{w})$ still exists by [F2] but the quadratic relation is not the group relation.

3.1F2F3F5step 1.1step 1.2step 2.1step 2.2∎

Assembly: part 1 is steps 1.1, 1.2 and 2.1, the specialization mechanism of part 2 is steps 1.1 and 2.2 (the case $Q_{s} = 1$ of the normalization [F3]), and part 3 is step 2.2 together with the base-change freeness of [F2]. No choice is used: the coefficient homomorphism is unique by [F5], both maps are defined on explicit generators, and no selection occurs. The two application pages named in the statement are reading pointers only; no item of this pair depends on them.

Generic Coxeter Hecke Algebras and the Standard Basis — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Unequal parameters in the dihedral cases: the odd-edge obstruction and the even-edge freedom

Example

Facts & Assumptions

Verification

Rank-one Hecke multiplication in both normalizations

Example

Facts & Assumptions

Verification

The complete S3 multiplication table in both normalizations

Example

Facts & Assumptions

Verification

Specialization of the generic Hecke algebra to the group ring

Example

Facts & Assumptions

Verification

Sources