ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-10-08

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

88 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

George Lusztig, Lectures on Hecke Algebras with Unequal Parameters (MIT Fall 1999 lecture notes, arXiv:math/0108172v1) (standard reference, not scraped)
George Lusztig, Lectures on Hecke Algebras with Unequal Parameters (MIT Fall 1999 lecture notes, arXiv:math/0108172v1) (standard reference, not scraped)