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Quadratic Hecke normalizations: S=qT with Q=q^2, the opposite-sign form, and the Soergel-calculus and Kazhdan-Lusztig conversions

Example

Let $S = {s}$ and let $H$ be the generic Hecke algebra over $R = Z [v^{\pm 1}]$ with generator $T = T_{s}$ and the single relation $(T - v) (T + v^{- 1}) = 0$ , equivalently $T^{2} = (v - v^{- 1}) T + 1$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, The standard basis of the generic Hecke algebra and base change).

The four generators and relations. Put $q : = v$ , $Q : = q^{2} = v^{2}$ , and

$S : = q T, H^{-} : = - T, T^{E W} : = - q^{- 1} T .$

Then, in $H$ ,

$S^{2} = (Q - 1) S + Q, (H^{-})^{2} = (v^{- 1} - v) H^{-} + 1, (T^{E W} + 1) (T^{E W} - q^{- 2}) = 0,$

i.e. $(T^{E W})^{2} = (q^{- 2} - 1) T^{E W} + q^{- 2}$ . Moreover ${1, T}$ , ${1, S}$ , ${1, H^{-}}$ and ${1, T^{E W}}$ are each $R$ -bases of $H$ , and the four normalizations are interconverted by

$T = q^{- 1} S = - H^{-} = - q T^{E W}, S = q T, T^{E W} = - q^{- 1} T,$

which are mutually inverse changes of generators. The first displayed relation is the multiplicative convention " $S = q T$ , $Q = q^{2}$ " (The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization (4)); the second is the opposite-sign normalization and the third is the quadratic relation of Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary (2).

The rank-two consistency check. For $S = {s, t}$ with $m = m (s, t) < \infty$ and the single parameter $v_{s} = v_{t} = v$ (a legitimate specialization; equality of the two parameters is forced when $m$ is odd by Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy), the same substitutions preserve the braid relation $T_{s} T_{t} T_{s} \dots = T_{t} T_{s} T_{t} \dots$ (both sides have $m$ factors, so they acquire the same factor $q^{m}$ , $(- 1)^{m}$ or $(- q^{- 1})^{m}$ ), and on the standard bases the substitutions are diagonal:

$S_{w} = q^{ℓ (w)} T_{w}, H_{w}^{-} = (- 1)^{ℓ (w)} T_{w}, T_{w}^{E W} = (- q^{- 1})^{ℓ (w)} T_{w} .$

In particular both alternating products have the same sign and the same multiplicative factor, including for odd-length braid words.

Kazhdan–Lusztig conversion (rank one). Let $T^{K L}$ be the generator of the presentation with the single relation $(T^{K L})^{2} = (q^{- 2} - 1) T^{K L} + q^{- 2}$ , so that $T = - q T^{K L}$ . Then the substitution sends $T^{K L}$ to $T^{E W} = - q^{- 1} T$ , and with $H^{K L} : = q T^{K L}$ the corresponding element $q T^{E W} = - T = H^{-}$ satisfies $(H^{K L})^{2} = (q^{- 1} - q) H^{K L} + 1$ ; this is the rank-one instance of the conversion $H_{x} = v^{ℓ (x)} T_{x}^{K L}$ and $h_{y, x} = v^{ℓ (x) - ℓ (y)} P_{y, x} (v^{- 2})$ recorded in the Hodge-theory source. The conversion is stated here so that coefficients are compared only after it; no canonical basis, Kazhdan–Lusztig polynomial or positivity statement is constructed in this example.

Facts & Assumptions

Given: The rank-one Hecke datum $S = {s}$ , the ring $R = Z [v^{\pm 1}]$ , the algebra $H$ with generator $T = T_{s}$ and relation $(T - v) (T + v^{- 1}) = 0$ , the parameters $q = v$ , $Q = q^{2}$ , and the substituted generators $S = q T$ , $H^{-} = - T$ , $T^{E W} = - q^{- 1} T$ ; in the rank-two part, the system $S = {s, t}$ with $m = m (s, t) < \infty$ and $v_{s} = v_{t} = v$ .

[F1]

In the generic Hecke algebra the normalized relation is $T_{s}^{2} = (v_{s} - v_{s}^{- 1}) T_{s} + 1$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F2]

The multiplicative generators $S_{s} : = v_{s} T_{s}$ satisfy $(S_{s} - Q_{s}) (S_{s} + 1) = 0$ with $Q_{s} = v_{s}^{2}$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F3]

The opposite-sign generators $H_{s} : = - T_{s}$ satisfy $H_{s}^{2} = (v_{s}^{- 1} - v_{s}) H_{s} + 1$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F4]

The Soergel-calculus generators $T_{s}^{E W} : = - v_{s}^{- 1} T_{s}$ satisfy $(T_{s}^{E W} + 1) (T_{s}^{E W} - v_{s}^{- 2}) = 0$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F5]

The four normalizations are interconverted by $T_{s} = v_{s}^{- 1} S_{s} = - H_{s} = - v_{s} T_{s}^{E W}$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F6]

In the single-parameter normalization, $H_{x} = v^{ℓ (x)} T_{x}^{K L}$ ; for a supplied finite expansion $C_{x} = v^{ℓ (x)} \sum_{y} P_{y, x} (v^{- 2}) T_{y}^{K L}$ , its $H_{y}$ -coefficients are $h_{y, x} = v^{ℓ (x) - ℓ (y)} P_{y, x} (v^{- 2})$ . (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary)

[F7]

The induced scalar action of an $R$ -algebra makes it an $R$ -module and its multiplication $R$ -bilinear, so multiplication by a unit of $R$ is an $R$ -linear bijection. (Algebras over a commutative ring, central structure maps, and algebra homomorphisms)

Verification

1.1F1F2F3F4F5F7

The four relations of part 1 are the relations [F1]–[F4] specialized to $v_{s} = v$ : the normalized relation is the defining relation $T^{2} = (v - v^{- 1}) T + 1$ , and the other three read $S^{2} = (Q - 1) S + Q$ , $(H^{-})^{2} = (v^{- 1} - v) H^{-} + 1$ and $(T^{E W} + 1) (T^{E W} - q^{- 2}) = 0$ with $Q = q^{2}$ . The interconversions are [F5] at $v_{s} = v$ . In rank one the standard basis of The standard basis of the generic Hecke algebra and base change is ${1, T}$ , and ${1, S}$ , ${1, H^{-}}$ , ${1, T^{E W}}$ are obtained by fixing $1$ and scaling the other basis vector $T$ by the units $q$ , $- 1$ , $- q^{- 1}$ , respectively. For each such unit $a$ , the $R$ -linear map $1 \mapsto 1$ , $T \mapsto a T$ has inverse $1 \mapsto 1$ , $T \mapsto a^{- 1} T$ , hence carries a basis to a basis by [F7].

2.1F2F3F4F5step 1.1

For the rank-two check, each of the three substitutions multiplies every generator by one constant: $S_{s} \mapsto v$ , $H_{s} \mapsto - 1$ , $T_{s}^{E W} \mapsto - v^{- 1}$ . Hence an alternating product of $m$ factors acquires the same constant factor, $v^{m}$ , $(- 1)^{m}$ or $(- v^{- 1})^{m}$ , on both sides of the braid relation, so the braid relation is preserved by each substitution; this is exactly the content of the corresponding change of generators in [F2]–[F5]. For the diagonal formulas, the substituted standard-basis element is the product of the images of the generators along a reduced expression, $S_{w} = S_{s_{1}} \dots S_{s_{k}} = v^{k} T_{s_{1}} \dots T_{s_{k}} = v^{ℓ (w)} T_{w}$ , and similarly $H_{w}^{-} = (- 1)^{ℓ (w)} T_{w}$ and $T_{w}^{E W} = (- v^{- 1})^{ℓ (w)} T_{w}$ , using the product rule and independence of the standard basis in Reduced-word independence of T_w and the length-multiplication rules (1).

2.2F1F5F6step 1.1

For the rank-one Kazhdan–Lusztig conversion, set $T^{K L} : = - q^{- 1} T$ , which is part 1's $T^{E W}$ ; since $T = - q T^{K L}$ and the relation for $T$ holds, the substitution is consistent, and $T^{K L}$ satisfies $(T^{K L})^{2} = q^{- 2} T^{2} = q^{- 2} ((v - v^{- 1}) T + 1) = (q^{- 2} - 1) T^{K L} + q^{- 2}$ by the computation of step 1.1. With $H^{K L} : = q T^{K L} = - T = H^{-}$ one gets $(H^{K L})^{2} = (H^{-})^{2} = (v^{- 1} - v) H^{-} + 1 = (q^{- 1} - q) H^{K L} + 1$ , the rank-one case of the recorded conversion [F6] with $ℓ (s) = 1$ .

3.1given∎

Scope and choice: this example compares four quadratic normalizations and verifies the exponent conversions of the standard bases in one rank-two family; it constructs no canonical basis, no Kazhdan–Lusztig polynomial and no positivity or bar-invariance statement, all of which remain in their designated proof homes, and the general coefficient conversion [F6] is supplied by the seam lemma. Every computation is a substitution in $R = Z [v^{\pm 1}]$ , and no choice is used.

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Sources

George Lusztig, Hecke Algebras with Unequal Parameters (revised book text, arXiv:math/0208154v2) (standard reference, not scraped)
Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2) (standard reference, not scraped)
Ben Elias and Geordie Williamson, Soergel calculus (arXiv:1309.0865v1) (standard reference, not scraped)