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Parabolic freeness and bases of induced Hecke modules

Statement

Let $(S, m)$ , $W$ , $ℓ$ , $R$ , and $H$ be as in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups and Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy, and let $J \subseteq S$ . Let $H_{J}$ be the candidate submodule and let induction and restriction be as in Parabolic Hecke subalgebras, induction, and restriction. By Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, $W_{J}$ is the Coxeter group on $J$ with the restricted matrix. The inherited parameters define its Hecke algebra over $R$ by the same presentation.

Use the coset convention of Left and right cosets $g H$ and $H g$ of a subgroup: the right cosets are $W_{J} d$ and the left cosets are $d W_{J}$ . Define

D_{R} (J) : = {d \in W : ℓ (s d) > ℓ (d) for every s \in J}, D_{L} (J) : = {d \in W : ℓ (d s) > ℓ (d) for every s \in J} .

These are respectively the unique minimal representatives of $W_{J} \ W$ and $W / W_{J}$ .

The candidate $H_{J} = ⨁_{u \in W_{J}} R T_{u}$ is the subalgebra of $H$ generated by $T_{j}$ for $j \in J$ . The homomorphism from the restricted-system Hecke algebra that sends each generator $T_{j}$ to the same element of $H$ is injective, with image $H_{J}$ .
There are direct-sum decompositions

H = ⨁_{d \in D_{R} (J)} H_{J} T_{d} as a left H_{J} -module, H = ⨁_{d \in D_{L} (J)} T_{d} H_{J} as a right H_{J} -module .

Thus the displayed $T_{d}$ are free module bases on the indicated sides. These assertions hold for arbitrary, possibly infinite, $W$ .

If $M$ is a left $H_{J}$ -module that is free over $R$ with basis $B$ , then $H \otimes_{H_{J}} M$ is free over $R$ with basis

{T_{d} \otimes b : d \in D_{L} (J), b \in B} .

For every left $H$ -module $N$ , the maps

{Hom}_{H} (H \otimes_{H_{J}} M, N) ⟶ {Hom}_{H_{J}} (M, {Res}_{J}^{W} N), F ⟼ [m \mapsto F (1 \otimes m)],

and

g ⟼ [h \otimes m \mapsto h g (m)]

are inverse bijections. In particular, induction is left adjoint to restriction. If $W$ is finite, the two bases of $H$ over $H_{J}$ have cardinality $[W : W_{J}]$ .

Facts & Assumptions

Given: A finite Coxeter matrix $(S, m)$ , its group $W$ and length $ℓ$ , the generic Hecke algebra $H$ over $R$ , a subset $J \subseteq S$ , a left $H_{J}$ -module $M$ , and a left $H$ -module $N$ .

[F1]

The candidate parabolic module is $H_{J} = ⨁_{u \in W_{J}} R T_{u}$ ; induction is the balanced tensor product $H \otimes_{H_{J}} M$ , with $(h k) \otimes m = h \otimes (k m)$ , and restriction uses the inclusion into $H$ . (Parabolic Hecke subalgebras, induction, and restriction)

[F2]

For any finite Coxeter system and any commutative specialization of its universal parameter ring, the standard elements $T_{w}$ form a basis of the Hecke algebra. (The standard basis of the generic Hecke algebra and base change)

[F3]

The reduced product $T_{w}$ is well defined; the left and right length-multiplication rules hold for every $s \in S, w \in W$ . (Reduced-word independence of T_w and the length-multiplication rules)

[F4]

Every right coset $W_{J} w$ has a unique minimal element $d$ and a unique factorization $w = u d$ with $u \in W_{J}$ ; then $ℓ (w) = ℓ (u) + ℓ (d)$ . Every left coset $w W_{J}$ has a unique minimal element $d$ and a unique factorization $w = d u$ with the same length additivity. (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification)

[F5]

Under the library's coset convention, $g H$ is a left coset and $H g$ is a right coset. (Left and right cosets $g H$ and $H g$ of a subgroup)

[F6]

The restricted system on $W_{J}$ has the same braid labels on $J$ , and a Hecke algebra is the presented algebra with the quadratic and finite braid relations. Parameters constant on odd-edge components of $S$ remain constant along odd-edge paths in $J$ , so they specialize the restricted-system parameter ring to $R$ . (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

Proof

Given: The data and hypotheses of the Statement.

1.1F1F3algebra

Let $K_{J} : = ⨁_{u \in W_{J}} R T_{u}$ . If $u \in W_{J}$ and $j \in J$ , the left and right length-multiplication rules in [F3] express $T_{j} T_{u}$ and $T_{u} T_{j}$ as linear combinations of $T_{j u}, T_{u}$ and $T_{u j}, T_{u}$ respectively; since $j u, u j \in W_{J}$ , both products lie in $K_{J}$ . Each $T_{u}$ is a product of the generators indexed by $J$ , so $K_{J}$ is contained in the $R$ -subalgebra they generate; conversely it is closed under multiplication by every such generator and contains $1$ , so every word in those generators lies in $K_{J}$ . Hence $K_{J}$ is exactly that generated subalgebra.

1.2F1F2F3F4F5

For each $w \in W$ , [F4] gives a unique factorization $w = u d$ with $u \in W_{J}$ and $d \in D_{R} (J)$ and $ℓ (w) = ℓ (u) + ℓ (d)$ . Thus $W_{J} d$ is the right coset represented by $d$ under [F5]. Choose reduced expressions for $u$ and $d$ ; their concatenation has length $ℓ (u) + ℓ (d) = ℓ (u d)$ and so is reduced, giving $T_{w} = T_{u} T_{d}$ by [F3]. Thus the elements of $H_{J} T_{d}$ span $H$ . If a finite sum $\sum_{d} h_{d} T_{d} = 0$ with $h_{d} \in H_{J}$ is given, expand each $h_{d}$ as a finite sum $\sum_{u \in W_{J}} r_{u, d} T_{u}$ using [F1]. The resulting sum is $\sum_{u, d} r_{u, d} T_{u d} = 0$ . Unique factorization makes the elements $u d$ distinct for distinct pairs $(u, d)$ ; linear independence of the standard basis [F2] forces every $r_{u, d} = 0$ . This proves the left-module direct sum and its basis.

1.3F1F2F3F4F5

For each $w \in W$ , the left-coset part of [F4] gives a unique factorization $w = d u$ with $d \in D_{L} (J)$ , $u \in W_{J}$ , and $ℓ (w) = ℓ (d) + ℓ (u)$ . By [F5], $d W_{J}$ is the left coset represented by $d$ . Reduced expressions for $d$ followed by $u$ concatenate to a reduced expression of $d u$ , so $T_{w} = T_{d} T_{u}$ by [F3]. Hence the elements of $T_{d} H_{J}$ span $H$ . If $\sum_{d} T_{d} h_{d} = 0$ , expand $h_{d} = \sum_{u \in W_{J}} r_{d, u} T_{u}$ ; unique left-coset factorization makes all $d u$ distinct, and [F2] then forces every coefficient to vanish. This proves the right-module direct sum and its basis.

2.1F1F2F4F6step 1.1

By [F4], $(W_{J}, J)$ is a Coxeter system. Any odd-edge path in $J$ is also an odd-edge path in $S$ , so the inherited parameters agree along every odd component of $J$ and give a ring map from the restricted system's universal parameter ring to $R$ . Thus [F6] supplies a homomorphism from the restricted Hecke presentation to $H$ sending each $T_{j}$ to $T_{j}$ . Apply the standard-basis theorem [F2] to the restricted system and this parameter map: its standard elements $T_{u}$ form an $R$ -basis. Their images are the correspondingly named elements of the ambient $H$ -basis, so the map is injective; its image is $K_{J}$ by step 1.1 and because every $u \in W_{J}$ has a word in $J$ . Thus the candidate in [F1] is an embedded subalgebra, proving part 1.

2.2F1step 1.3constructalgebra

By step 1.3, every $h \in H$ has a unique finite expansion $h = \sum_{d \in D_{L} (J)} T_{d} k_{d}$ with $k_{d} \in H_{J}$ . Define $Φ : ⨁_{d \in D_{L} (J)} M \to H \otimes_{H_{J}} M$ by $(m_{d})_{d} \mapsto \sum_{d} T_{d} \otimes m_{d}$ , and define $Ψ (h \otimes m) : = (k_{d} m)_{d}$ . The latter respects the balancing relation: the coefficients of $h k$ are $k_{d} k$ , so $Ψ (h k \otimes m) = (k_{d} (k m))_{d} = Ψ (h \otimes k m)$ . Uniqueness of the expansion shows $Ψ Φ$ and $Φ Ψ$ are identities. Thus $H \otimes_{H_{J}} M ≅ ⨁_{d \in D_{L} (J)} M$ ; if $B$ is an $R$ -basis of $M$ , the displayed tensors form an $R$ -basis.

3.1F1step 2.2algebra

For an $H_{J}$ -linear map $g : M \to {Res}_{J}^{W} N$ , the formula $\tilde{g} (h \otimes m) = h g (m)$ is balanced because $h k g (m) = h g (k m)$ , and it is $H$ -linear. Conversely, if $F : H \otimes_{H_{J}} M \to N$ is $H$ -linear, $m \mapsto F (1 \otimes m)$ is $H_{J}$ -linear since $1 \otimes k m = k \otimes m$ . The two constructions are inverse: evaluation at $1 \otimes m$ recovers $g (m)$ , and $H$ -linearity gives $F (h \otimes m) = h F (1 \otimes m)$ . This proves the adjunction.

4.1F2F4step 1.2step 1.3step 3.1∎

The coset decompositions, their independence arguments, and the adjunction used no finiteness of $W$ and no choice of coset representatives: [F4] supplies each unique minimal representative. If $W$ is finite, each coset has $∣ W_{J} ∣$ elements, the cosets partition $W$ , and the number of representatives is $∣ W ∣ / ∣ W_{J} ∣ = [W : W_{J}]$ . All coefficient arguments use standard bases and remain valid after any commutative base change of $R$ because the standard basis in [F2] does.

Depends on

Used by

Inducing the rank-one Hecke module into type A2 Example

Cited to discharge well-definedness by Parabolic Hecke subalgebras, induction, and restriction.

Dependency tree · two levels

47 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

Christoph Schönnenbeck, Representations of Iwahori-Hecke Algebras Induced from Parabolic Subalgebras (RWTH Aachen Master's thesis, 2016) (standard reference, not scraped)
George Lusztig, Hecke Algebras with Unequal Parameters (revised book text, arXiv:math/0208154v2) (standard reference, not scraped)