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Level one recovers the finite type-A Hecke algebra

Example

Let $R_{1} = Z [Q^{\pm 1}, u^{\pm 1}]$ . Let $H_{Q} (n) = R_{1} \otimes_{Z [v^{\pm 1}]} H_{v} (n)$ be the finite type-A Hecke algebra obtained from The generic type-A Hecke algebra by $v \mapsto Q$ . At cyclotomic level $r = 1$ , the Ariki–Koike algebra $H_{1, n}$ is isomorphic to $H_{Q} (n)$ by the map fixing each $T_{i}$ , and its basis is ${T_{w} : w \in S_{n}}$ .

Facts & Assumptions

Given: $n \geq 1$ , the multiplicative level-one presentation over $R_{1}$ , and the finite type-A Hecke algebra $H_{Q} (n)$ .

[F1]

The multiplicative cyclotomic Hecke algebra: the cyclotomic quotient imposes $p (X_{1}) = X_{1} - u = 0$ .

[F2]

The multiplicative cyclotomic Hecke algebra: the affine recursion is $X_{i + 1} = Q^{- 1} T_{i} X_{i} T_{i}$ .

[F3]

The multiplicative cyclotomic Hecke algebra: $T_{i}^{- 1} = Q^{- 1} (T_{i} - (Q - 1))$ .

[F4]

The generic type-A Hecke algebra: specializing $H_{v} (n)$ by $v \mapsto Q$ gives the finite type-A presentation over $R_{1}$ with quadratic relation $(T_{i} - Q) (T_{i} + 1) = 0$ .

[F5]

The standard basis of the generic type-A Hecke algebra: the reduced-word elements $T_{w}$ form a basis of $H_{v} (n)$ over $Z [v^{\pm 1}]$ .

[F6]

The Ariki–Koike PBW basis and generic splitness: the elements $X_{1}^{a_{1}} \dots X_{n}^{a_{n}} T_{w}$ , $0 \leq a_{j} < r$ , form an $R$ -basis of $H_{r, n}$ .

Verification

Given: The algebras and presentations in the example, with the homomorphism on generators $T_{i} \mapsto T_{i}$ .

1.1F1F2F3F4algebra

At $r = 1$ , [F1] gives $p (X_{1}) = X_{1} - u = 0$ , so $X_{1} = u$ and $X_{1}^{- 1} = u^{- 1}$ . By the affine recursion [F2], each $X_{i + 1}$ lies in the subalgebra generated by $T_{1}, \dots, T_{n - 1}$ if $X_{i}$ does. Also $X_{i + 1}^{- 1} = Q T_{i}^{- 1} X_{i}^{- 1} T_{i}^{- 1}$ , which lies in the same subalgebra by [F3]. The generator relations agree with the specialization [F4], so the assignment $T_{i} \mapsto T_{i}$ defines a surjective homomorphism $H_{Q} (n) \to H_{1, n}$ .

2.1F5F6step 1.1algebra∎

By [F5], $H_{v} (n)$ is the direct sum of the coefficient modules $Z [v^{\pm 1}] T_{w}$ . Tensoring this explicit direct sum with $R_{1}$ shows that $H_{Q} (n)$ is free over $R_{1}$ with basis the images of $T_{w}$ . By [F6] with $r = 1$ , the same family is an $R_{1}$ -basis of $H_{1, n}$ . The surjection of step 1.1 sends the source basis to this target basis, so its kernel is zero and it is an isomorphism.

Depends on

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