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The multiplicative cyclotomic Hecke algebra

Definition

Fix integers $r, n \geq 1$ . Put $R = Z [Q^{\pm 1}, u_{1}^{\pm 1}, \dots, u_{r}^{\pm 1}]$ and $p (X) = \prod_{a = 1}^{r} (X - u_{a})$ . Let $A_{n}^{a f f}$ be the unital associative $R$ -algebra on generators $T_{1}, \dots, T_{n - 1}$ and $X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}$ , with central $R$ -action, subject to

$(T_{i} - Q) (T_{i} + 1) = 0, T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}, T_{i} T_{j} = T_{j} T_{i} (∣ i - j ∣ > 1),$

$X_{j} X_{k} = X_{k} X_{j}, T_{i} X_{j} = X_{j} T_{i} (j \notin {i, i + 1}), T_{i} X_{i} T_{i} = Q X_{i + 1} .$

Here the braid relation is imposed for $1 \leq i < n - 1$ and the distant commutation relation for $1 \leq i, j < n$ . Define the multiplicative cyclotomic Hecke algebra of level $r$ and rank $n$ by

$H_{r, n} = A_{n}^{a f f} / (p (X_{1})),$

where $(p (X_{1}))$ denotes the two-sided ideal it generates. The unit $Q$ and the quadratic relation make every $T_{i}$ invertible, with $T_{i}^{- 1} = Q^{- 1} (T_{i} - (Q - 1))$ . The cyclotomic relation also gives the polynomial inverse

$X_{1}^{- 1} = (- 1)^{r + 1} (u_{1} \dots u_{r})^{- 1} (X_{1}^{r - 1} - e_{1} X_{1}^{r - 2} + \dots + (- 1)^{r - 1} e_{r - 1}),$

where $e_{k}$ is the $k$ th elementary symmetric polynomial in $u_{1}, \dots, u_{r}$ . The presentation already includes inverse generators; this formula records that the quotient does not require a separate inverse parameter. For $i \geq 1$ , write $X_{i + 1} = Q^{- 1} T_{i} X_{i} T_{i}$ when using the recursion determined by the affine relation.

This is the multiplicative presentation. It is not the additive degenerate cyclotomic presentation obtained after setting a parameter equal to $1$ and changing generators.

Facts & Assumptions

Given: The integers, a level $r \geq 1$ , a rank $n \geq 1$ , the Laurent polynomial ring $R$ , and the displayed generators and relations.

[F1]

The integers form a commutative ring, and the Laurent polynomial construction over a commutative ring has the stated universal property and has its Laurent monomials as a basis. (The integers form a commutative ring, Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields)

[F2]

A free associative algebra on a set of generators exists over a commutative coefficient ring; quotienting by a two-sided ideal gives the universal algebra satisfying the specified relations. (The free associative R-algebra on a set and descent of relations)

[F3]

A unital associative $R$ -algebra is a unital ring with central structural image of $R$ . (Algebras over a commutative ring, central structure maps, and algebra homomorphisms)

Verification

technique · direct

1.1F1F2F3

The displayed Laurent polynomial ring is a commutative ring, and each of $Q, u_{1}, \dots, u_{r}$ is a unit by [F1]. The free associative algebra over $R$ on the displayed generator set is a unital associative $R$ -algebra by [F2] and [F3]; quotienting by the two-sided ideal generated by the displayed relations therefore constructs $A_{n}^{a f f}$ and then $H_{r, n}$ as unital associative $R$ -algebras.

1.2F2F3

An assignment of the symbols $T_{i}, X_{j}, X_{j}^{- 1}$ to elements of a unital associative $R$ -algebra extends uniquely to the free algebra. It factors through the relation ideal exactly when it satisfies the displayed relations and the two inverse equations, so the quotient has the asserted universal property. This construction also works for $n = 1$ , when the $T_{i}$ relations are empty.

1.3givenF1algebra

Expanding $(T_{i} - Q) (T_{i} + 1) = 0$ gives $T_{i}^{2} = (Q - 1) T_{i} + Q$ , hence $T_{i} (T_{i} - (Q - 1)) = (T_{i} - (Q - 1)) T_{i} = Q$ and the stated inverse formula. Expanding $p (X_{1}) = 0$ gives $X_{1}^{r} - e_{1} X_{1}^{r - 1} + \dots + (- 1)^{r} e_{r} = 0$ , with $e_{r} = u_{1} \dots u_{r}$ a unit; multiplying by $X_{1}^{- 1}$ and solving gives the displayed inverse polynomial.

2.1step 1.3algebra

If $p (X_{1}) = 0$ and $X_{1}$ is invertible, its constant term is a unit because $(- 1)^{r} e_{r}$ is a unit. The recursive affine relation carries invertibility along the tower: $X_{i + 1} = Q^{- 1} T_{i} X_{i} T_{i}$ is a product of units whenever $X_{i}$ is a unit. Thus the unit-parameter convention is compatible with every generator in the quotient.

3.1step 1.1step 1.2step 1.3step 2.1∎

The construction and inverse calculations establish the definition. The named theorem The Ariki–Koike PBW basis and generic splitness ↗ will prove that this quotient is free of rank $r^{n} n!$ over $R$ , so in particular it is nonzero; that property is not assumed in defining it. No choice is used.

Depends on

Used by

Dependency tree · two levels

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Sources

Georges Neaime, Geodesic normal forms and Hecke algebras for the complex reflection groups G(de,e,n), arXiv:1810.12053v1 (standard reference, not scraped)