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Cyclotomic Jucys–Murphy generators and their transport relations

Statement

In $H_{r, n}$ define

$L_{1} = X_{1}, L_{i} = Q^{1 - i} T_{i - 1} \dots T_{1} X_{1} T_{1} \dots T_{i - 1} (2 \leq i \leq n) .$

Then $L_{i} = X_{i}$ for every $i$ , so the $L_{i}$ commute and are units. For $1 \leq i < n$ they satisfy

$T_{i} L_{i} T_{i} = Q L_{i + 1}, T_{i} L_{i} = L_{i + 1} T_{i} - (Q - 1) L_{i + 1}, T_{i} L_{i + 1} = L_{i} T_{i} + (Q - 1) L_{i + 1},$

and $T_{i} L_{j} = L_{j} T_{i}$ when $j \notin {i, i + 1}$ . More generally, for every polynomial $f \in R [X_{1}, \dots, X_{n}]$ ,

$T_{i} f (L_{1}, \dots, L_{n}) = f (s_{i} L_{1}, \dots, s_{i} L_{n}) T_{i} - (Q - 1) L_{i + 1} \frac{f (L_{1}, \dots, L_{n}) - f (s_{i} L_{1}, \dots, s_{i} L_{n})}{L_{i} - L_{i + 1}},$

where $s_{i}$ interchanges the $i$ th and $(i + 1)$ st variables and the displayed divided difference is a polynomial. In particular the cyclotomic relation transports as

$T_{i} p (L_{i}) = p (L_{i + 1}) T_{i} - (Q - 1) L_{i + 1} \frac{p (L_{i}) - p (L_{i + 1})}{L_{i} - L_{i + 1}} .$

When $p (L_{i}) = 0$ , this gives $p (L_{i + 1}) T_{i} = (Q - 1) L_{i + 1} (p (L_{i}) - p (L_{i + 1})) / (L_{i} - L_{i + 1})$ ; the cyclotomic quotient guarantees this only for $i = 1$ . It does not impose $p (L_{i}) = 0$ for every $i$ .

Facts & Assumptions

Given: The algebra and affine relations of The multiplicative cyclotomic Hecke algebra, its level polynomial $p$ , and the finite type-A Hecke standard basis of The standard basis of the generic Hecke algebra and base change.

[F1]

$X_{1}, \dots, X_{n}$ commute, $T_{i} X_{i} T_{i} = Q X_{i + 1}$ , and $T_{i} X_{j} = X_{j} T_{i}$ for $j \notin {i, i + 1}$ . (The multiplicative cyclotomic Hecke algebra)

[F2]

$T_{i}^{2} = (Q - 1) T_{i} + Q$ , $Q$ is a unit, and therefore $T_{i}^{- 1} = Q^{- 1} (T_{i} - (Q - 1))$ . (The multiplicative cyclotomic Hecke algebra)

[F3]

For a generic type-A Hecke algebra, the finite Hecke words $T_{w}$ indexed by $w \in S_{n}$ are linearly independent over its universal coefficient ring. (The standard basis of the generic Hecke algebra and base change)

Proof

Given: The affine generators and relations, the cyclotomic quotient, and the elements $L_{i}$ in the statement.

1.1givenF1

The recursion in the definition gives $L_{1} = X_{1}$ . If $L_{i} = X_{i}$ , then $L_{i + 1} = Q^{- 1} T_{i} L_{i} T_{i} = Q^{- 1} T_{i} X_{i} T_{i} = X_{i + 1}$ by [F1]. Induction proves $L_{i} = X_{i}$ for all $i$ , and the commutation and unit claims follow from the affine presentation.

1.2F1F2algebra

The equality $T_{i} L_{i} T_{i} = Q L_{i + 1}$ is [F1]. Right-multiplying it by $T_{i}^{- 1}$ and using [F2] gives $T_{i} L_{i} = Q L_{i + 1} T_{i}^{- 1} = L_{i + 1} (T_{i} - (Q - 1))$ . Left-multiplying by $T_{i}^{- 1}$ gives $L_{i} T_{i} = (T_{i} - (Q - 1)) L_{i + 1}$ , which is equivalent to the displayed relation for $T_{i} L_{i + 1}$ . The distant relation is [F1].

1.3F2F3algebra

The failure of a common cyclotomic polynomial already occurs at level $r = 1$ , rank $n = 2$ . Here $X_{1} = u_{1}$ and $L_{2} = Q^{- 1} T_{1} X_{1} T_{1} = u_{1} Q^{- 1} T_{1}^{2} = u_{1} + (Q - 1) u_{1} Q^{- 1} T_{1}$ . There is an algebra map from this presented quotient to the finite type-A Hecke algebra sending $T_{1}$ to its generator, $X_{1}$ to $u_{1}$ , and $X_{2}$ to $Q^{- 1} u_{1} T_{1}^{2}$ : the affine relation holds by construction, the $X_{j}$ commute because $X_{1}$ maps to a scalar, and all inverse generators map to units. Under this map, $p (L_{2}) = L_{2} - u_{1}$ becomes $(Q - 1) u_{1} Q^{- 1} T_{1}$ , which is nonzero by the finite Hecke basis [F3] and the nonzero coefficient in the rational function field. Thus $p (L_{2})$ was already nonzero in the cyclotomic quotient. The initial cyclotomic relation gives a transport identity, not the same polynomial relation on every $L_{i}$ .

2.1step 1.1step 1.2algebra

Put $δ_{i} (f) = (f - s_{i} f) / (X_{i} - X_{i + 1})$ . For a monomial $X_{i}^{a} X_{i + 1}^{b}$ with $a, b \geq 0$ , the numerator is divisible by $X_{i} - X_{i + 1}$ : if $a \geq b$ it equals $X_{i}^{b} X_{i + 1}^{b} (X_{i}^{a - b} - X_{i + 1}^{a - b})$ , and if $b > a$ interchange the two variables. Thus $δ_{i} (f)$ is a polynomial for every polynomial $f$ . The generator cases of step 1.2 give $T_{i} f = s_{i} (f) T_{i} - (Q - 1) X_{i + 1} δ_{i} (f)$ for $f = X_{i}, X_{i + 1}$ ; it is immediate for every other variable and for coefficients in $R$ . The identity $δ_{i} (f g) = δ_{i} (f) g + s_{i} (f) δ_{i} (g)$ then extends the same formula from $f, g$ to $f g$ , and linearity proves it for every polynomial. Replacing $X_{j}$ by $L_{j}$ using step 1.1 yields the stated transport rule.

3.1givenstep 2.1algebra

Take $f (X_{i}) = p (X_{i})$ in the general transport formula. It gives the first displayed identity for every $i$ . For $i = 1$ , $p (L_{1}) = p (X_{1}) = 0$ by definition, so the identity specializes to the stated conditional relation. The quotient by $L_{i} - L_{i + 1}$ is a divided difference polynomial, not an inverse of that element. No cancellation of $T_{i}$ or of a content difference is involved.

4.1step 1.1step 1.2step 2.1step 3.1step 1.3∎

Steps 1.1–3.1 establish the commuting generators and all claimed transport identities, including the explicit non-common-polynomial example.

Depends on

Used by

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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)