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Bounded ordered monomials span the cyclotomic Hecke quotients

Statement

Over any commutative base ring $A$ with invertible nondegenerate parameter, use the defining relations of Affine type-A presentations and charged cyclotomic quotients, writing $q_{H}$ for the nondegenerate Hecke parameter. For a monic cyclotomic polynomial $f$ of degree $ℓ > 0$ , with unit constant term in the nondegenerate branch, the quotient by $f (X_{1}) = 0$ is spanned by $X_{1}^{a_{1}} \dots X_{d}^{a_{d}} T_{w}, 0 \leq a_{r} < ℓ, w \in S_{d},$ and the degenerate quotient by $f (x_{1}) = 0$ is spanned by $x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w, 0 \leq a_{r} < ℓ, w \in S_{d} .$ In particular, over a field $F$ each quotient has dimension at most $ℓ^{d} d!$ . This applies to $f (t) = \prod_{i \in I} (t - q_{H}^{i})^{λ_{i}}$ and $f (t) = \prod_{i \in I} (t - i)^{λ_{i}}$ in the two finite-quantum-characteristic branches, with $ℓ = \sum_{i} λ_{i}$ . No independence and no separate relation $f (X_{r}) = 0$ or $f (x_{r}) = 0$ for $r > 1$ is asserted. If $ℓ = 0$ and $d > 0$ , the imposed relation $1 = 0$ gives the zero algebra with empty spanning family; for $d = 0$ the algebra is $A$ with the singleton empty monomial.

Facts & Assumptions

Given: The two presentations of Affine type-A presentations and charged cyclotomic quotients over $A$ , a monic cyclotomic polynomial $f$ of degree $ℓ > 0$ , and the quotients of the Statement.

[F1]

In the nondegenerate presentation the mixed relation is $T_{r} X_{r} T_{r} = q_{H} X_{r + 1}$ with $q_{H}$ invertible, $T_{r}^{2} = (q_{H} - 1) T_{r} + q_{H}$ , the distant and braid relations hold, and $T_{r}$ is invertible with $T_{r}^{- 1} = q_{H}^{- 1} (T_{r} - q_{H} + 1)$ ; in the degenerate presentation $s_{r} x_{r} = x_{r + 1} s_{r} - 1$ , $s_{r} x_{r + 1} = x_{r} s_{r} + 1$ , $s_{r} x_{t} = x_{t} s_{r}$ for $t \notin {r, r + 1}$ , and the Coxeter relations hold (Affine type-A presentations and charged cyclotomic quotients, The generic type-A Hecke algebra).

[F2]

The $T_{r}$ generate a quotient of the generic type-A Hecke algebra with standard basis ${T_{w} : w \in S_{d}}$ , and the $s_{r}$ generate the quotient of $A [S_{d}]$ with basis $S_{d}$ ; finite straightening in the Coxeter group reduces each word in the $T_{r}$ (respectively $s_{r}$ ) to a linear combination of the standard words (The standard basis of the generic type-A Hecke algebra, The symmetric group has the Coxeter presentation).

Proof

technique · direct

Given: The two presentations over $A$ , the cyclotomic quotient by

$f (X_{1}) = 0$ (nondegenerate) or $f (x_{1}) = 0$ (degenerate), and the data of

[F1]–[F2].

1.1F1F2algebra

Elementary straightening of polynomials past a simple generator. In the nondegenerate branch the mixed relation gives, for a polynomial $p$ in the commuting variables, $T_{r} p = (s_{r} p) T_{r} - (q_{H} - 1) X_{r + 1} \frac{p - s_{r} p}{X_{r} - X_{r + 1}},$ and in the degenerate branch $s_{r} p = (s_{r} p) s_{r} - \frac{p - s_{r} p}{x_{r} - x_{r + 1}} .$ Both divided differences are polynomials: for $p = X_{r}^{m}$ the first identity is $T_{r} X_{r}^{m} = X_{r + 1}^{m} T_{r} - (q_{H} - 1) (X_{r}^{m - 1} + X_{r}^{m - 2} X_{r + 1} + \dots + X_{r + 1}^{m - 1}) X_{r + 1}$ , which follows from [F1] by induction on $m$ , and the second is $s_{r} x_{r}^{m} = x_{r + 1}^{m} s_{r} - \sum_{k = 0}^{m - 1} x_{r}^{k} x_{r + 1}^{m - 1 - k}$ , again by induction; the general case follows from the divided-difference product rule for the two variable replacements $X_{r} \leftrightarrow X_{r + 1}$ and $x_{r} \leftrightarrow x_{r + 1}$ . Consequently any word in the generators and polynomial variables rewrites as an $A$ -linear combination of ordered monomials $X^{a} T_{w}$ in the nondegenerate branch (the $T$ -word reduced by [F2], polynomial factors collected by commutation) and of $x^{a} w$ in the degenerate branch. The nondegenerate straightening preserves total polynomial degree; the degenerate one lowers degree by one in the correction term.

2.1F1step 1.1algebra

The nondegenerate quotient is generated by positive powers. In the nondegenerate branch the relation $f (X_{1}) = 0$ has nonzero constant term $a_{0}$ , so $X_{1}^{- 1} = - a_{0}^{- 1} (X_{1}^{ℓ - 1} + a_{ℓ - 1} X_{1}^{ℓ - 2} + \dots + a_{1})$ is a polynomial in $X_{1}$ ; and from $T_{r} X_{r} T_{r} = q_{H} X_{r + 1}$ one gets $X_{r + 1}^{- 1} = q_{H} T_{r}^{- 1} X_{r}^{- 1} T_{r}^{- 1}$ by inverting the equation. Starting from the displayed polynomial expression for $X_{1}^{- 1}$ and iterating this forward recursion expresses every $X_{r}^{- 1}$ through positive powers of $X_{1}$ and the $T$ 's, since $T_{r}^{- 1} = q_{H}^{- 1} (T_{r} - q_{H} + 1)$ is a polynomial in $T_{r}$ . Therefore the quotient is already generated, as an $A$ -module, by words in $X_{1}, \dots, X_{d}$ and the $T_{r}$ , and step 1.1 applies to them.

2.2F1F2step 1.1algebra

The transported relations in the nondegenerate branch. Put $R_{1} : = f (X_{1})$ and $R_{r + 1} : = T_{r} R_{r} T_{r}$ ; each $R_{r}$ is zero in the quotient. We claim by induction on $r$ that ordered straightening writes $R_{r} = q_{H}^{r - 1} X_{r}^{ℓ} + \sum_{a, w} c_{a, w} X_{1}^{a_{1}} \dots X_{r}^{a_{r}} T_{w},$ where $w \in S_{r}$ , $\sum_{j} a_{j} \leq ℓ$ , $a_{r} < ℓ$ , and no variable of index above $r$ occurs. The case $r = 1$ is the definition, with constant coefficient rearranged. For the induction step, direct summation of the mixed relation gives, for $m \geq 1$ , $T_{r} X_{r}^{m} T_{r} = q_{H} X_{r + 1}^{m} - (q_{H} - 1) \sum_{k = 1}^{m - 1} X_{r}^{k} X_{r + 1}^{m - k} T_{r},$ and for $m = 0$ one uses $T_{r}^{2} = (q_{H} - 1) T_{r} + q_{H}$ . Transporting the leading term $q_{H}^{r - 1} X_{r}^{ℓ}$ therefore produces exactly $q_{H}^{r} X_{r + 1}^{ℓ}$ plus terms of $X_{r + 1}$ -degree below $ℓ$ . Each remainder monomial has $X_{r}$ -degree $a_{r} < ℓ$ ; variables of index below $r$ commute with $T_{r}$ and with $X_{r + 1}$ , and $T_{w} \in H_{r}$ for $w \in S_{r}$ , so applying the displayed formula to $X_{r}^{a_{r}}$ and then straightening the finite words $T_{r} T_{w} T_{r}$ , $T_{w} T_{r}$ by [F2] produces terms of $X_{r + 1}$ -degree at most $a_{r} < ℓ$ and total polynomial degree at most $ℓ$ . This proves the claim.

2.3F1F2step 1.1algebra

The transported relations in the degenerate branch. Put $R_{1} : = f (x_{1})$ and $R_{r + 1} : = s_{r} R_{r} s_{r}$ . Straightening and induction on $r$ show $R_{r} = x_{r}^{ℓ} + \sum_{a, w} c_{a, w} x^{a} w,$ where $w \in S_{r}$ , all occurring variables have index at most $r$ , and $\sum_{j} a_{j} < ℓ$ : conjugation transports $x_{r}^{m}$ by $s_{r} x_{r}^{m} s_{r} = x_{r + 1}^{m} - \sum_{k = 0}^{m - 1} x_{r}^{k} x_{r + 1}^{m - 1 - k} s_{r},$ so the leading term $x_{r}^{ℓ}$ produces $x_{r + 1}^{ℓ}$ plus correction terms of total degree $ℓ - 1$ , and every old remainder, when straightened, stays of degree below $ℓ$ .

3.1step 2.2algebra

Termination of the nondegenerate reduction. Order exponent vectors lexicographically as $(a_{d}, a_{d - 1}, \dots, a_{1})$ , larger first differing coordinate larger; this is a well-order on $N^{d}$ preserved by addition. Suppose $a_{r} \geq ℓ$ for some $r$ and some spanning monomial $X^{a} T_{w}$ . Multiplying the zero relation $R_{r}$ of step 2.2 on the left by $X^{a - ℓ e_{r}}$ and on the right by $T_{w}$ gives a relation whose leading term is $q_{H}^{r - 1} X^{a} T_{w}$ and whose remaining terms all have the same coordinates above $r$ and a strictly smaller $r$ -th coordinate (increases in coordinates below $r$ do not affect the lexicographic comparison), with $w \in S_{r}$ and finite straightening changing no dot exponents. Since $q_{H}^{r - 1}$ is a unit, this solves for $X^{a} T_{w}$ as a combination of spanning monomials with strictly smaller exponent vector; well-founded induction on the lexicographic order terminates and expresses every element using only monomials with every $a_{r} < ℓ$ . Hence the bounded ordered monomials span the nondegenerate quotient, over $A$ and after any base change.

3.2step 2.3algebra

Termination of the degenerate reduction. If $a_{r} \geq ℓ$ , multiply the zero relation $R_{r}$ of step 2.3 on the left by $x^{a - ℓ e_{r}}$ and on the right by $w$ ; the leading term is $x^{a} w$ , and every other term has total degree strictly below $\sum_{j} a_{j}$ (all other coefficients come from the remainder of $R_{r}$ , of degree $< ℓ$ ). Induction on the total degree $\sum_{j} a_{j}$ therefore expresses every element in terms of monomials $x^{a} w$ with all $a_{r} < ℓ$ . This also shows that no relation $f (x_{r}) = 0$ for $r > 1$ is claimed: at level one $f (t) = t$ , and $x_{1} = 0$ gives $x_{2} = s_{1}$ , which is exactly the form of the reduced spanning family.

4.1step 2.1step 3.1step 3.2∎

Conclusion. Steps 1.1, 2.1 and 3.1 give the spanning statement for the nondegenerate quotient by $f (X_{1}) = 0$ , and steps 1.1, 2.3 and 3.2 for the degenerate quotient by $f (x_{1}) = 0$ , with all exponents $a_{r} < ℓ$ in both branches and no independence asserted. Since the spanning families have at most $ℓ^{d} d!$ elements, each quotient has dimension at most $ℓ^{d} d!$ over a field. The specialisations $f (t) = \prod_{i} (t - q_{H}^{i})^{λ_{i}}$ and $f (t) = \prod_{i} (t - i)^{λ_{i}}$ are instances of the two branches with $ℓ = \sum_{i} λ_{i}$ . For $ℓ = 0$ the imposed relation is $1 = 0$ , giving the zero algebra for $d > 0$ and $A$ for $d = 0$ , as stated.

Remarks

The lemma is a pure spanning statement: no independence of the bounded ordered monomials is asserted here (the later cyclotomic basis theorem supplies it in the finite-dimensional setting). The two straightening identities are the only place where the difference between the nondegenerate and degenerate branches matters, and the reduction terminates by a well-order in the nondegenerate case and by total degree in the degenerate case. No choice principle is used.

Depends on

Used by

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Sources

Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v4 (standard reference, not scraped)