The centre
The centre $\mathrm{Z}(A)$ of a graded algebra is the graded subalgebra of elements commuting with everything. Write $\mathrm{Z}_n(A) = \mathrm{Z}(A) \cap A_n$ for its degree-$n$ part. A nonzero central element of positive degree makes the centre non-trivial, meaning $\mathrm{Z}(A) \neq k$.
Families with non-trivial centre
The table concerns the general member of each family in characteristic zero and includes those with nonzero recorded central elements in degrees 1–10. The dimensions are supported by the finite-field checks described below. Special parameter values can have larger centres. Its entries are the dimensions $\dim_k \mathrm{Z}_n(A)$, counting all independent central elements in degree $n$, including products of elements of smaller degree. The last column records full Hilbert series where justified by a description of the centre; a dash means no such series is recorded here.
| family | $\dim_k \mathrm{Z}_{1}(A)$ | $\dim_k \mathrm{Z}_{2}(A)$ | $\dim_k \mathrm{Z}_{3}(A)$ | $\dim_k \mathrm{Z}_{4}(A)$ | $\dim_k \mathrm{Z}_{5}(A)$ | $\dim_k \mathrm{Z}_{6}(A)$ | $\dim_k \mathrm{Z}_{7}(A)$ | $\dim_k \mathrm{Z}_{8}(A)$ | $\dim_k \mathrm{Z}_{9}(A)$ | $\dim_k \mathrm{Z}_{10}(A)$ | $H_{\mathrm{Z}(A)}(t)$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| commutative | 4 | 10 | 20 | 35 | 56 | 84 | 120 | 165 | 220 | 286 | $\frac{1}{(1-t)^4}$ |
| Sklyanin | 0 | 2 | 0 | 3 | 0 | 4 | 0 | 5 | 0 | 6 | $\frac{1}{(1-t^2)^2}$ |
| Clifford | 0 | 4 | 0 | 10 | 0 | 20 | 0 | 35 | 0 | 56 | $\frac{1}{(1-t^2)^4}$ |
| central extension of Sklyanin | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | $\frac{1}{(1-t)(1-t^3)}$ |
| Shelton–Tingey | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 8 | 0 | 8 | — |
| Caines | 0 | 2 | 0 | 3 | 0 | 4 | 0 | 5 | 0 | 6 | — |
| Generalized Clifford 1 ($\alpha = \beta$) | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 8 | 0 | 8 | — |
| Generalized Clifford 2 | 0 | 2 | 0 | 5 | 0 | 8 | 0 | 14 | 0 | 20 | — |
| \(\mathrm{S}_\infty\) twist | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 3 | 0 | 3 | — |
| \(\mathrm{S}_{d,i}\) | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 3 | 0 | 3 | — |
| \(\mathrm{S}_\infty\) | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 3 | 0 | 0 | — |
| Sklyanin twist | 0 | 2 | 0 | 3 | 0 | 4 | 0 | 5 | 0 | 6 | $\frac{1}{(1-t^2)^2}$ |
| Goetz–Kirkman–Moore–Vashaw R | 0 | 2 | 0 | 6 | 0 | 12 | 0 | 23 | 0 | 38 | — |
| Goetz–Kirkman–Moore–Vashaw S | 0 | 1 | 0 | 4 | 0 | 5 | 0 | 11 | 0 | 14 | — |
| Goetz–Kirkman–Moore–Vashaw T | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 5 | 0 | 0 | — |
| Zhang twist of a Sklyanin central extension | 0 | 1 | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | — |
For Generalized Clifford 1, the row uses $\alpha = \beta$. The choices with $\alpha = -\beta$ instead give dimensions $(0,0,0,3,0,0,0,13,0,0)$ in degrees 1–10 in the same checks.
All other families with recorded centre dimensions have no nonzero central elements in degrees 1–10 at the parameters used in the computations. This does not rule out central elements in higher degrees or at special parameter values. Centre dimensions are not recorded for Generalized Clifford 3 or $\mathrm{S}_{d,i}$ twist, so neither is included in this vanishing statement.
Hilbert series from the literature
For Sklyanin, the centre is a polynomial ring on two degree-2 generators when the elliptic translation has infinite order; see Levasseur–Smith, Proposition 6.12. The same description holds for the Sklyanin cocycle twist under the infinite-order hypothesis, by Davies, Corollary 5.11. These descriptions give the recorded series $1/(1-t^2)^2$.
For Clifford, the centre is a polynomial ring on four degree-2 generators, giving $1/(1-t^2)^4$; see Ingalls, Proposition 2.2.
For the untwisted central extension of Sklyanin, Walton, Proposition V.10 identifies the centre as a polynomial ring on generators of degrees 1 and 3 when the elliptic translation has infinite order. This gives $1/((1-t)(1-t^3))$.
Caines, Proposition 3.4.4 proves the series $1/(1-t^2)^2$ for the particular member $a=b=c=d=1$. That proposition is stated for this specialisation, so it is not recorded as a theorem about the general member in the table. For the general member of Caines, further checks at the same five primes give dimensions $(0,7,0,8,0,9)$ in degrees 11–16, using its ordered monomial basis and exact modular linear algebra. Thus its computed coefficients agree with $1/(1-t^2)^2$ through degree 16.
Why does the twisted central extension have no central linear form?
A central extension by a degree-1 element does have a nonzero $\mathrm{Z}_1(A)$ by definition. The family previously labelled “central extension of Sklyanin twist” is a Zhang twist of a central extension; see Remark 5.13 of the classification paper. In its displayed presentation, $x_3x_j=-x_jx_3$ for $j=1,2,4$. Thus twisting makes the distinguished generator normal rather than central, while $x_3^2$ is central. The general member has $\mathrm{Z}_1(A)=0$, consistently with the table.
What do the first ten degrees tell us?
The centre need not be generated by its parts in degrees 1–4. For example, the computations for Shelton–Tingey give two central elements in degree 6, although the only nonzero positive-degree part below degree 6 is in degree 4. These sextics therefore cannot be products of central elements of smaller positive degree.
Nor does a computation through degree 10 determine the full centre or its Hilbert series. New central generators can occur in higher degrees; a description of the whole centre requires a proof that the proposed generators and relations are complete.
How it was computed
The dimensions through degree 10 were checked with Macaulay2 at parameter samples over five prime fields, with primes $60013,90001,120049,150001,180001$, respecting the prescribed roots of unity. The five samples agree for all 54 families with recorded data and reproduce the previously recorded dimensions in degrees 2–4. Noncommutative Gröbner bases and the algebra’s Hilbert function were checked through degree 11, covering the products needed to test centrality in degree 10. All six choices of the replaced relation in $\mathrm{S}_{d,i}$ give the same dimensions in these checks. Both normalizations $\alpha_2=0,1$ in Generalized Clifford 2 also give the same dimensions. These specialisations provide evidence for generic dimensions; agreement at five primes alone does not prove genericity, and special parameter choices can have larger centres. Generalized Clifford 3 is regular only on its parameter constraint locus, and generic centre dimensions on that locus are not recorded here.