Reading the figure
Paired bars compare original quotient dimension N minus two with whole-source dimension twice d, for N equal to two to the d and d from one through five. The data table separately records representation carrier dimensions.
Mathematical labels
Original quotient \(N-2\)
Whole source \(2d\)
| d | Ambient rank | Host symplectic dimension | Source symplectic dimension | Host representation dimension | Source representation dimension |
|---|---|---|---|---|---|
| 1 | 2 | 0 | 2 | 1 | 2 |
| 2 | 4 | 2 | 4 | 2 | 4 |
| 3 | 8 | 6 | 6 | 8 | 8 |
| 4 | 16 | 14 | 8 | 128 | 16 |
| 5 | 32 | 30 | 10 | 32768 | 32 |
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
d=1 N=2 d=2 N=4 d=3 N=8 d=4 N=16 d=5 N=32 05 10 15 20 25 30 Symplectic dimension over F₂ O rigin al q u o tie n t N − 2 W h ole s o u r c e 2 d
Original figure caption
Compare symplectic dimensions first. The original low-rank quotient remains too small.
The host symplectic quotient has dimension \(N-2\) over \({\mathbb F}_2\); the supplied whole source of \(d\) planes has dimension \(2d\). Its original-type embedding is possible by dimension exactly when \(d\ge3\).
| \(N\) | host quotient | source quotient | host carrier | source carrier |
|---|---|---|---|---|
| 2 | 0 | 2 | 1 | 2 |
| 4 | 2 | 4 | 2 | 4 |
| 8 | 6 | 6 | 8 | 8 |
| 16 | 14 | 8 | 128 | 16 |
Scope
The low-rank obstruction concerns the original maximal-simple-character carrier. Enlarged actual arithmetic carriers repair ranks 2 and 4 without changing this quotient theorem.