What this page covers
The multilinear identities of $M_{3}(F)$ in degree $d$ form a representation of the symmetric group $S_d$, which decomposes into Specht modules $S^\lambda$ with multiplicities $m_\lambda$, $0\le m_\lambda\le f^\lambda$. The database stores one row for every partition $\lambda$ of $d$ and every family below, with a status that says what is actually proved. A partition that is not stored is not claimed: absence is never a zero.
Rows: 4747, of which 1451 are exact. Database version: 2026-11-10.
Ordinary identities
| degree | stored | partitions of $d$ | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|---|
| 6 | 11 | 11 (complete) | 11 | 0 | 0 | 0 | 0 | open |
| 7 | 15 | 15 (complete) | 15 | 0 | 0 | 0 | 0 | open |
| 8 | 22 | 22 (complete) | 22 | 0 | 0 | 0 | 0 | open |
| 9 | 30 | 30 (complete) | 30 | 0 | 0 | 0 | 0 | open |
| 10 | 42 | 42 (complete) | 42 | 0 | 0 | 0 | 0 | open |
| 11 | 56 | 56 (complete) | 56 | 0 | 0 | 0 | 0 | open |
| 12 | 77 | 77 (complete) | 77 | 0 | 0 | 0 | 0 | open |
| 13 | 101 | 101 (complete) | 101 | 0 | 0 | 0 | 0 | open |
| 14 | 135 | 135 (complete) | 26 | 0 | 0 | 109 | 0 | open |
| 15 | 176 | 176 (complete) | 32 | 0 | 0 | 144 | 0 | open |
| 16 | 231 | 231 (complete) | 40 | 0 | 0 | 191 | 0 | open |
| 17 | 297 | 297 (complete) | 49 | 0 | 0 | 248 | 0 | open |
| 18 | 385 | 385 (complete) | 71 | 0 | 0 | 314 | 0 | open |
| 19 | 20 | 490 | 7 | 0 | 0 | 13 | 0 | open |
| 20 | 17 | 627 | 6 | 0 | 0 | 11 | 0 | open |
| 21 | 15 | 792 | 5 | 0 | 0 | 10 | 0 | open |
| 22 | 14 | 1002 | 5 | 0 | 0 | 9 | 0 | open |
| 23 | 11 | 1255 | 5 | 0 | 0 | 6 | 0 | open |
| 24 | 10 | 1575 | 4 | 0 | 0 | 6 | 0 | open |
| 25 | 9 | 1958 | 4 | 0 | 0 | 5 | 0 | open |
| 26 | 9 | 2436 | 4 | 0 | 0 | 5 | 0 | open |
| 27 | 9 | 3010 | 4 | 0 | 0 | 5 | 0 | open |
| 28 | 9 | 3718 | 4 | 0 | 0 | 5 | 0 | open |
| 29 | 7 | 4565 | 4 | 0 | 0 | 3 | 0 | open |
| 30 | 7 | 5604 | 4 | 0 | 0 | 3 | 0 | open |
| 31 | 6 | 6842 | 3 | 0 | 0 | 3 | 0 | open |
Even-support (A-)identities
| degree | stored | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|
| 6 | 6 | 6 | 0 | 0 | 0 | 0 | open |
| 7 | 8 | 8 | 0 | 0 | 0 | 0 | open |
| 8 | 12 | 12 | 0 | 0 | 0 | 0 | open |
| 9 | 16 | 16 | 0 | 0 | 0 | 0 | open |
| 10 | 22 | 22 | 0 | 0 | 0 | 0 | open |
| 11 | 30 | 30 | 0 | 0 | 0 | 0 | open |
Consequences of the standard polynomial
| degree | stored | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|
| 6 | 11 | 11 | 0 | 0 | 0 | 0 | open |
| 7 | 15 | 12 | 0 | 0 | 3 | 0 | open |
| 8 | 22 | 22 | 0 | 0 | 0 | 0 | open |
| 9 | 30 | 13 | 0 | 0 | 17 | 0 | open |
| 10 | 42 | 5 | 0 | 0 | 37 | 0 | open |
| 11 | 56 | 5 | 0 | 0 | 51 | 0 | open |
| 12 | 77 | 1 | 0 | 0 | 76 | 0 | open |
| 13 | 101 | 1 | 0 | 0 | 100 | 0 | open |
| 14 | 135 | 1 | 0 | 0 | 134 | 0 | open |
Proper identities
| degree | stored | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|
| 6 | 11 | 11 | 0 | 0 | 0 | 0 | open |
| 7 | 15 | 15 | 0 | 0 | 0 | 0 | open |
| 8 | 22 | 22 | 0 | 0 | 0 | 0 | open |
| 9 | 30 | 30 | 0 | 0 | 0 | 0 | open |
| 10 | 42 | 42 | 0 | 0 | 0 | 0 | open |
| 11 | 56 | 49 | 0 | 0 | 7 | 0 | open |
| 12 | 77 | 24 | 0 | 0 | 53 | 0 | open |
| 13 | 101 | 20 | 0 | 0 | 81 | 0 | open |
| 14 | 135 | 27 | 0 | 0 | 108 | 0 | open |
| 15 | 176 | 28 | 0 | 54 | 94 | 0 | open |
Central polynomials modulo identities (positive multiplicities)
| degree | stored | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|
| 8 | 3 | 3 | 0 | 0 | 0 | 0 | open |
| 9 | 8 | 5 | 0 | 0 | 3 | 0 | open |
| 10 | 7 | 3 | 0 | 0 | 4 | 0 | open |
| 11 | 6 | 3 | 1 | 0 | 2 | 0 | open |
| 12 | 6 | 4 | 2 | 0 | 0 | 0 | open |
| 13 | 2 | 2 | 0 | 0 | 0 | 0 | open |
| 14 | 3 | 2 | 1 | 0 | 0 | 0 | open |
| 15 | 2 | 2 | 0 | 0 | 0 | 0 | open |
| 16 | 2 | 0 | 2 | 0 | 0 | 0 | open |
Weak identities for traceless matrices
Weak identities of the pair $(M_{3},\mathfrak{sl}_{3})$: polynomials that vanish when every variable is replaced by a traceless matrix. These are not ordinary identities of $M_{3}(F)$.
| degree | stored | exact | probable | upper bound only | lower bound only | unchecked | open the table |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 0 | 0 | 0 | open |
| 2 | 2 | 2 | 0 | 0 | 0 | 0 | open |
| 3 | 3 | 3 | 0 | 0 | 0 | 0 | open |
| 4 | 5 | 4 | 0 | 0 | 0 | 1 | open |
| 5 | 7 | 6 | 0 | 0 | 0 | 1 | open |
| 6 | 11 | 5 | 0 | 4 | 0 | 2 | open |
| 7 | 15 | 2 | 0 | 9 | 2 | 2 | open |
| 8 | 22 | 3 | 0 | 11 | 7 | 1 | open |
| 9 | 30 | 3 | 0 | 2 | 24 | 1 | open |
| 10 | 42 | 4 | 0 | 2 | 36 | 0 | open |
| 11 | 56 | 6 | 0 | 2 | 48 | 0 | open |
| 12 | 77 | 9 | 0 | 2 | 66 | 0 | open |
| 13 | 101 | 14 | 0 | 2 | 85 | 0 | open |
| 14 | 135 | 21 | 0 | 2 | 112 | 0 | open |
| 15 | 176 | 32 | 0 | 2 | 142 | 0 | open |
| 16 | 231 | 47 | 0 | 2 | 182 | 0 | open |
| 17 | 297 | 69 | 0 | 2 | 226 | 0 | open |
| 18 | 385 | 99 | 0 | 2 | 284 | 0 | open |
| 19 | 20 | 5 | 0 | 2 | 13 | 0 | open |
| 20 | 17 | 4 | 0 | 2 | 11 | 0 | open |
| 21 | 15 | 3 | 0 | 2 | 10 | 0 | open |
| 22 | 14 | 3 | 0 | 2 | 9 | 0 | open |
| 23 | 11 | 3 | 0 | 2 | 6 | 0 | open |
| 24 | 10 | 2 | 0 | 2 | 6 | 0 | open |
| 25 | 9 | 2 | 0 | 2 | 5 | 0 | open |
| 26 | 9 | 2 | 0 | 2 | 5 | 0 | open |
| 27 | 9 | 2 | 0 | 2 | 5 | 0 | open |
| 28 | 9 | 2 | 0 | 2 | 5 | 0 | open |
| 29 | 7 | 2 | 0 | 2 | 3 | 0 | open |
| 30 | 7 | 2 | 0 | 2 | 3 | 0 | open |
| 31 | 6 | 1 | 0 | 2 | 3 | 0 | open |
Download
- m3.csv and m3.json: all rows of this page (columns: order, degree, type, partition, specht_dim, multiplicity, status, lower_bound, upper_bound)
- matrix-algebras-all.csv and .json: every algebra and family
- central-records.csv and .json: bounds on the central quotient, zeros included
Status legend
- exact: certified equality, from matching certified lower and upper bounds. A zero is certified only in this status.
- probable: exact upper bound plus a randomized test with an explicit one-sided error bound. Not a proof.
- lower bound and upper bound: only the stated inequality is certified.
- unchecked: no certified direction; do not use the number.
License and how to cite
The data are released under CC BY 4.0. Please cite the method paper and the database version: W. Schützer and H. L. Talpo, Certified computation of low-degree polynomial identities of matrix algebras, manuscript, 2026; W. Schützer, Matrix Algebras Identity Tables, database of 2026-11-10, Universidade Federal de São Carlos, www.dm.ufscar.br/profs/waldeck/pi/, DOI 10.5281/zenodo.23807359.