What this page covers
The multilinear identities of the upper triangular matrix algebra $UT_r(F)$ in each degree $d$ form a representation of $S_d$. The database records the multiplicity of every irreducible representation, and every value here is exact. One row per partition of $d$, orders 1 to 6, degrees 1 to 31.
Identities of the upper triangular algebra
| order $r$ | degrees | rows | exact | open |
|---|---|---|---|---|
| $UT_{1}$ | 1–31 | 1868 | 1868 | open |
| $UT_{2}$ | 1–31 | 1868 | 1868 | open |
| $UT_{3}$ | 1–31 | 1868 | 1868 | open |
| $UT_{4}$ | 1–31 | 1868 | 1868 | open |
| $UT_{5}$ | 1–31 | 1868 | 1868 | open |
| $UT_{6}$ | 1–31 | 1868 | 1868 | open |
Download
- ut.csv and ut.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.