Certified multiplicities of polynomial identities and central polynomials of matrix algebras

This dataset consists of a SQLite database (matrix-algebras.db) and its documentation (matrix-algebras.txt). It records the multiplicities of the irreducible S_d-modules in spaces of multilinear polynomial identities and central polynomials of the full matrix algebras Mₙ(F), F a field of characteristic zero, for n ≤ 6 and degrees up to 31. Each entry is indexed by the matrix order n, the degree d, a family code and a partition λ of d. The families are:• f: ordinary polynomial identities of Mₙ• s: consequences of the standard polynomial s₂ₙ• p: proper polynomial identities• e: identities supported on even permutations (A-identities)• t: weak identities of the pair (Mₙ, slₙ)• c: central polynomials modulo identities• u: identities of the upper triangular matrix algebra UTₙ Every value is either exact or a certified interval [L, U] of proved lower and upper bounds (table certified_bounds), with a reference to the audit that certifies it. Values that rest on probabilistic tests are kept separate, with explicit error bounds (table central_probable_observations), and never replace certified bounds. The file matrix-algebras.txt describes the schema and the family codes and gives example queries. The certified results include the complete ordinary cocharacters of M₃ in degrees 8 to 10, of M₄ in degrees 10 to 12 and of M₅ in degrees 12 and 13, and all central multiplicities of M₂ through degree 8. They also include central polynomials of degree 12 for M₄ and of degree 16 for M₅, in the components (2⁶) and (2⁸). These show that Formanek's conjectured minimal degree (n²+3n−2)/2 for central polynomials fails for n = 4 and n = 5. The data accompany the paper W. Schützer and H. L. Talpo, "Certified Computation of Low-Degree Polynomial Identities of Matrix Algebras" (preprint, 2026). The database can also be browsed at https://www.dm.ufscar.br/profs/waldeck/pi.