What the database records
Let $F$ be a field of characteristic zero and let $V_d$ be the space of multilinear polynomials of degree $d$ in $x_1,\dots,x_d$. The symmetric group $S_d$ permutes the variables, so $V_d$ is a copy of the regular representation $FS_d$. The identities of the matrix algebra $M_r(F)$ in $V_d$ form an $S_d$-submodule, and therefore $$V_d\cap T(M_r(F))\;\cong\;\bigoplus_{\lambda\vdash d} m_\lambda\, S^\lambda ,\qquad 0\le m_\lambda\le f^\lambda=\dim S^\lambda .$$ The database stores one row for every tuple (order $r$, degree $d$, type, partition $\lambda$). The row holds the Specht dimension $f^\lambda$, the recorded multiplicity $m_\lambda$, and a status that says what is actually proved about that number.
How to read a row
| Column | Meaning |
|---|---|
| Partition $\lambda$ | The Specht component, written in partition notation, for example $(2^2,1^3)$ for $(2,2,1,1,1)$, with exponents counting repeated parts. In the results table partitions sort in the natural lexicographic order. |
| Specht dim $f^\lambda$ | Dimension of $S^\lambda$, from the hook length formula. It is also the largest possible multiplicity. |
| Multiplicity $m_\lambda$ | The recorded value. Its meaning depends only on the status. A prefix shows the direction: $\ge$ for a certified lower bound, $\le$ for a certified upper bound. |
| Status | See the legend below. |
| Lower, Upper | The independently certified bounds on $m_\lambda$ from the reconciliation table. Equal bounds mean the value is exact. A dash means nothing is recorded there. Not every component has an entry. |
Status legend
Types
| Code | Family |
|---|---|
f | Ordinary identities: multiplicity of $S^\lambda$ in $V_d\cap T(M_r(F))$. |
e | Even-support ($A$-)identities, the analogue for the alternating group. Conjugate partitions are tied together. |
s | Consequences of the standard polynomial $s_{2r}$, which is an identity of $M_r(F)$. These are bounded above by the corresponding ordinary multiplicities. |
p | Proper identities: multiplicity in the intersection of $T(M_r(F))$ with the multilinear proper polynomials. Related to the ordinary values by the exact Pieri transforms. |
c | Nontrivial central polynomials modulo identities: multiplicity of $S^\lambda$ in the multilinear central polynomials of $M_r(F)$ modulo $T(M_r(F))$. Only positive occurrences are stored. |
u | Identities of the upper triangular algebra $UT_r(F)$, diagonal matrices included. These multiplicities come from the upper-triangular Hilbert series formula with exact Schur coefficients, and all are exact. |
t | Weak identities for traceless matrices: polynomials that vanish whenever every variable is replaced by a traceless matrix. These are identities of the pair $(M_r,\mathfrak{sl}_r)$, not ordinary identities. |
Coverage
Computed from the loaded data. Each cell gives the degrees present and the number of rows.
Cautions
The database records computational claims, not proofs. Treat a multiplicity as established only when its status is exact, and cite the certificate behind it. A recorded zero is certified only when the status is exact. A value marked unchecked may even lie below a proved upper bound. The Theory and trust tab explains why the exact values can be believed and shows exactly what is claimed in every cell.
Download and permanent links
Every value can be downloaded and cited. Each matrix algebra has its own static page, with a summary of every family and degree, a stable address, and machine-readable dataset markup for search engines.
| Algebra | Page | Data |
|---|---|---|
| $M_2(F)$ | algebra/m2.html | CSV · JSON |
| $M_3(F)$ | algebra/m3.html | CSV · JSON |
| $M_4(F)$ | algebra/m4.html | CSV · JSON |
| $M_5(F)$ | algebra/m5.html | CSV · JSON |
| $M_6(F)$ | algebra/m6.html | CSV · JSON |
| Upper triangular $UT_r(F)$ | algebra/ut.html | CSV · JSON |
| Everything | CSV · JSON | |
| Central determination records | CSV · JSON |
Columns: order, degree, type, partition, Specht dimension, multiplicity, status, lower bound, upper bound.
The status travels with the value: please do not use a multiplicity without it. A partition that is not in the file is not claimed.
The same links for one table are available from the Identities tab: #identities?ord=4°ree=13&type=f.
License and how to cite
The data in this database, that is the tables on this site and the data files behind them, are released under the Creative Commons Attribution 4.0 International license (CC BY 4.0). You may copy, redistribute and build on them, also commercially, provided that you give appropriate credit, link to the license and indicate any changes. Please keep the status of each value with the value: an exact multiplicity and a bound are different claims.
To cite the database, please give both 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 , Universidade Federal de São Carlos. www.dm.ufscar.br/profs/waldeck/pi, DOI 10.5281/zenodo.23807359. Please cite this DOI for the database as a whole.
Acknowledgements
This project spans more than ten years and has used several kinds of computing devices, from personal computers to the Cluster UFSCar Service. It is currently carried out on a workstation owned by the author, with an Intel Core i5-13400F processor (10 physical cores, 16 threads), about 30 GiB of main memory, and an NVIDIA GeForce RTX 4060 Ti with 16 GiB of device memory and 4352 CUDA cores. Waldeck Schützer created the historical computations and the database.
An AI assistant contributed to computational development, validation, bibliographic searches, and editing. The authors retain responsibility for the mathematical arguments and final claims. Exact claims rely on the stated proofs and auditable certificates. The probabilistic claims retain their explicit one-sided error bounds.
How the tables were made
Each multiplicity is computed one symmetric group component at a time, in a space of dimension $f^\lambda$ rather than the full multilinear component of dimension $d!$. The method follows Bondari and the highest-weight formulation of Benanti, Demmel, Drensky and Koev, and adds the certification discipline described below. The reasoning behind it, a worked example and the evidence are on the Theory and trust tab.
Which tool for which family
| Type | Main tool | Key sources |
|---|---|---|
Ordinary identities f |
Bondari slice and highest-weight evaluation. After depolarization, one generic diagonal matrix with colored matrix paths gives exact positive certificates for row-symmetric candidates. | Bondari; Benanti–Demmel–Drensky–Koev; Drensky–La Scala; Drensky–Kostadinov; Procesi |
Even-support e |
Pass to the alternating group through the explicit decomposition of $FS_d$ and $FA_d$. An ordinary prerequisite plus a row-alternating remainder reduces the test to distinct matrix units in each row. | Henke–Regev; Regev; Gonçalves–Schützer–Talpo |
Standard consequences s |
Explicit highest-weight consequences of $s_{2r}$, with the first layers above $s_{2r}$ taken from the literature. | Amitsur–Levitzki; Olsson–Regev; Benanti–Drensky; Schützer |
Proper p |
Exact Pieri transforms between ordinary and proper multiplicities, and finite interval propagation. | this project |
Central c |
The same evaluation engine with target “scalar” instead of “zero”, measured modulo the identities. | Bondari; Drensky–Kostadinov |
| Boundary and large degree | Bergman’s classification of identities with small outside degree gives closed formulas for boundary multiplicities, with exact certificates for the support question. | Bergman; this project |
Triangular u and weak t |
Upper-triangular Hilbert series with exact Schur coefficients, and weak identities for traceless matrices. | Drensky 2021 (weak identities) |
Papers and sources
Every reference of the underlying paper is listed here, grouped by the role it plays.
This project
- W. Schützer and H. L. Talpo, Certified computation of low-degree polynomial identities of matrix algebras. Manuscript, 2026. Methods, exact modular certificates, and the tables for orders 2 through 6.
- W. Schützer, Consequences of the standard polynomial in degrees $n+3$ and $n+4$. Manuscript, 2026. The complete characters of $V_{n+3}\cap\langle s_n\rangle^T$ and $V_{n+4}\cap\langle s_n\rangle^T$.
- D. J. Gonçalves, W. Schützer and H. L. Talpo, $A$-identities for the $2\times2$ matrix algebra. Arch. Math. (Basel) 106 (2016), 417–429.
- W. Schützer, Certified multiplicities of polynomial identities and central polynomials of matrix algebras. Dataset, Zenodo, 2026. doi:10.5281/zenodo.23807359. This database, with a DOI for the whole dataset.
Classical theory of identities of matrices
- S. A. Amitsur and J. Levitzki, Minimal identities for algebras. Proc. Amer. Math. Soc. 1 (1950), 449–463. The standard polynomial $s_{2r}$ is an identity of $M_r(F)$, and there is none of smaller degree.
- U. Leron, Multilinear identities of the matrix ring. Trans. Amer. Math. Soc. 183 (1973), 175–202. For $r>2$, all identities of degree $2r+1$ follow from $s_{2r}$.
- C. Procesi, The invariant theory of $n\times n$ matrices. Adv. Math. 19 (1976), 306–381. Multilinear trace identities and the trace-quotient bound.
- A. Regev, Young tableaux and P.I. algebras. Astérisque 87–88 (1981), 335–352. Cocharacters and the structure of multilinear components.
- G. M. Bergman, Wild automorphisms of free p.i. algebras, and some new identities. Unpublished manuscript, 1979. author’s manuscript. Identities with small outside degree, behind the boundary multiplicities.
- U. Vishne, Polynomial identities of $M_2(G)$. Comm. Algebra 30 (2002), 443–454. author’s version. An independent representation-theoretic computation of the degree-8 identities of $M_3(F)$.
- V. Drensky, Weak polynomial identities and their applications. Commun. Math. 29 (2021), 291–324. Survey, including the open basis problem for $r>2$ and the weak identities.
- S. Rosset, A new proof of the Amitsur–Levitski identity. Israel J. Math. 23 (1976), 187–188. Drensky and Kasparian used ideas of this proof for the central polynomial of $M_3(F)$.
- V. Drensky and A. Kasparian, Some polynomial identities of matrix algebras. C. R. Acad. Bulg. Sci. 36 (1983), 565–568. Boundary multiplicities, used with Bergman’s classification for the support question.
- V. Drensky and A. Kasparian, Polynomial identities of eighth degree for $3\times3$ matrices. Annuaire Univ. Sofia, Fac. Math. Mécan. 77 (1983), 175–195. The degree-8 identities of $M_3(F)$.
- V. Drensky and A. Kasparian, A new central polynomial for $3\times3$ matrices. Comm. Algebra 13 (1985), 745–752. Central polynomials of $M_3(F)$.
- E. Formanek, Invariants and the ring of generic matrices. J. Algebra 89 (1984), 178–223. Comparison of ordinary and mixed trace cocharacters, behind the trace-quotient bound.
- E. Formanek, A conjecture of Regev about the Capelli polynomial. J. Algebra 109 (1987), 93–114. Regev’s polynomials, which are central for $M_n(F)$.
- A. Berele, Homogeneous polynomial identities. Israel J. Math. 42 (1982), 258–272. With Drensky, the bridge between multilinear and homogeneous identities.
- V. Drensky, Codimensions of $T$-ideals and Hilbert series of relatively free algebras. J. Algebra 91 (1984), 1–17. The proper decomposition for unital algebras and the cocharacters of $M_2(F)$.
- R. G. Swan, An application of graph theory to algebra. Proc. Amer. Math. Soc. 14 (1963), 367–373. The Euler-trail reduction of the standard polynomial.
Consequences of the standard polynomial
- J. B. Olsson and A. Regev, On the $T$-ideal generated by a standard identity. Israel J. Math. 26 (1977), 97–104. The first layer above $s_m$.
- F. Benanti and V. Drensky, On the consequences of the standard polynomial. Comm. Algebra 26 (1998), no. 12, 4243–4275. The second layer above $s_m$, used in degree 12 for $M_5$.
Computational methods
- S. Bondari, Constructing the polynomial identities and central identities of degree $<9$ of $3\times3$ matrices. Linear Algebra Appl. 258 (1997), 233–249. doi:10.1016/S0024-3795(96)00208-X. The multilinear slice and the first computer determination of the identities and central polynomials of $M_3(F)$ through degree 8.
- S. Bondari, Constructing the Identities and the Central Identities of Degree $<9$ of the $n\times n$ Matrices. Ph.D. thesis, Iowa State University, 1993.
- F. Benanti, J. Demmel, V. Drensky and P. Koev, Computational approach to polynomial identities of matrices, a survey. In: Ring Theory: Polynomial Identities and Combinatorial Methods (Pantelleria, 2001), Lecture Notes in Pure and Appl. Math. 235, Dekker, 2003, 141–178. The highest-weight method and the historical tables.
- V. Drensky and R. La Scala, Defining relations of low degree of invariants of two $4\times4$ matrices. Internat. J. Algebra Comput. 19 (2009), 107–127. arXiv:0708.3583. Specializing one generic matrix to a diagonal one.
- V. Drensky and B. Kostadinov, Central polynomials of minimal degree for matrices. Preprint, 2026, version 2. arXiv:2601.07750. Recent calculations of central polynomials with the diagonal specialization.
- M. R. Bremner, S. Madariaga and L. A. Peresi, Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions. Comment. Math. Univ. Carolin. 57 (2016), no. 4, 413–452. Constructive Young-tableau coordinates for the Wedderburn blocks.
- S. Boumova and V. Drensky, Cocharacters of polynomial identities of upper triangular matrices. Preprint, 2011. arXiv:1101.4503. The method for the cocharacters behind the upper-triangular tables.
Even-support and alternating-group methods
- A. Henke and A. Regev, Explicit decompositions of the group algebras $FS_n$ and $FA_n$. In: Polynomial Identities and Combinatorial Methods (Pantelleria, 2001), Lecture Notes in Pure and Appl. Math. 235, Dekker, 2003, 329–357. The decomposition behind the reduction for $A$-identities.
- A. Regev, Double centralizing theorems for the alternating groups. J. Algebra 250 (2002), 335–352. arXiv:math/0106072. Comparison of $S_d$ and $A_d$ tensor-power actions.
- A. Regev and Y. Roichman, Permutation statistics on the alternating group. Adv. Appl. Math. 33 (2004), 676–709. doi:10.1016/j.aam.2003.10.004. Related alternating-group statistics. They do not supply identity multiplicities.
- A. Henke and A. Regev, $A$-codimensions and $A$-cocharacters. Israel J. Math. 133 (2003), 339–355. The $A$-identities, the local $A$-codimension bound and the conjectured least degree $2r+2$.
- D. J. Gonçalves and P. Koshlukov, $A$-identities for upper triangular matrices: a question of Henke and Regev. Israel J. Math. 186 (2011), 407–426. Cited in the manuscript in connection with the Henke–Regev question.
Randomized testing
- N. Saxena, Progress on polynomial identity testing. Bull. EATCS 99 (2009), 49–79. ECCC TR09-101. The Schwartz–Zippel lemma behind the status “probable”.
- J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 (1980), 701–717.
- R. Zippel, Probabilistic algorithms for sparse polynomials. In: Symbolic and Algebraic Computation, Lecture Notes in Computer Science 72, Springer, 1979, 216–226.
Are we really mapping out the identities?
The short answer is yes, with two precise meanings. By mapping we mean computing the isomorphism type of the space of identities as a representation of the symmetric group, one irreducible component at a time. By certified we mean that a value is backed by a proof that anyone can re-check from stated finite data, and that every number which is not backed that way is labelled as a bound, a probable value, or unchecked. This page explains the theory behind the tables, shows one complete example by hand, and lists the evidence.
What is claimed, cell by cell
This map is computed in your browser from the same data as the tables, so it can never drift from them. A cell is colored by its weakest stored component: it is green only if everything stored there is exact. A blank cell means that nothing is stored and nothing is claimed.
What exactly is being mapped
Fix a matrix order $r$ and a degree $d$. The multilinear polynomials $x_{\sigma(1)}\cdots x_{\sigma(d)}$, $\sigma\in S_d$, form a basis of the space $V_d$, so $\dim V_d=d!$. The symmetric group acts by permuting the variables, and the identities $I_{r,d}=V_d\cap T(M_r(F))$ form an $S_d$-submodule. In characteristic zero such a module is a direct sum of Specht modules, and the whole object is described by the numbers $$I_{r,d}\;\cong\;\bigoplus_{\lambda\vdash d} m_{r,d}(\lambda)\,S^\lambda,\qquad 0\le m_{r,d}(\lambda)\le f^\lambda .$$ These numbers determine the dimension of the identity space, $\sum_\lambda m_{r,d}(\lambda)f^\lambda$ (not the sum of the multiplicities), the codimension $d!-\dim I_{r,d}$, and the cocharacter of $M_r(F)$, which has $f^\lambda-m_{r,d}(\lambda)$ copies of $S^\lambda$. The multiplicities are independent of the characteristic-zero field $F$.
The database therefore records the shape of the identity space, not explicit generating polynomials. Writing down a basis of $T(M_r(F))$ is a different problem, and it is open for $r>2$, already for $r=3$ [Drensky 2021]. The tables answer the question that comes first: how large is the space of identities in each degree, and which representations does it contain? That is also the question that tells you where new generators must appear.
Why a finite computation settles an infinite-looking question
There is even an exact closed-form oracle for the whole problem. Let $g,h\in S_d$ act on the coefficient vector and put $G_{g,h}=r^{\,c(\gamma^{-1}k\gamma k^{-1})}$ with $k=g^{-1}h$, where $c$ counts cycles of a permutation in $S_{d+1}$, the element $k$ is regarded as fixing $d+1$, and $\gamma=(1\,2\,\cdots\,d{+}1)$. Then $a^{\mathsf T}Ga$ equals the sum of the squares of the values of $P_a$ over all matrix-unit substitutions, so $P_a$ is an identity exactly when $a^{\mathsf T}Ga=0$, and the multiplicity of an irreducible representation is its dimension minus the rank of the corresponding block. This is a finite, exact characterization. It is too expensive to run directly in the large cases, which is why the certificates below are used, but it shows that nothing in the problem is approximate in principle.
Two-sided certificates
A value $m$ is accepted as exact only when an upper and a lower bound meet. The two bounds come from completely different arguments, so a mistake in one cannot silently agree with a mistake in the other.
Upper bound from evaluation
Evaluate on matrix-unit tuples and compute a rank. A minor that is nonzero modulo a prime $p$ is the reduction of a nonzero integer minor, hence also nonzero over $\mathbb Q$ and over every field of characteristic zero. So the rank over $\mathbb Q$ is at least the size $q$ of that minor, and $m_{r,d}(\lambda)\le f^\lambda-q$. A bad prime can only make the rank look smaller, which weakens the bound and never falsifies it. Adding more substitutions can only lower the bound.
Lower bound from construction
Exhibit identities and show they are independent. Sources include the consequences of the standard polynomial $s_{2r}$ (Amitsur–Levitzki; Olsson–Regev, Benanti–Drensky and the companion paper for the first layers above it), Bergman’s classification of identities with small outside degree, explicit rational certificates verified on matrix units, the trace-quotient bound through Procesi, and degree-to-degree propagation: multiplying by a new variable is injective, which transports lower bounds from degree $d$ to $d+1$.
What each status guarantees
Two further rules protect the reader. Absence from a table is never read as multiplicity zero: a partition that is not stored is simply not claimed. And a positive nullity that merely equals a known upper estimate is not called sharp until an independent identity bound closes it.
Central polynomials use the same engine
Let $Z_{r,d}\supseteq I_{r,d}$ be the multilinear polynomials whose values on $M_r(F)$ are scalar matrices. The set of central polynomials that are not identities is not a vector space, so the database counts the quotient $Z_{r,d}/I_{r,d}$, with multiplicities $z_{r,d}(\lambda)$. For a component whose identity multiplicity is known, form the evaluation matrix of the commutators $[f(A_1,\dots,A_d),e_{ij}]$ and let $R$ be its exact rank. Then $$0\le z_{r,d}(\lambda)\le f^\lambda-R-m_{r,d}(\lambda),$$ and equality holds as soon as a basis of the kernel is verified to take scalar values on every tuple of matrix units. An upper bound of zero proves that no central polynomial occurs, with no candidate to verify. Positive values certify central polynomials that are genuinely not identities.
Two positive central entries come with their own certificates. For $M_4$ in degree 12 the shape $(2^6)$ has multiplicity 1, with an explicit integer witness that has been verified by an independent reimplementation. For $M_5$ in degree 16 the shape $(2^8)$ is recorded with multiplicity at least 1: centrality over characteristic zero was verified exactly by an exhaustive enumeration (run on a GPU, every batch hashed and audited), and the fact that the polynomial is not an identity was replayed modulo the prime 1009 in two different bases. The database stores this entry as a lower bound, because the matching upper bound is weak (at most 20), so the exact multiplicity is not claimed.
A worked example: $M_3$ in degree 9
The matrix algebra $M_2$ is used in this project only to validate the algorithms. Its identities and cocharacters are known in closed form (Formanek, Drensky), so the values the programs produce for $M_2$ can be compared with a formula, and they agree wherever this was checked. It cannot show how the method behaves when no formula is available. A representative case is $M_3$ in degree 9: the first degree in which the standard polynomial $s_6$ does not generate all identities (Drensky), so the two bounds really come from different sources. The space $V_9$ has dimension $9!=362\,880$, which is split into 30 slices, one for each partition of 9. The slice of $\lambda$ has dimension $f^\lambda$, and for each one the program has to prove two inequalities for the multiplicity $m=m_{3,9}(\lambda)$.
| Shape $\lambda$ | $f^\lambda$ | Lower bound $m\ge$ | Upper bound $m\le$ | Result |
|---|---|---|---|---|
| $(7,2)$ | 27 | 0, trivially | 0. Here $d-\lambda_1=2<r$, and Bergman’s boundary theorem says no identity of $M_3$ lives on such a shape. | exact $m=0$ |
| $(6,3)$ | 48 | 1, from the boundary formula ($d-\lambda_1=r$): the identities of that boundary form a known module | 1, from the same formula | exact $m=1$ |
| $(5,1^4)$ | 70 | 12, from the consequences of $s_6$ (an exact rational certificate of their character) | 12: the evaluation matrix on matrix units has nullity 12 in this slice, found by a rank computation modulo a prime, which bounds the rational nullity | exact $m=12$ |
| $(5,4)$ | 42 | 1, from an explicit integer identity of this weight (a highest-weight polynomial with 60 terms, checked on matrix units). The certified consequences of $s_6$ only give 0 here | 1: a minor of size $41=42-1$ of the evaluation matrix is nonzero modulo a prime | exact $m=1$ |
The last row is the point of the example. The upper bound alone would only say that there is at most one copy of $S^{(5,4)}$ among the identities, and $s_6$ alone would say that nothing is known to exist. Neither statement is a value. Only when an explicit identity shows $m\ge1$ do the two bounds meet, and the cell becomes exact. Had the explicit identity been missing, the stored entry would be the pair $0\le m\le1$ with the status upper bound, and it would not be used as a zero. For many other shapes of this degree the consequences of $s_6$ are only part of the identities. For $(4,3,2)$, for instance, they give at least 2 copies and the exact multiplicity is 8, so the remaining ones come from the other constructions listed on the Methods and papers tab, and the audit file of each cell says which.
The smallest case by hand: $M_2$ in degree 4 (a validation, not a result)
Here $r=2$ and $d=4$, so $V_4$ has the $24$ words $x_{\sigma(1)}x_{\sigma(2)}x_{\sigma(3)}x_{\sigma(4)}$. A polynomial $P=\sum_\sigma a_\sigma x_{\sigma(1)}\cdots x_{\sigma(4)}$ is an identity if and only if it vanishes on all $2^{8}=256$ tuples of matrix units. Each tuple gives equations in the $a_\sigma$. For instance, take $(x_1,x_2,x_3,x_4)=(e_{11},e_{12},e_{22},e_{21})$. A word is nonzero only if its factors chain together, and exactly four words survive:
so this single substitution says $a_{1234}+a_{2341}=0$ and $a_{3412}+a_{4123}=0$. For the standard polynomial the two permutations in each pair differ by a 4-cycle, which is odd, so their signs are opposite and both equations hold. Stacking the equations from all $256$ tuples gives a matrix with $24$ columns, and a computer-algebra rank over $\mathbb Q$ gives
Already $18$ well-chosen tuples out of the $256$ reach rank $23$ (an exact rank over $\mathbb Q$, so this is a rigorous certificate on its own), illustrating that far fewer substitutions than $r^{2d}$ suffice. The Gram-matrix oracle of the previous section agrees: the $24\times24$ matrix $G$ has rank $23$, its kernel is again the sign vector, and it equals the sum of $EE^{\mathsf T}$ over the $256$ tuples, where $E$ is the $24\times4$ matrix of values of the words, as the theory predicts.
Read off the multiplicities. The identity space is one-dimensional and spanned by the alternating polynomial, so it is a copy of the sign representation $S^{(1^4)}$. The lower bound comes from Amitsur–Levitzki, which says $s_4$ is an identity of $M_2(F)$. The upper bound comes from the rank $23$. They meet, so the table is exact in every component:
The same computation with the target changed from “zero” to “scalar” finds the degree-4 central polynomials. For two traceless $2\times2$ matrices $A,B$ one has $AB+BA=\operatorname{tr}(AB)\,I$, so $[x_1,x_2][x_3,x_4]+[x_3,x_4][x_1,x_2]$ takes only scalar values on $M_2(F)$, and it is not an identity. The quotient $Z_{2,4}/I_{2,4}$ is two-dimensional, one copy of $S^{(2,2)}$, which is the single positive entry in the central tables for this degree.
Evidence that the machinery is right
The theory is proved in the paper, so correctness of the theorems does not rest on experiment. The checks below target a different risk: bugs, bad data and misread conventions. They fall into independent recomputation, agreement with classical theorems, internal controls, and consistency of the database itself.
| Check | Outcome |
|---|---|
| Independent recomputation of earlier ordinary data. Two separately performed modular evaluations, for $M_4$ and $M_5$ in degrees 11 and 12 (56, 56, 70 and 77 components). | Identical evaluation matrices in all 259 of 259 components (256 needed a matrix, 3 had a zero target). Every requested rank was certified. |
| Independent even-support evaluations for $M_2$ in degrees 6 and 7 and $M_4$ in degree 11, including positive multiplicities and self-conjugate partitions. | 43 of 43 components agree. |
| Exact verification ladder for $M_2$, degrees 4 through 8. Candidate identities are built over $\mathbb Q$ and verified on Euler-trail and vertex-orbit representatives of the matrix-unit multisets. | All 60 reference components agree and are recorded as certified sharp. |
| Trace-quotient audit (Procesi). Where the trace target is zero the whole source block lies in the identities. | 51 components certified exact (19 for $M_2$ in degree 10, 29 for $M_2$ in degree 11, one for $M_3$ in degree 10, two for $M_3$ in degree 11). The 35 other agreements have positive target and are not claimed sharp. |
| Complete coefficient verification of the new positive certificates: 24 ordinary and 5 even-support entries. | All 3,709 candidate identity checks completed (3,582 and 127), covering 10,203,573,418 coefficient contributions. One candidate per ordinary entry was rebuilt independently from the multilinear definition at every output position, with positive and negative controls (the standard polynomial and a commutator). |
| Colored diagonal coefficients for three $M_4$ degree-14 candidates, compared with the multilinear definition. | All 7,436 binary word coefficients agree, zeros included. Deliberately perturbed candidates were rejected by both coefficient calculations. |
| Literal Euler-word sums at the first, middle and last retained tuple of each of the five even-support exact entries. | Fifteen independent controls, all reproducing every modular tableau entry with zero integer residual. |
| Classical results reproduced. Amitsur–Levitzki, Leron (degree $2r+1$), Vishne and Bondari for $M_3$ below degree 9, Benanti–Demmel–Drensky–Koev in degree $2r+2$ for $r=3,4,5$, and Bergman’s boundary multiplicities. | Agreement throughout. For instance, all 97 recorded boundary entries agree with the boundary theorem (a check that the proof itself does not use). The degree-9 $M_3$ table of the survey is clarified, since sampled positive nullities there are upper bounds only. |
| Necessity of the hypotheses. The reductions were also run where their hypotheses fail: a counterexample for direct depolarization of an even projection, no row symmetry, small characteristic, an unjustified sign twist, an insufficient modulus. | Each breaks the algorithm, so the stated hypotheses are not decorative. |
| Consistency of the database itself (run by the site update script on the current database file). Order-to-order monotonicity, $s\le f$, $p\le f$, $f\le u$ and the central records. | 74,294 order comparisons, 2,177 standard-polynomial, 3,057 proper and 8,336 upper-triangular comparisons, and 205 central comparisons: no violation. |
Exactness of a value rests on its two certificates, not on these samples. The samples supplement the complete calculations, they do not replace them.
Checks you can watch run
These execute in your browser every time this tab opens.
Questions a skeptic should ask
Did you just run a program and trust the output?
No. Every exact value is the meeting point of two proofs: an integer rank certificate for the upper bound and a verified family of identities for the lower bound. The program only proposes certificates. What counts is the certificate itself, an explicit minor or an explicit family of identities, which can be re-checked in exact integer or rational arithmetic without trusting the program.
Is it not suspicious that you work modulo a prime?
It would be if the prime were used to decide anything in the “wrong” direction. It is only used to exhibit a nonzero minor. If a determinant is nonzero modulo $p$, its integer value is nonzero, so the rank over $\mathbb Q$ is at least that large. An unlucky prime can only hide rank, which weakens an upper bound and never produces a false one. For positive lower-bound certificates the candidate identities are recovered exactly from large primes and then verified over the integers.
Matrix units only? Surely identities must hold for all matrices.
Multilinearity does the work: a multilinear polynomial that vanishes on every tuple of basis elements $e_{ij}$ vanishes on every tuple of matrices. For central polynomials the same argument applies to “scalar value” instead of “zero”.
What if the code has a bug?
That is the reason for the independent evidence above: a second modular evaluation reproduced 259 of 259 earlier components, literal sums over Euler words reproduced the modular tableau entries, one candidate per entry was rebuilt straight from the multilinear definition, controls with known identities and non-identities behave correctly, and the classical theorems come out right. No finite list of checks proves a program correct, and this page does not claim that. But the claims that are published as exact rest on certificates that can be recomputed independently of the original code.
Are any of the numbers random?
Only the status probable, and it is never promoted. Such a value comes with an exact upper bound and a one-sided randomized test whose error probability is bounded explicitly, with a term for the sampling of coefficients and one for the choice of primes. A single nonzero evaluation would refute it with certainty. Currently one ordinary value in the database has this status.
Where are the actual generating polynomials?
The tables record the multiplicities, which determine dimension and representation type, not explicit bases. Finding a generating set for $T(M_r(F))$ is open for $r>2$. Some identities are named in the paper (for example, those of Bergman at the boundary and the consequences of $s_{2r}$), but the database does not purport to list generators.
Is the method new?
The framework is not: Bondari’s slice and the highest-weight method of Benanti, Demmel, Drensky and Koev are the foundation, and the one-diagonal-matrix specialization goes back to Drensky–La Scala and is used by Drensky–Kostadinov. Bergman’s classification of boundary identities is also not claimed as new. What the project adds is the certification discipline, the even-support reduction with its parity split, exact tableau recovery, the degree-to-degree propagation of bounds, and the new exact results, including a proof that a proposed support criterion fails for $M_5$.
What is still open?
A lot, and it is shown above in the claims map rather than hidden. In the ordinary tables up to degree 18, 4,475 components are not yet exact (4,475 unchecked or bounded on one side): $M_{2}$ in degrees 14 to 18, $M_{3}$ in degrees 14 to 18, $M_{4}$ in degrees 13 to 18, $M_{5}$ in degrees 14 to 18 and $M_{6}$ in degrees 14 to 18. Beyond degree 18 only selected boundary components are stored, and a missing partition means no claim. The cost of verification grows quickly: the largest slice has dimension 21,450 in degree 13, 69,498 in degree 14 and 16,336,320 in degree 18.
Does a pattern in the data prove a theorem?
No, and the paper warns about it. A natural “support criterion” for which partitions can occur in the identities holds for orders 2, 3 and 4 but is false for order 5: it is proved that $m_{5,20}((11,9))=0$, that no two-variable identity of $M_5$ has degree below 20, and that the least eventual support threshold is 21. For order 6 the criterion is proved from degree 30 on, and in degree 29 the single partition $(20,9)$ remains undecided.
Was AI involved?
Yes, and it is disclosed in the paper: an AI assistant contributed to computational development, validation, bibliographic searches and editing. The authors retain responsibility for the mathematical arguments and the final claims, and the exact claims rely on stated proofs and auditable certificates. That is precisely why the certificates, rather than any assistant’s say-so, are the thing to check.
Conjectures and theorems that the data meet
The tables are finite, so they cannot prove a statement about every matrix order or every degree. They can do three useful things: check a prediction in the range they cover, refute a prediction with a single certified entry, and show exactly where a prediction has not been tested. This page sorts the statements that touch the database into those groups. Every table below is computed in your browser from the loaded data. The text around the tables was written by hand and names the source of each statement.
At a glance
| Statement | Source | Outcome |
|---|---|---|
| The least degree of a central polynomial of $M_n(F)$ is $(n^2+3n-2)/2$. | Formanek, restated by Drensky and Kostadinov | agrees for $n=2,3$. refuted for $n=4$: a central polynomial of degree 12 exists, the prediction is 13. refuted for $n=5$: a central polynomial of degree 16 exists, the prediction is 19. undecided for $n=6$. |
| For every $r\ge2$ the shape $(2^{2r-2})$ occurs in the central polynomials of $M_r(F)$, in degree $4(r-1)$. | Conjecture of the author (W. Schützer) | confirmed for $r=2,3,4,5$, with a certified record for each of $(2^2)$, $(2^4)$, $(2^6)$, $(2^8)$. For $r=1$ ($M_1=F$) the statement is trivial. open for $r\ge6$. |
| The least degree of a central polynomial of $M_r(F)$ is $4(r-1)$. | Conjecture of the author (W. Schützer) | agrees for $r=2,3$ (with the literature below degree 6 for $r=3$). consistent for $r=4,5$: degrees 8 to 11 ($M_4$) and 10 to 13 ($M_5$) hold no central polynomial, but lower degrees are not stored and $M_5$ in degrees 14 and 15 is only partly covered. open for $r\ge6$. |
| The least degree of a nonzero $A$-identity of $M_r(F)$ is $2r+2$. | Henke and Regev | agrees for $r=2$. refuted for $r=3$ (least degree 11) and $r=4$ (least degree 15). |
| For every $r>2$ the least degree of a nonzero $A$-identity of $M_r(F)$ is $4r-1$. The case $r=2$ (least degree 6) is exceptional, because of the exceptional nature of the alternating group $A_6$. | Conjecture of the author (W. Schützer) | agrees for $r=3$ (least degree 11) and $r=4$ (least degree 15). upper bound met for $r=5$ (degree 19) and $r=6$ (degree 23): the database records a certified $A$-identity there, but the degrees below are not fully covered. open for $r\ge7$. |
| The identities of $M_r(F)$ in degrees $2r+1,\dots,2r+4$ are the consequences of $s_{2r}$. | Leron, Benanti and Drensky, Benanti–Demmel–Drensky–Koev, and the companion paper on degrees $n+3$ and $n+4$ | confirmed for every shape of $M_4$ in degrees 9 to 12 and of $M_5$ in degrees 11 to 13. undecided in higher degrees. false in general: $M_2$ in degree 5, $M_3$ in degree 9. |
| For $n\ge2$ (some sources: $n>2$) the identities of $M_n(F)$ follow from $s_{2n}$ and the identity of algebraicity $a_n$, of degree $n(n+3)/2$. | Razmyslov; see Drensky, Weak polynomial identities and their applications | false for $n=3$ (Okhitin; Domokos). refuted for $n=4$ and $n=5$: two-row identities of degree 14 and 20 that cannot come from $s_{2n}$ and $a_n$. |
| The least degree of a central polynomial of $M_n(F)$ is $n^2$ (a common belief, recorded by Drensky and Kostadinov). | Folklore; Formanek’s construction gives degree $n^2$ | agrees for $n=2$. false for $n=3,4,5$: central polynomials of degree 8, 12 and 16 exist. |
| The last obstruction to positive identity multiplicities always has two rows. | Conjecture of the companion manuscript | open. Proved for $r\le5$. |
Formanek’s conjecture on central polynomials
A multilinear polynomial is central for $M_n(F)$ if it takes only scalar values on $M_n(F)$ and is not itself an identity. Kaplansky asked whether such polynomials exist, and Formanek and Razmyslov showed that they do. Formanek then conjectured that the least degree of a central polynomial of $M_n(F)$ is $$\frac{n^2+3n-2}{2}\;=\;4,\;8,\;13,\;19,\;26\qquad\text{for } n=2,3,4,5,6 .$$ It is known to hold for $n=2$ and $n=3$ (Drensky and Kostadinov). The best general upper bounds are of order $n^2$, for example $(n-1)^2+4$ for $n>2$ (Drensky), which gives 13 for $n=4$.
In the database a central polynomial shows up as a positive multiplicity in the central quotient, the central polynomials modulo the identities. The table gives, for each order, the least degree where such a multiplicity is certified, and the degrees in which every shape is certified to have multiplicity zero.
Degrees below the first stored degree of an order are not claimed by the database. The statement for $n=3$ therefore combines the zeros in the table with the literature below degree 6.
The case $n=4$
The database records a certified central polynomial of degree 12 for $M_4(F)$, in the shape $(2^6)$ with multiplicity 1. The shape has six rows, so the polynomial involves six variables, each of degree two. It is the subject of a computer-assisted theorem in the companion manuscript, which gives an explicit integer witness, and its verification has been repeated with an independent reimplementation of the reduction and of the evaluation. To the author’s knowledge the result is new. It shows that Formanek’s value 13 is not the minimum for $n=4$: the minimum is at most 12.
This does not contradict the theorem of Drensky and Kostadinov that $M_4(F)$ has no central polynomial of degree at most 12 in two variables, since the new polynomial has six. It also leaves the exact minimum open: the table shows that degrees 8 through 11 contain no central polynomial, but nothing is stored below degree 7, and the data alone do not exclude central polynomials there.
The case $n=5$
The database records a certified central polynomial of degree 16 for $M_5(F)$, in the shape $(2^8)$ with multiplicity at least 1 (see the section on the shape $(2^{2r-2})$ below). It shows that Formanek’s value 19 is not the minimum for $n=5$ either: the minimum is at most 16. The table shows that degrees 10 to 13 contain no central polynomial; degrees 14 and 15 are only partly covered, and nothing is stored below degree 9.
The case $n=6$
For $M_6$ the stored degrees are only partly covered and no positive multiplicity has been found, so the prediction 26 is untested. Since the formula fails for $n=4$ and $n=5$, the value 26 is better read as a target to be tested than as an expected answer.
Coverage behind the table
For one order, the central quotient in each degree. A degree is complete when every one of its $p(d)$ partitions has a certified value.
A conjecture on the shape $(2^{2r-2})$
The central polynomials found so far for $M_2$, $M_3$, $M_4$ and $M_5$ suggest a pattern. The conjecture of the author is that, for every $r\ge2$, the shape $(2^{2r-2})$ (that is, $2r-2$ rows of length two, of size $4(r-1)$) occurs in the central quotient of $M_r(F)$: there is a multilinear central polynomial of degree $4(r-1)$ in $2r-2$ variables, each used twice, that is not an identity. The conjecture also says that $4(r-1)$ is the least degree of a central polynomial of $M_r(F)$. For $r=1$ the algebra is the field, every polynomial is central and the statement is empty. If true, the least degree grows linearly in $r$, whereas Formanek’s formula and the known upper bounds are quadratic.
The first three cases are the degree-4 polynomial of $M_2$, a degree-8 polynomial of $M_3$ and the degree-12 polynomial of $M_4$ discussed above. The case $r=5$ is now settled: the database records a central polynomial of degree 16 for $M_5(F)$ in the shape $(2^8)$, with multiplicity at least 1. The record rests on a computer-assisted certificate: the polynomial is written in the same Capelli-type form as the degree-12 one, its centrality over characteristic zero is verified exactly by an exhaustive enumeration (carried out on a GPU, with every batch hashed and audited), and the fact that it is not an identity is proved by a replayed evaluation modulo a prime, in two different bases. The record does not prove that the multiplicity is exactly 1 (the stored upper bound is weak), although the modular computations always gave exactly one central line modulo the identities in the family $(2^m)$, for $m=2r-2$ and for no smaller $m$.
The data also say something, but little, about the second half. Every shape of $M_4$ in degrees 8 to 11, and every shape of $M_5$ in degrees 10 to 13, is certified to carry no central polynomial. So within the stored range no central polynomial of $M_4$ or $M_5$ has degree below $4(r-1)$, except possibly degree 14 or 15 for $M_5$, which are only partly covered. Nothing is stored below degree 7 for $M_4$ or below degree 9 for $M_5$, so the minimality is open for $r\ge4$. The case $r=6$ requires degree 20 and the shape $(2^{10})$, which no computation has reached yet.
Razmyslov’s conjecture on a basis of identities
By Amitsur–Levitzki, $s_{2n}$ is an identity of $M_n(F)$. Razmyslov conjectured that in characteristic 0 it generates all identities together with a single further one, the identity of algebraicity $a_n$ (the Cayley–Hamilton-type identity that expresses algebraicity of degree $n$ over the center). It has degree $N_n=n(n+3)/2$ and uses one variable of degree $n(n+1)/2$. The statement is false for $n=3$ (Okhitin 1986; Domokos 1995) and was open for $n\ge4$ (Drensky).
The data decide it for $n=4$ and $n=5$ by a short argument. In degree $N_n$ the consequences of $a_n$ come from linear substitutions, so every two-row component of the module they generate has $\lambda_1\ge n(n+1)/2$. Two-variable consequences of $s_{2n}$ have degree at least $L(2n)=10,16,22$ for $n=3,4,5$, which exceeds $N_n=9,14,20$ (the counting of distinct words used in the manuscript for $s_{10}$). Hence a two-row identity of degree $N_n$ with $\lambda_1\lt n(n+1)/2$ is not a consequence of $s_{2n}$ and $a_n$. The table lists such shapes. The check for $n=3$ reproduces the known counterexample.
For $n=5$ the three shapes are certified in the companion manuscript (positive multiplicities in degree 20 with explicit integer polynomials); the database stores no positive lower bound for them yet.
A consequence: the least degree of an identity of $M_4(F)$ that does not follow from $s_8$ is 13 or 14, and for $M_5(F)$ it lies between 14 and 20, since the companion manuscript shows that everything in degrees up to 12 for $M_4$ and up to 13 for $M_5$ does follow from $s_{2r}$.
The Henke–Regev conjecture on $A$-identities
An $A$-identity is a polynomial whose words are supported on even permutations, that is, an identity of the algebra
with the alternating group in place of the symmetric group. The database stores these under type e.
Henke and Regev conjectured that the least degree of a nonzero $A$-identity of $M_r(F)$ is $2r+2$. For $r=2$ the least degree is
6 (Gonçalves, Schützer and Talpo), which fits. The table shows what the data say for the other orders. For $r=3$ the first
certified positive multiplicity sits in degree 11, with every stored shape of degrees 8 to 10 certified zero, so the prediction 8 is
refuted. For $r=4$ everything stored up to degree 14 is zero, which refutes the prediction 10, and the manuscript proves that 15 is the least degree. The table shows the same for $r=5$ as far as the data go.
A counterexample for $r=6$ was already given by Gonçalves and Koshlukov.
The values 6, 11 and 15 for $r=2,3,4$ suggest a different pattern. The conjecture of the author is that, for every $r>2$, the least degree of a nonzero $A$-identity of $M_r(F)$ is $4r-1$, that is 11, 15, 19, 23 for $r=3,4,5,6$, and that the case $r=2$, where the least degree is 6 and not 7, is exceptional because of the exceptional nature of the alternating group $A_6$. The least degree would then grow like $4r$, against the prediction $2r+2$ of Henke and Regev. The data agree for $r=3$ and $r=4$, where $4r-1$ is the proved minimum. For $r=5$ and $r=6$ the database records a certified positive $A$-multiplicity in degree $19$ (shape $(5^3,4)$) and in degree $23$ (shape $(6^3,4,1)$), so $4r-1$ is at least an upper bound for the least degree. Whether nothing smaller occurs is open: the degrees below are only partly covered, and for $r=5$ the data certify zero only through degree 13. Degree $4r-1$ is a candidate for the least degree, not a certified one, outside $r=3,4$.
For $r=4$ the data are zero through degree 14 and positive in degree 15, so 15 is the least degree. For $r=5$ and $r=6$ the first column shows the lowest degree with a certified positive value, not a proved minimum.
How much of the identities do the consequences of $s_{2r}$ explain?
By the Amitsur–Levitzki theorem $s_{2r}$ is an identity of $M_r(F)$, so every element of the $T$-ideal $\langle s_{2r}\rangle^T$
is an identity of $M_r(F)$. The question is how far this one identity goes. Write $d=2r+e$ and call $e$ the excess.
The $S_d$-character of the consequences of $s_n$ in degree $n+e$ is known for small $e$: Olsson and Regev settled $e=1$,
Benanti and Drensky settled $e=2$, and the author’s companion paper determines the complete characters for $e=3$ and $e=4$,
for every $n\ge1$. These characters are the multiplicities stored under type s, taken at $n=2r$. They are lower bounds for
the multiplicities of the identities (type f). Independent evaluations on matrices give upper bounds. Where the two meet,
every identity in that component is a consequence of $s_{2r}$.
The sequence of known conclusions is as follows. Leron proved that for $r>2$ every identity of degree $2r+1$ follows from $s_{2r}$. Benanti, Demmel, Drensky and Koev proved the same for degree $2r+2$ when $r=3,4,5$. The companion paper continues to degrees 11 and 12 for $M_4(F)$ and degree 13 for $M_5(F)$, that is, to excess 3 and 4 for $r=4$ and excess 3 for $r=5$. The database holds both sides of each comparison, and the table below compares them shape by shape.
A green row says that all $p(d)$ shapes of that degree have equal exact values, so every multilinear identity of the algebra in that degree follows from $s_{2r}$. For $M_4(F)$ this covers degrees 9 to 12 and for $M_5(F)$ degrees 11 to 13, in particular all identities of degree 11 of $M_4(F)$. The rows with undecided shapes mark where the evaluation bounds or the consequence characters still leave an interval, so the data neither confirm nor deny the claim there. An undecided row does not contradict a theorem. Leron’s theorem covers degree $2r+1$ for $r>2$, for example, although the database does not yet certify every shape of those rows.
The claim is not true in general. For $M_2(F)$ the Hall identity $[[x,y]^2,z]$ in degree 5 is needed together with $s_4$ (Razmyslov, Drensky), and degree 9 of $M_3(F)$ already contains identities that $s_6$ does not generate (Drensky). So how many degrees follow from the standard polynomial is something to establish shape by shape, as the table does, and not something to assume. No basis of $T(M_r(F))$ is known for $r>2$.
Support thresholds and the two-row conjecture
Bergman’s classification and the boundary multiplicities of Drensky and Kasparian reduce positivity of identity multiplicities in all sufficiently large degrees to one finite degree. The companion manuscript determines the exact thresholds $\tau_r$ for $r=2,3,4,5$ as $5,\,9,\,14,\,21$, and for $r=6$ shows $27\le\tau_6\le30$. The order-5 value comes from a zero multiplicity of the two-row shape $(11,9)$ in degree 20. This also gives that the least degree of a two-variable identity of $M_5(F)$ is 20, and it disproves a prescribed value $N_r$, which the thresholds were once expected to equal.
In every proved case the last obstruction is a two-row shape, and the manuscript conjectures that this holds for all $r\ge2$. The conjecture is open. The numbers in this paragraph come from the manuscript and its certificates and are not recomputed here.
Separately, the degree-9 data for $M_3$ settle the exactness conjecture in Section 5.2 of Benanti, Demmel, Drensky and Koev and correct a discrepancy in one of their printed tables. Both are discussed in the manuscript.
Other statements the work touches
Identities of degree $2n+2$. Drensky and Kostadinov (arXiv:2601.07750) recall, without attribution, that for $n>2$ all identities of $M_n(F)$ of degree $2n+2$ follow from $s_{2n}$. This was proved for $n=3$ (Drensky and Kasparian) and for $n=4,5$ (Benanti, Demmel, Drensky and Koev). The companion manuscript reproduces $n=5$ and goes further: degree $2n+3$ for $n=4,5$ and $2n+4$ for $n=4$. Degree $2n+3$ already fails for $n=3$ (Okhitin).
Two-variable polynomials of $M_4$. Drensky and Kostadinov proved that $M_4(F)$ has no central polynomial and no identity in two variables of degree at most 12. The database agrees and extends this: every two-row shape of degree 13 is certified zero for identities, the least two-variable identity has degree 14, and no two-variable central polynomial exists through degree 14. Degree 15 is open at $(10,5)$, $(9,6)$ and $(8,7)$, and $(12,4)$ in degree 16 carries a probable, not certified, candidate.
Order 6. The database holds a scalar candidate modulo a prime in the shape $(2^{10})$ of degree 20 for $M_6$, which would continue the pattern $4(r-1)$. It is a candidate only: centrality and the non-identity property are not certified. Formanek’s value for $n=6$ is 26.
Regev’s polynomial. Formanek’s central polynomial built from Regev’s construction is a single sum $D_S$ over blocks of lengths $1,3,\dots,2n-1$ in two sets of variables (Procesi, arXiv:2102.10657).
What the data cannot say
A finite table refutes a universal statement by one entry but can only support it by absence of counterexamples. Three limits apply to everything above. First, the database stores only the degrees and shapes that have been computed, and each order has its own range. A statement about “no polynomial of degree below $d$” is only as strong as the lowest stored degree and the coverage of every degree above it, which the tables display. Second, the status of each value says what is proved: exact values rest on two certificates, probable values on a probabilistic test, and the others are bounds. The tables above count only certified values as evidence. Third, the refutations at $n=4$ and $n=5$ and in the $A$-identity conjecture rest on explicit witnesses or certificates that anyone can re-check from the files the manuscript describes.
Identity tables
Central polynomial tables
A multilinear central polynomial of $M_r(F)$ takes only scalar values on $M_r(F)$. The tables give the multiplicity of $S^\lambda$ in the central polynomials modulo the identities, so identities themselves contribute nothing. Only positive multiplicities are stored in the tables. Bounds on the central quotient, zeros included, are kept separately and shown in the records section below.
Central determination records
For each partition examined, the lower and upper bounds on the multiplicity of $S^\lambda$ in the central quotient. Equal bounds give an exact value, and a zero here means the partition contributes no central polynomial. A gap means only an interval is certified so far.
Codimension sequences
The database in numbers
Everything on this page is computed in your browser from the loaded tables, so it is always in step with the data. Cells, dots and bars are interactive: hover for the numbers, click a heatmap cell to open the table behind it.
Certification coverage
Growth with the degree
Shape map
Use the slider or the play button to step through the degrees, and watch the cursor move across the certification heatmap above. Every partition $\lambda\vdash d$ is placed by its largest part and its number of parts. Ordinary identities of $M_r$ fill the tall and wide corners of the map completely, while the thin hook-like band near the origin is where the cocharacter lives.