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Math-3280-A1 · revision r1 · on Math-3280

A division-free partition formula for elementary symmetric functions of a sum of matrices

Full solution claimed · LLM-generated · Submitted by Wentao Li · Wentao Li · gpt 6.1 sol

Unverified claim of a full solution. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: full solution claimed

A finite division-free decorated-partition formula gives elementary symmetric functions of a sum of arbitrary noncommuting matrices over every field, with integral specialization to commutative rings and all recorded boundary cases.

Limitations: The source paper's separate positivity conjecture is outside the recorded target.; No new mathematical priority is claimed; same-model audits and exact checks are not independent human or formal verification.; Unverified; no independent community acceptance or formal certification asserted.

Contributor’s completion estimate: 100%. Solved (LLM claim), 100% of the entire recorded catalogue formula target, using established Amitsur and Newton-Girard mathematics. Not a probability of correctness.

Manuscript

Contents
  1. A division-free partition formula for elementary symmetric functions of a sum of matrices
  2. Abstract
  3. Original target and established context
  4. Primitive words and the integral product identity
  5. The finite partition expression
  6. Boundary cases.
  7. The characteristic-zero trace partition formula
  8. Examples and checks of order, signs, and repetition
  9. Verification, scope, and contributions
  10. Exact computation.
  11. Full-solution scope declaration.
  12. Process and attribution.
  13. Catalogue provenance.
  14. Completion Estimate
  15. References

A division-free partition formula for elementary symmetric functions of a sum of matrices

OpenAI GPT 6.1 Sol (Max reasoning)

This research was conducted under the direction and oversight of Wentao Li.

12 October 2026

Abstract

We give a finite sum indexed by integer partitions of $k$ for $S_k(A_1+\cdots+A_N)$, using only elementary symmetric functions of ordered matrix words. It applies over every field, including positive characteristic, with no assumption that the matrices commute. The answer organizes the established Amitsur cyclic-word formula by the total degrees of its selected invariants. A self-contained proof and exact checks are supplied.

Original target and established context

For $A\in M_n(F)$, where $F$ is any field, set \[ S_j(A)=\sum_{\substack{I\subseteq\{1,\ldots,n\}\\|I|=j}} \det A[I,I],\qquad S_0(A)=1. \] Thus $S_n(A)=\det A$, $S_1(A)=\operatorname{tr} A$, and \begin{equation}\label{wsref-0-64-65-74-2d-67-65-6e-65-72-61-74-69-6e-67} \det(I_n-tA)=\sum_{j=0}^n(-1)^jS_j(A)t^j. \end{equation} For convenience set $S_j(A)=0$ for $j>n$. Determinants of zero-sized matrices and empty products are $1$.

The recorded target [catalogue,costas] is to find a closed expression organized by integer partitions for \[ S_k(A_1+\cdots+A_N),\qquad 0\leq N\leq k\leq n, \] in matrix quantities built from the $A_i$, permitting elementary symmetric functions of their ordered words. Every matrix tuple over every field is included; the empty sum is the zero matrix, and the empty partition is allowed when $k=0$. No commutativity hypothesis is present.

Costas-Santos [costas, §§1,5] gives low-degree formulas and asks for a general closed expression involving partitions and elementary symmetric functions of words. The arXiv record dates the first submission to 16 December 2006 and version 5 to 17 September 2009; the later date displayed by the HTML conversion is not a new mathematical publication.

Amitsur's 1980 paper [amitsur] addresses characteristic coefficients of sums in terms of coefficients of words. We consulted its publisher abstract and bibliographic record, but could not access its full text. The full primary paper of Reutenauer and Schützenberger [rs] was visually inspected: its theorem on p. 300 and equations $6$--$7$ on p. 301 give the cyclic-word formula over a commutative ring; its p. 302 addendum credits Amitsur. The reference request [mo] was a bibliographic lead. The present answer applies this established mathematics to the exact recorded target and gives its partition organization; no novelty of the underlying identities is claimed. This does not revise the catalogue's original open source label.

Primitive words and the integral product identity

A word $w=(i_1,\ldots,i_d)$ in the alphabet $\{1,\ldots,N\}$ has length $|w|=d$ and matrix value $A_w=A_{i_1}\cdots A_{i_d}$, in precisely that order. A nonempty word is primitive if it is not a power $v^r$ with $r>1$. Identify words only under cyclic rotation, and choose the lexicographically least representative of each primitive cyclic class. Let $\mathcal C_N$ be these representatives and $\mathcal C_{N,k}=\{w\in\mathcal C_N:|w|\leq k\}$. These sets depend only on the alphabet, not on the matrix entries. When $N=0$ there are no nonempty words.
Lemma (Cyclic invariance): For matrices $X,Y$ of the same size over any commutative ring $R$, $\det(I-uXY)=\det(I-uYX)$ in $R[u]$. Consequently $S_j(A_w)$ is independent of the cyclic representative of $w$.
Proof: Eliminate either off-diagonal block in $\left(\begin{smallmatrix}I&uX\\Y&I\end{smallmatrix}\right)$ by multiplication with a block triangular matrix with identity diagonal blocks. The two resulting determinants are $\det(I-uXY)$ and $\det(I-uYX)$. Compare coefficients using (det-generating). Every one-step cyclic rotation has the form $XY\mapsto YX$. ∎
Lemma (Amitsur product, with integral specialization): For any finite tuple of $n\times n$ matrices over a commutative ring $R$, \begin{equation}\label{wsref-0-61-6d-69-74-73-75-72-2d-70-72-6f-64-75-63-74} \det\left(I_n-t\sum_{i=1}^N A_i\right) =\prod_{w\in\mathcal C_N}\det(I_n-t^{|w|}A_w) \quad\hbox{in }R[[t]]. \end{equation} The product is formal: its coefficients through degree $k$ depend only on the finite set $\mathcal C_{N,k}$.
Proof: First use generic matrices $X_1,\ldots,X_N$ whose entries are distinct indeterminates, and let $P=\mathbb Z[x_{i,a,b}]$ and $Q=\mathbb Q[x_{i,a,b}]$. Put $B=\sum_iX_i$. For any matrix $M$ over $Q$, the scalar formal series $f(t)=\det(I-tM)$ has constant term $1$. Differentiating the determinant by multilinearity, and using the adjugate identity, gives \[ \frac{f'(t)}{f(t)}=-\operatorname{tr}\left(M(I-tM)^{-1}\right) =-\sum_{m\geq1}\operatorname{tr}(M^m)t^{m-1}. \] Termwise integration with zero constant term yields \begin{equation}\label{wsref-0-6c-6f-67-2d-64-65-74-65-72-6d-69-6e-61-6e-74} \log\det(I-tM)=-\sum_{m\geq1}\frac{\operatorname{tr}(M^m)}{m}t^m. \end{equation} All operations here are formal over a rational algebra.

Expand $B^m$ as the sum of all ordered words of length $m$. The cyclic orbit of such a word has size $d$ for a unique divisor $d$ of $m$: the rotations fixing the word form a subgroup of $\mathbb Z/m\mathbb Z$, so the least positive period $d$ divides $m$. The word is the $(m/d)$-th power of a primitive word of length $d$. Each primitive cyclic class of length $d$ produces exactly one such orbit, with exactly $d$ distinct linear words. Their traces agree by cyclic invariance of trace. Therefore \begin{equation}\label{wsref-0-6f-72-62-69-74-2d-74-72-61-63-65} \operatorname{tr}(B^m)=\sum_{d\mid m}d \sum_{w\in\mathcal C_N,\ |w|=d} \operatorname{tr}(X_w^{m/d}). \end{equation} Using $m=dr$ in (log-determinant) gives \[ \begin{aligned} \log\det(I-tB) &=-\sum_{w\in\mathcal C_N}\sum_{r\geq1} \frac{\operatorname{tr}(X_w^r)}r t^{r|w|}\\ &=\sum_{w\in\mathcal C_N}\log\det(I-t^{|w|}X_w). \end{aligned} \] Each coefficient receives only finitely many contributions. Formal exponentiation proves (amitsur-product) over $Q$.

Both sides of that identity have coefficients in $P$: each right-hand factor is a determinant with integer polynomial coefficients, and only finitely many factors affect a fixed degree. Since $P\hookrightarrow Q$ is injective, equality already holds coefficient by coefficient in $P$. Now specialize the indeterminates to the entries of any given matrices over $R$ via the homomorphism $P\to R$. This proves the identity over $R$, including finite fields and rings of positive characteristic. If $N=0$, both sides are $1$, with an empty product on the right. ∎

The finite partition expression

Write $\lambda\vdash k$ for an integer partition $\lambda=(\lambda_1\geq\cdots\geq\lambda_\ell>0)$ of $k$. For each such partition let $D_\lambda(N,n)$ be the following finite, explicitly enumerable set of tuples: \begin{equation}\label{wsref-0-64-65-63-6f-72-61-74-69-6f-6e-73} \begin{split} D_\lambda(N,n)=\bigl\{((w_1,r_1),\ldots,(w_\ell,r_\ell)): & w_h\in\mathcal C_{N,k},\quad 1\leq r_h\leq n,\quad r_h|w_h|=\lambda_h;\\ &w_h\ne w_j\ (h\ne j);\\ &w_h<w_{h+1}\text{ whenever }\lambda_h=\lambda_{h+1}\bigr\}. \end{split} \end{equation} The last comparison uses lexicographic order and fixes the order inside each block of equal parts. For the empty partition, $D_\varnothing(N,n)$ contains the empty tuple. An empty inner sum is $0$.
Theorem (Closed partition formula): For every field $F$, every $A_1,\ldots,A_N\in M_n(F)$, and $0\leq N\leq k\leq n$, one has \begin{equation}\label{wsref-0-70-61-72-74-69-74-69-6f-6e-2d-66-6f-72-6d-75-6c-61} \boxed{\displaystyle S_k\left(\sum_{i=1}^N A_i\right)= \sum_{\lambda\vdash k} \sum_{((w_h,r_h))_{h=1}^{\ell}\in D_\lambda(N,n)} (-1)^{\,k+\sum_{h=1}^{\ell}r_h} \prod_{h=1}^{\ell}S_{r_h}(A_{w_h}).} \end{equation} The same formula holds for every finite $N$ and $k\geq0$ over any commutative ring, with the convention $S_k=0$ for $k>n$.
Proof: In the product (amitsur-product), the factor for $w$ is \[ 1+\sum_{r=1}^n(-1)^r S_r(A_w)t^{r|w|}. \] To obtain degree $k$, choose a nonconstant term from finitely many distinct factors, each supplying an index $r_h$. The selected degrees $r_h|w_h|$ have sum $k$. Sort them into an integer partition $\lambda$, and sort the representatives within equal-degree blocks. This is a bijection between product selections and the tuples in (decorations); no factorial or extra cyclic-orbit multiplicity is needed. Conversely every such tuple specifies one selection. Its coefficient is $(-1)^{\sum_h r_h}\prod_h S_{r_h}(A_{w_h})$. The degree-$k$ coefficient on the left is $(-1)^k S_k(\sum_iA_i)$, so multiplication by $(-1)^k$ proves (partition-formula). ∎

This is a closed finite expression rather than a recurrence: for each part $s$, enumerate primitive representatives of lengths $d\mid s$ with $d\leq k$, then take $r=s/d\leq n$ and enforce the displayed distinctness and ordering rules. Every scalar invariant in the answer is an $S_r$ of an ordered word in the input matrices. Each part records its total degree $r|w|$ in those matrices. Cyclic rotation is allowed; reversal and other permutations of letters are not identified.

Boundary cases.
For $k=0$ the only partition and decoration are empty, so the answer is $1$; within the recorded range this forces $N=0$. For $N=0$ and $k>0$ every decoration set is empty, giving $0$. For $N=1$ the only primitive representative is the single letter $1$: only $\lambda=(k)$ contributes when $k>0$, giving $S_k(A_1)$ with sign $(-1)^{2k}=1$. The case $k=n$ is the required determinant formula. If $n=0$ is permitted, the sole recorded case is again $N=k=0$. Singular matrices and zero input matrices need no separate hypothesis.

The characteristic-zero trace partition formula

For comparison, when $F$ has characteristic zero define \[ p_j=\sum_{(i_1,\ldots,i_j)\in\{1,\ldots,N\}^j} S_1(A_{i_1}\cdots A_{i_j}) =\operatorname{tr}\left(\left(\sum_i A_i\right)^j\right), \qquad z_\lambda=\prod_{j\geq1}j^{m_j(\lambda)}m_j(\lambda)!. \] Here $m_j(\lambda)$ is the multiplicity of the part $j$ and $\ell(\lambda)=\sum_jm_j(\lambda)$. The familiar Newton--Girard partition expression is \begin{equation}\label{wsref-0-74-72-61-63-65-2d-70-61-72-74-69-74-69-6f-6e} S_k\left(\sum_i A_i\right)= \sum_{\lambda\vdash k} \frac{(-1)^{k-\ell(\lambda)}}{z_\lambda} \prod_{j\geq1}p_j^{m_j(\lambda)}. \end{equation} Indeed, substitute $-t$ in (log-determinant) and expand \[ \det(I+tB)=\prod_{j\geq1} \exp\left(\frac{(-1)^{j-1}p_j}{j}t^j\right). \] Choosing multiplicity $m_j$ supplies the denominator $j^{m_j}m_j!$ and sign $(-1)^{\sum_j(j-1)m_j}=(-1)^{k-\ell(\lambda)}$. This proves (trace-partition) without commuting any matrix letters. Equation (partition-formula) supplies the arbitrary-characteristic answer. For example, over $\mathbb F_2$ the matrices $I_2$ and $0_2$ have identical traces of every positive power, but their $S_2$ values are $1$ and $0$. Ordinary traces of words alone cannot distinguish this one-letter example; higher $S_r$ are essential.

Examples and checks of order, signs, and repetition

For $n\geq2$ the degree-two specialization is \begin{equation}\label{wsref-0-64-65-67-72-65-65-2d-74-77-6f} S_2\left(\sum_iA_i\right) =\sum_i S_2(A_i)+\sum_{i<j} \left(S_1(A_i)S_1(A_j)-S_1(A_iA_j)\right). \end{equation} For $A=E_{12}$ and $B=E_{21}$ in dimension $2$, $AB=E_{11}$ and $BA=E_{22}$, so the pair does not commute. Equation (degree-two) gives $S_2(A+B)=-1$, agreeing with the determinant of $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$; over $\mathbb F_2$ this value is $1$.

For two letters the primitive representatives through length $4$ are \[ A,\ B;\quad AB;\quad AAB,\ ABB;\quad AAAB,\ AABB,\ ABBB. \] In particular $ABAB$ is periodic, and is accounted for by $S_2(AB)$ in the factor belonging to $AB$. For $n\geq4$ the formula gives \begin{equation}\label{wsref-0-64-65-67-72-65-65-2d-66-6f-75-72} \begin{aligned} S_4(A+B)={}&\sum_{\substack{a,b\geq0\\a+b=4}}S_a(A)S_b(B) -S_1(AB)\sum_{\substack{a,b\geq0\\a+b=2}}S_a(A)S_b(B)\\ &+\left(S_1(A^2B)+S_1(AB^2)\right)\left(S_1(A)+S_1(B)\right)\\ &-S_1(A^3B)-S_1(A^2B^2)-S_1(AB^3)+S_2(AB). \end{aligned} \end{equation} This recovers the source's fourth-degree invariant language in every characteristic.

Order beyond cyclic rotation matters already for $N=k=n=3$. Take $A_1=E_{12}$, $A_2=E_{23}$, $A_3=E_{31}$. Then $S_1(A_1A_2A_3)=1$ and $S_1(A_1A_3A_2)=0$. These are distinct primitive cyclic classes. In degree $3$ they are the only possibly nonzero terms of (partition-formula); their sum is $1$, which equals $S_3(A_1+A_2+A_3)$ over every field.

Verification, scope, and contributions

Exact computation.
Run python3 scripts/check_matrix_sum_3027.py from the repository root. The checker uses Python 3.13.2, its standard library, and seed $3027$. Its oracle sums principal minors using the Leibniz determinant, separately from both the decorated-partition sum and the primitive-word product. All $2293$ coefficient cases and $469$ auxiliary checks passed. The comparisons cover integer examples through dimension $6$ and every $0\leq N\leq n$; rational examples; finite fields $\mathbb F_2,\mathbb F_3,\mathbb F_5,\mathbb F_7$ through dimension $5$; every one of the $256$ ordered pairs of $2\times2$ matrices over $\mathbb F_2$; and small generic polynomial pairs in dimensions $2$ and $3$ with respectively $8$ and $18$ independent variables. Additional examples over $\mathbb Z/4,\mathbb Z/6,\mathbb Z/8$ test ring specialization. Some checks extend past degree $n$, where coefficients must vanish. The auxiliary checks cover primitive-word counts, periodic trace multiplicities, cyclic invariance, and the trace obstruction. The program also checks (trace-partition) over the rationals.
Full-solution scope declaration.
Theorem Theorem (Closed partition formula) answers the entire recorded target: an explicit finite integer-partition expression in the permitted invariants, for every matrix tuple, every field, and every $0\leq N\leq k\leq n$, including all boundary cases. Lemmas Lemma (Cyclic invariance) and Lemma (Amitsur product, with integral specialization) prove the needed identities; the proof uses only elementary determinants, formal series, finite cyclic orbit counting, and specialization of integer polynomials. The established Amitsur and Newton--Girard machinery is attributed and re-proved here. There is no asserted remaining mathematical gap for this target. The separate positivity conjecture in the source is not part of catalogue 3027's recorded question and is not addressed by this answer.
Process and attribution.
OpenAI GPT 6.1 Sol (Max reasoning) generated the proof, manuscript, and verification code. Research directed and overseen by Wentao Li. No other model was used. Filled mandatory Prompts A and B, draft and audit snapshots, source-access notes, and the exact output report are retained in the local research directory research/matrix-sum-3027-2026-10-12/, outside this repository. The checker is included with this contribution; those research notes are not asserted to have been publicly posted. The audit is a same-model review. No independent human proof verification or theorem-prover formalization has been reported; the proofs remain unverified in those senses. Computation does not replace the general proof.
Catalogue provenance.
The target and identity metadata derive from ULAM.AI and the UnsolvedMath Contributors [catalogue], under CC BY 4.0. This separately attributed attempt adds the formula, proof, and checks; it preserves the imported target and source status. Source/literature and repository collision checks were performed on 12 October 2026. No earlier 3027 attempt or matching pull request was found in the checked snapshot and searches; this is not a claim of an exhaustive historical literature search or new mathematical priority.

Completion Estimate

COMPLETION: 100% Theorem Theorem (Closed partition formula) supplies the requested closed partition expression for all recorded fields, matrix tuples, parameter values, and boundary cases. This is a full-solution claim for the recorded target using established mathematics, subject to the verification limits stated above.

References