A double Sylvester determinant

1. Problem Statement

The paper investigates the properties of a specific matrix $W$ constructed from two $(n+1) \times (n+1)$ matrices, $A$ and $B$, over a commutative ring $K$.

• Construction of $W$:

  • Let $n, k$ be non-negative integers with $k \leq n$.
  • Let $P_k$ be the set of all $k$-element subsets of $\{1, 2, \dots, n\}$.
  • For any subset $S \subseteq \{1, \dots, n\}$, let $S^+ = S \cup \{n+1\}$.
  • The matrix $W$ is a $|P_k| \times |P_k|$ matrix (size $\binom{n}{k} \times \binom{n}{k}$).
  • The entry at position $(I, J)$ (where $I, J \in P_k$) is defined as:
    $$W_{I,J} = \det(\text{sub}_{I^+}^{J^+} A) \cdot \det(\text{sub}_{I^+}^{J^+} B)$$
  • Here, $\text{sub}_{I^+}^{J^+} A$ denotes the $(k+1) \times (k+1)$ submatrix of $A$ formed by rows $I^+$ and columns $J^+$. Crucially, these minors must include the last row ($n+1$) and the last column ($n+1$).

• The Core Question: What can be said about the determinant $\det(W)$?

  • In the general case, $\det(W)$ is a highly complex, irreducible polynomial in the entries of $A$ and $B$ with no obvious factorization.
  • The paper seeks to identify conditions under which $\det(W)$ factors into simpler terms involving $\det(A)$ and $\det(B)$.

2. Methodology

The author employs a combination of classical linear algebra, commutative algebra, and polynomial ring theory. The proofs are indirect, relying on the properties of Unique Factorization Domains (UFDs) rather than direct combinatorial expansion.

• Reduction to Polynomial Rings: To prove divisibility properties in a general commutative ring, the author assumes without loss of generality (WLOG) that the ring $K$ is a polynomial ring over $\mathbb{Z}$ with indeterminates representing the matrix entries. This allows the use of algebraic properties like irreducibility and coprimality.
• Localization: The proofs utilize the localization of rings at multiplicative subsets (specifically powers of the bottom-right entry $a_{n+1,n+1}$) to transform the problem into a setting where certain entries are invertible, simplifying matrix operations.
• Gauss's Lemma and Content: The concept of the "content" of a polynomial (the GCD of its coefficients) is used to establish that specific polynomials (like $\det A$) are irreducible and coprime to others.
• Linear Algebra over Integral Domains: A key lemma (Proposition 3.7) states that if a matrix $W$ over an integral domain $M$ has a non-zero vector $u$ in its kernel ($Wu=0$), then $\det(W) = 0$. The proof strategy involves constructing such a vector $u$ in a quotient ring.
• Cauchy-Binet Formula: Lemma 3.8 (a version of the Cauchy-Binet formula) is used to expand the determinant of a product of matrices, which is essential for deriving the kernel relation $Wu=0$.

3. Key Contributions and Results

The paper establishes two main theorems regarding the factorization of $\det(W)$:

Theorem 2.1: Divisibility by $\det(A)$

• Condition: The bottom-right entry of matrix $B$ is zero ($b_{n+1,n+1} = 0$).
• Result: $\det(A)$ divides $\det(W)$.
  $$\det(A) \mid \det(W)$$
• Mechanism: The proof constructs a specific non-zero vector $u$ in the quotient ring $K/(\det A)$ such that $Wu = 0$. By Proposition 3.7, this implies $\det(W) \equiv 0 \pmod{\det A}$.

Theorem 2.2: Divisibility by $\det(A)\det(B)$

• Condition: The bottom-right entries of both matrices are zero ($a_{n+1,n+1} = 0$ and $b_{n+1,n+1} = 0$).
• Result: The product of the determinants divides $\det(W)$.
  $$(\det A)(\det B) \mid \det(W)$$
• Mechanism: Since the roles of $A$ and $B$ are symmetric in the construction of $W$, Theorem 2.1 implies $\det(A) \mid \det(W)$ and $\det(B) \mid \det(W)$. The proof then demonstrates that $\det(A)$ and $\det(B)$ are coprime polynomials in the generic case (due to disjoint sets of variables and content 1). In a UFD, if two coprime elements divide a third, their product must also divide it.

Generalization of Previous Work

• The paper generalizes a result by Olver and Grinberg (2018) concerning $\binom{n}{2} \times \binom{n}{2}$ determinants arising from the $n$-body problem. By setting $k=2$ and specific boundary conditions, the previous result is recovered as a special case of Theorem 2.2.

4. Significance and Limitations

• Theoretical Significance:

  • The results extend Sylvester's classical determinant identity (which relates minors of a matrix to the determinant of the matrix itself) to a "double" setting involving two matrices.
  • It provides a rigorous algebraic framework for understanding when complex determinants of compound matrices factorize.
  • It connects combinatorial matrix theory with the structural properties of polynomial rings and integral domains.

• Limitations and Open Questions:

  • The Quotient: While the paper proves that $\det(W)$ is a multiple of $\det(A)$ (or $\det(A)\det(B)$), it does not provide an explicit formula for the quotient $\frac{\det(W)}{\det(A)}$ or $\frac{\det(W)}{\det(A)\det(B)}$. The author notes that this quotient remains a "mysterious" and complicated polynomial.
  • Proof Directness: The proofs rely heavily on abstract algebraic machinery (localization, UFDs). The author poses the question of whether a more direct, "combinatorial" proof exists that stays within the original ring $K$ without invoking polynomial rings.
  • Generalization to $m$ Rows/Columns: The paper asks whether these results can be extended to matrices where the minors use the last $m$ rows and columns (for $m > 1$), rather than just the last row and column.

Summary

Darij Grinberg's paper resolves the factorization properties of a specific "double" compound determinant. By leveraging the algebraic structure of polynomial rings, the author proves that if the bottom-right entries of the input matrices vanish, the determinant of the constructed matrix $W$ is divisible by the determinant of the input matrices. This work bridges classical determinant identities with modern algebraic techniques, offering new insights into the factorization of complex matrix polynomials while leaving the explicit form of the remaining factors as an open problem.