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Statement

Notation (printed pp. 1--2). For a primitive $n$-th root of unity $\zeta$, the cyclotomic matroid of order $n$, $\mu_n$, is the matroid on the $n$ vectors $Z_n=\{1,\zeta,\zeta^2,\ldots,\zeta^{n-1}\}$ of the $\mathbb Q$-vector space $\mathbb Q(\zeta)$; its rank is $\phi(n)$. For a $d$-dimensional simplicial complex $\Delta^d$ with $\widetilde H_{d-1}(\Delta,\mathbb F)=0$ and exactly $n$ facets, the simplicial matroid $\mathcal S(\Delta^d,\mathbb F)$ is the matroid on the $n$ facets represented over $\mathbb F$ by the columns of the boundary map $\partial_d:\widetilde C_d(\Delta^d,\mathbb F)\to\widetilde C_{d-1}(\Delta^d,\mathbb F)$. The complex $\Delta^{r-1}_{n_1,\ldots,n_r}$ is the simplicial join of $0$-dimensional complexes $\Delta^0_{n_1},\ldots,\Delta^0_{n_r}$, where $\Delta^0_{n_i}$ consists of $n_i$ disjoint vertices; a face takes at most one vertex from each $\Delta^0_{n_i}$. The dual $M^*$ of a matroid $M$ is the matroid whose bases are the complements of the bases of $M$ (p. 1).

Theorem 1 (printed p. 2). "Let $n=p_1^{m_1}\cdots p_r^{m_r}$, with $p_1,\ldots,p_r$ distinct primes and $m_1,\ldots,m_r$ positive integers.

Then the following two matroids representable over $\mathbb Q$ are dual:

• The cyclotomic matroid $\mu_n$.
• The direct sum of $p_1^{m_1-1}\cdots p_r^{m_r-1}$ copies of $\mathcal S(\Delta^{r-1}_{p_1,\ldots,p_r},\mathbb Q)$."

The two ground sets both have $n$ elements: $\Delta^{r-1}_{p_1,\ldots,p_r}$ has $p_1\cdots p_r$ facets, one for each choice of a vertex from every $\Delta^0_{p_i}$, and the direct sum has $p_1^{m_1-1}\cdots p_r^{m_r-1}$ copies of it. The statement leaves the matching of the ground sets implicit; the proof (pp. 3--4) makes it by splitting $Z_n$ into $p_1^{m_1-1}\cdots p_r^{m_r-1}$ blocks, each a copy of $\mu_{p_1\cdots p_r}$, and, for square-free $n$, through the Chinese Remainder Theorem, which matches a facet, one residue modulo each $p_i$, with the root $\zeta^j$ whose exponent has those residues. This is a filing observation, not a review verdict.

Consequences recorded in the paper. Remark 6 (p. 5): if $n$ is divisible by at most two primes, the complex $\Delta^{r-1}_{p_1,p_2}$ is a graph, the complete bipartite graph $K_{p_1,p_2}$ (p. 3), and $\mu_n$ is cographic; if $n$ is odd, $\mu_{2n}$ is the parallel extension of $\mu_n$ with one parallel copy of each ground-set element, so that for odd primes $p,q$ the matroid $\mu_{2pq}$ is the cographic matroid of the graph obtained from $K_{p,q}$ by doubling every edge. Remark 5 (p. 5) records, citing Johnsen, that the primitive $n$-th roots of unity form a $\mathbb Q$-basis of $\mathbb Q(\zeta)$ if and only if $n$ is square-free, and identifies the corresponding basis of the simplicial matroid for square-free $n$ as a union of vertex stars, a contractible complex.

Source. Jeremy L. Martin and Victor Reiner, "Cyclotomic and simplicial matroids," arXiv:math/0402206v1 (2004), published in Israel J. Math. 150 (2005), 229--240; Theorem 1 on printed p. 2 of the arXiv preprint. Labels and pages here are the preprint's; the edition read is identified in the source digest.

Read depth. Claims checked: the definitions (pp. 1--2), Theorem 1, Lemma 3 (p. 3) and Remarks 5 and 6 (p. 5) were read clause by clause on the page images of the preprint. The proof (pp. 3--4) was read for structure only; nothing here is independently reviewed.

Proof pointer

§ 2, pp. 3--4. Lemma 3 (p. 3) says that a matroid whose ground set splits into parts whose restricted ranks add up to the full rank is the direct sum of the restrictions. With $s=p_1\cdots p_r$ and $t=n/s$, the sets $E_j=\{\zeta^j,\zeta^{j+t},\ldots,\zeta^{j+(s-1)t}\}$, $0\le j\le t-1$, partition $Z_n$, each restriction is isomorphic to $\mu_s$, and $\phi(n)=t\,\phi(s)$, so $\mu_n$ is the direct sum of $t$ copies of $\mu_s$. Since duality commutes with direct sums, it remains to treat square-free $n$. There the tensor product over the primes $p\mid n$ of the two-term complexes $\mathbb Q\to\mathbb Q^p$ is the augmented cochain complex of the join $\Delta_{p_1,\ldots,p_r}$; by the Künneth formula it is exact except at the top, where the Chinese Remainder Theorem identifies the map onto the top cohomology with $\mathbb Q^n\to\mathbb Q(\zeta)$, $e_j\mapsto\zeta^j$. So the last coboundary spans the kernel of that map, and its transpose, whose columns represent the simplicial matroid, represents the dual of $\mu_n$.

Dependencies

Lemma 3 (p. 3), which the paper calls a well-known general fact; the Künneth formula over $\mathbb Q$ and the Chinese Remainder Theorem. Remark 5 cites K. Johnsen, Lineare Abhängigkeiten von Einheitswurzeln, Elem. Math. 40 (1985), 57--59, which has no library card.

Bears on

• Problem 774: indirectly. The circuits of $\mu_n$ are the minimal rational linear relations among the $n$-th roots of unity; as circuits of a matroid are the complements of the hyperplanes of its dual, the theorem identifies them with complements of hyperplanes of the direct sum of simplicial matroids, which for $n$ with two distinct prime factors are the minimal edge cuts of a copy of $K_{p_1,p_2}$ (Remark 6). A $\mathbb Q$-linearly independent set of roots of unity is dissociated, but dissociation forbids only relations with coefficients in $\{-1,0,1\}$, so this dictionary describes a stronger condition. The theorem concerns roots of unity, not subsets of the natural numbers, and proves nothing about dissociated or proportionately dissociated sets.