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Hodge theory for combinatorial geometries. (English) Zbl 1442.14194
The notion of a matroid, which apriori was introduced as a structure characterizing independence of vector spaces and graphs [H. Whitney, Am. J. Math. 57, 509–533 (1935; Zbl 0012.00404)], is of fundamental importance in graph theory. A (finite) matroid \(M\) is given by a pair \((E,\ell)\), where \(E\) is a finite set, and \(\ell\) is a family of subsets of \(E\) (referred to as independent sets) which satisfy certain properties, namely the defining properties of a closure operator and the Mac Lane-Steinitz exchange property.
The chromatic polynomial \(\chi_G\) associated to a graph \(G\) (which is a function whose value at a positive integer \(q\), \(\chi_G(q)\) is the number of colorings of \(G\) using \(q\) colors) is generalised to the set up of matroids by Rota, and is known as the characteristic poynomial associated to a matroid \(M\) [G.-C. Rota, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 340–368 (1964; Zbl 0121.02406)], defined by \[ \chi_M(\lambda) = \sum_{\ell \subset E} (-1)^{|\ell|} \lambda^{\mathrm{crk}(\ell)} \] where the sum is over all subsets \(\ell \subseteq E\) and \(\mathrm{crk}(\ell)\) is the corank of \(\ell\) in \(M\).
A fundamental conjecture that was open in matroid theory, stated in 70’s proposes that the absolute values of coefficients of the characteristic polynomial associated to a matroid \(M\) form a log-concave sequence. More precisely, writing \(r + 1\) for the rank of \(M\), which corresponds to the maximum size of an independent set in the matroid, and letting \(w_k(M)\) denote the absolute value of the coefficient of \(\lambda^{r-k+1}\) in the characteristic polynomial of \(M\), the log-concavity would imply: \[ w_{k-1}(M)w_{k+1}(M) \leq w_k(M)^2 ~ \text{ for all } 1 \leq k \leq r. \] A related conjecture which concerns the number of independent subsets of E of given cardinality was formulated by D. J. A. Welsh [in: Proceedings of a conference of combinatorial mathematics and its applications, 1969. London: Academic Press. 291–306 (1971; Zbl 0233.05001)] and J. H. Mason [“Matroids: unimodal conjectures and Motzkin’s theorem”, in: Proceedings of a conference of combinatorial mathematics and its applications, 1972. London: Academic Press. 207–220 (1972)]. The authors prove all these conjectures in full generality by constructing a “cohomology ring” associated to \(M\) that satisfies the hard Lefschetz theorem and the Hodge-Riemann relations.

MSC:
14T15 Combinatorial aspects of tropical varieties
05A99 Enumerative combinatorics
05E99 Algebraic combinatorics
14F99 (Co)homology theory in algebraic geometry
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