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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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