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The Whitney algebra of a matroid. (English) Zbl 0964.05018
Although introduction of technical detail would make this review excessively long, we note that this very substantial paper of 48 printed pages does well serve as an excellent introduction not only to the Whitney algebra of a matroid \(M\), but also as a demonstration of how this algebra \(W\) associated with \(M\) in a rather intricate (but eventually also understood as natural) way does indeed permit algebraization of important questions which may then be answered in a manner which yields interesting properties of such algebras, thus permitting efficient implementation of further arguments via these properties. For example, it is shown that if a matroid \(M\) is representable, then no product of independent words (juxtapositions of finite collections of independent elements) is equal to zero in the Whitney algebra \(W\) of \(M\). Using the theory developed one is e.g. able to demonstrate “by calculation” that the Fano matroid is only representable over fields of characteristic two via a result which follows from the observation quoted. An added feature of historical interest is the last section which not only shows the path taken in the development of this research but provides one more record of the role played by a great mathematician (i.e., G.-C. Rota) in steering a very complicated set of intuitions to a finished structure that properly describes in an optimal way the interrelationship of a package of ideas not heretofore tuned quite so well.

05B35 Combinatorial aspects of matroids and geometric lattices
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17A99 General nonassociative rings
08A99 Algebraic structures
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Full Text: DOI
[1] Barnabei, M.; Brini, A.; Rota, G.-C., On the exterior calculus of invariant theory, J. algebra, 96, 120-160, (1985) · Zbl 0585.15005
[2] Birkhoff, G., Abstract linear dependence in lattices, Amer. J. math., 57, 800-804, (1935) · Zbl 0013.00104
[3] Crapo, H.; Rota, G.-C., The resolving bracket, (), 197-222 · Zbl 0927.68104
[4] Doubilet, P.; Rota, G.-C.; Stein, J., On the foundations of combinatorial theory: IX combinatorial methods in invariant theory, Stud. appl. math., 8, 185-215, (1974) · Zbl 0426.05009
[5] Dress, A.; Wenzel, W., Geometric algebra for combinatorial geometries, Adv. in math., 77, 1-36, (1989) · Zbl 0684.05013
[6] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate texts in mathematics, 150, (1995), Springer-Verlag New York
[7] Fenton, N.E., Matroid representations—an algebraic treatment, Quart. J. math. Oxford, 35, 263-280, (1984) · Zbl 0546.05019
[8] Graves, W., An algebra associated to a combinatorial geometry, Bull. amer. math. soc., 77, 757-761, (1971) · Zbl 0225.05020
[9] Grosshans, F.D.; Rota, G.-C.; Stein, J., Invariant theory and superalgebras, Regional conference series in mathematics, (1987), American Mathematical Society Providence · Zbl 0648.15020
[10] Mac Lane, S., Some interpretations of abstract linear dependence in terms of projective geometry, Amer. J. math., 58, 236-240, (1936) · JFM 62.0649.02
[11] Mac Lane, S., Categories for the working Mathematician, Graduate texts in mathematics, 5, (1971), Springer-Verlag New York
[12] LeClerc, B., On identities satisfied by minors of a matrix, Adv. in math., 100, 101-132, (1993) · Zbl 0804.05074
[13] Muir, T., The theory of determinants in the historical order of development, (1960), Dover New York · JFM 45.1242.04
[14] Orlik, P.; Solomon, L., Combinatorics and topology of hyperplanes, Invent. math., 56, 167-189, (1980) · Zbl 0432.14016
[15] Vamos, P., A necessary and sufficient condition for a matroid to be linear, (), 166-173 · Zbl 0374.05017
[16] White, N.L., The bracket ring and combinatorial geometry, (1971), Harvard University
[17] White, N.L., The bracket ring of a combinatorial geometry I, Trans. amer. math. soc., 202, 79-95, (1975) · Zbl 0303.05022
[18] White, N.L., The bracket ring of a combinatorial geometry II: unimodular geometries, Trans. amer. math. soc., 214, 233-248, (1975) · Zbl 0294.05013
[19] Whitney, H., On the abstract properties of linear dependence, Amer. J. math., 57, 509-533, (1935) · JFM 61.0073.03
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