Quotients of tangential 𝑘-blocks

Whittle, Geoffrey

doi:10.1090/S0002-9939-1988-0934895-4

Quotients of tangential $k$-blocks
HTML articles powered by AMS MathViewer

by Geoffrey Whittle PDF

Proc. Amer. Math. Soc. 102 (1988), 1088-1098 Request permission

Abstract:

A tangential $k$-block over $GF(q)$ is a simple matroid representaba over $GF(q)$ with critical exponent $k + 1$ for which every proper loopless minor has critical exponent at most $k$. Such matroids are of central importance in the critical problem of Crapo and Rota. In this paper we provide sufficient conditions for a quotient of a tangential $k$-block over $GF(q)$ to be also a tangential $k$-block over $GF(q)$. This enables us to show that there exist rank $r$ supersolvable tangential $k$-blocks over $GF(q)$ exactly when ${q^k} \geq r \geq k + 1$.

References

Tom Brylawski, Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975), 1–44. MR 357163, DOI 10.1090/S0002-9947-1975-0357163-6
Tom Brylawski and James Oxley, Several identities for the characteristic polynomial of a combinatorial geometry, Discrete Math. 31 (1980), no. 2, 161–170. MR 583215, DOI 10.1016/0012-365X(80)90032-1
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory: Combinatorial geometries, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR 0290980
Richard P. Stanley, Modular elements of geometric lattices, Algebra Universalis 1 (1971/72), 214–217. MR 295976, DOI 10.1007/BF02944981
R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. MR 309815, DOI 10.1007/BF02945028
D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112
Dominic Welsh, Colouring problems and matroids, Surveys in combinatorics (Proc. Seventh British Combinatorial Conf., Cambridge, 1979) London Math. Soc. Lecture Note Ser., vol. 38, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 229–257. MR 561312
D. J. A. Welsh, Colourings, flows and projective geometry, Nieuw Arch. Wisk. (3) 28 (1980), no. 2, 159–176. MR 582924
Geoff Whittle, Modularity in tangential $k$-blocks, J. Combin. Theory Ser. B 42 (1987), no. 1, 24–35. MR 872405, DOI 10.1016/0095-8956(87)90060-8

Some aspects of the critical problem for matroids