Quotients of tangential 𝑘-blocks

Whittle, Geoffrey

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

Quotients of tangential $k$-blocks
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by Geoffrey Whittle

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

DOI: https://doi.org/10.1090/S0002-9939-1988-0934895-4

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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 ${q^k} \geq r \geq k + 1$.

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Some aspects of the critical problem for matroids