The covering number of the elements of a matroid and generalized matrix functions
type: Item
instance of: scholarly article
title: The covering number of the elements of a matroid and generalized matrix functions
zbMATH Open document ID: 0889.05036
DOI: 10.1016/S0024-3795(98)90071-4
author: J. A. Dias Da Silva
author: Rosário Fernandes
author: Amélia Fonseca
published in: Linear Algebra and its Applications
publication date: 1998-05-29T00:00:00Z
review text: For a matroid \(M\) on a ground set \(S\) of cardinality \(m\) one defines a matroid \(M^{(k)}\) whose independent sets are the unions of \(k\) independent sets of \(M\). In particular, \(M^{(1)} = M\) and if there is no loop in \(M\) then \(M^{(m)}\) is the Boolean algebra on the \(m\)-element set \(S\). The covering number for an element \(x \in S\) is the least number \(s_x(M)\) such that \(x\) lies in all bases of \(M^{(s_x(M))}\), i.e., \(s\) is a coloop in \(M^{(s_x(M))}\). The maximum of all \(s_x(M)\) where \(x \in S\) is called the covering number of \(M\). Clearly, it is the least number \(k\) such that \(M^{(k)} = M^{(m)}\). The paper studies connections between the value of the covering number and other structure invariants of a matroid. As an application the authors partially describe the regular complex \(n\) by \(n\) matrices \(B\) that for a given \(n\) by \(n\) matrix \(A\) and character \(\chi\) of \(S_n\) leave the value at \(A\) of the immanent corresponding to \(\chi\) invariant. The immanent of an \(n\) by \(n\) matrix is given by the same polynomial in the matrix entries as the determinant, except that a monomial is weighted by the character value of \(\chi\) at the permutation corresponding to the monomial. In particular, the determinant is the immanent for the sign character. The authors also give extensions of this result to generalized matrix functions.
reviewed by: Volkmar Welker
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 52B40
zbMATH DE Number: 1116051
zbMATH Keywords: matroid
zbMATH Keywords: covering number
zbMATH Keywords: generalized matrix function
MaRDI profile type: Publication
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full work available at URL: s0024-3795(98)90071-4
OpenAlex ID: W4256715731
description: scientific article; zbMATH DE number 1116051