The covering number of the elements of a matroid and generalized matrix functions
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.
scientific article; zbMATH DE number 1116051
scientific article