Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line i
nsertion. In this work we revisit this problem. First we show that every totally positive matrix can be associated to a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.
We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra, traditionally
defined by generators and relations, can be seen as subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of quantum matrices as a sum over paths in the graph, each path being assigned an element of the quantum torus. The quantum matrices relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchons deleting-derivations algorithm. The second part of this paper is to apply the paths viewpoint to the theory of torus-invariant prime ideals of quantum matrices. We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only for char(K)=0 and q transcendental over Q. Our strategy is to show that the quantum minors in a given torus-invariant ideal form a Grobner basis.
We present a combinatorial method to determine the dimension of $C{H}$-strata in the algebra of $mtimes n$ quantum matrices $Oq$ as follows. To a given $C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the
dimension of the $C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $C{H}$-strata in $Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $C{H}$-strata in $Oq$.
We take a graph theoretic approach to the problem of finding generators for those prime ideals of $mathcal{O}_q(mathcal{M}_{m,n}(mathbb{K}))$ which are invariant under the torus action ($mathbb{K}^*)^{m+n}$. Launois cite{launois3} has shown that the
generators consist of certain quantum minors of the matrix of canonical generators of $mathcal{O}_q(mathcal{M}_{m,n}(mathbb{K}))$ and in cite{launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindstr{o}m cite{lind} and Gessel-Viennot~cite{gv} to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.
