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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.
Let $Lsubset mathbb{Z}^n$ be a lattice and $I_L=langle x^{bf u}-x^{bf v}: {bf u}-{bf v}in Lrangle$ be the corresponding lattice ideal in $Bbbk[x_1,ldots, x_n]$, where $Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I
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
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
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polyno
The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $mathbb Q[x_1, dots, x_n]$ to a corresponding ideal in $mathbb F_p[x_1,dots, x_n]$ where $p$ is a prime number; in other words, the textit{reduction modulo