I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem:
Let $A (n \times n)$ be a lower triangular Toeplitz matrix (that is, a Toeplitz matrix without the upper triangular part). The Smith invariant factors of $A([k-1,\dots,n],[1,\dots,k-1])$ are also Smith invariant factors of $A([k,\dots,n],[1,\dots,k])$ including multiplicities, where $A([i_1,\dots,i_2],[j_1,\dots,j_2])$ denotes the submatrix of A formed by $i_1$ to $i_2$th rows and $j_1$ to $j_2$th columns and $k\leq [n/2]$; or equivalently, the gcd of all m by m minors of $A([k-1,\dots,n],[1,\dots,k-1])$ equals the gcd of all m by m minors of $A([k,\dots,n],[1,\dots,k])$. $(m\leq k-1)$
Since Toeplitz matrix is rather special and this property (though only confirmed by maple) seems quite nice, I am wondering if anyone else has seen this or something similar before?