Betti numbers of determinantal ideals

Let R=k[x1,…,xn] be a polynomial ring and let I R be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number βi(R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k[x1,…,xn]/I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.