  We study probability measures on partitions based on symmetric Grothendieck
polynomials. These deformations of Schur polynomials introduced in the K-theory
of Grassmannians share many common properties. Our Grothendieck measures are
analogs of the Schur measures on partitions introduced by Okounkov
(arXiv:math/9907127 [math.RT]). Despite the similarity of determinantal
formulas for the probability weights of Schur and Grothendieck measures, we
demonstrate that Grothendieck measures are \emph{not} determinantal point
processes. This question is related to the principal minor assignment problem
in algebraic geometry, and we employ a determinantal test first obtained by
Nanson in 1897 for the $4\times4$ problem. We also propose a procedure for
getting Nanson-like determinantal tests for matrices of any size $n\ge4$ which
appear new for $n\ge 5$.
  By placing the Grothendieck measures into a new framework of tilted
biorthogonal ensembles generalizing a rich class of determinantal processes
introduced by Borodin (arXiv:math/9804027 [math.CA]), we identify Grothendieck
random partitions as a cross-section of a Schur process, a determinantal
process in two dimensions. This identification expresses the correlation
functions of Grothendieck measures through sums of Fredholm determinants, which
are not immediately suitable for asymptotic analysis. A more direct approach
allows us to obtain a limit shape result for the Grothendieck random
partitions. The limit shape curve is not particularly explicit as it arises as
a cross-section of the limit shape surface for the Schur process. The gradient
of this surface is expressed through the argument of a complex root of a cubic
equation.
