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Let $V$ be a $d$-dimensional vector space and $Fl_n$ be the space of $n$-step partial flags of linear subspaces in $V.$ In this talk I will describe the combinatorial structure of orbits of the diagonal action of $Gl(V)$ on the double partial flag variety $Fl_n\times Fl_n.$ The orbits are in bijection with $n\times n$ matrices with non-negative integer entries and the total sum of entries equal to $d.$ Boundary relations between orbits give rise to a Bruhat order on such matrices, and the convolution product on $Fl_n\times Fl_n$ gives rise to an associative product (different from matrix multiplication). I will define the Bruhat order and the product and prove some fundamental properties.

The talk is based on a joint project with Sergey Arkhipov, in which we study the $Gl(V)$-equivariant $K$-thery of the double partial flag variety endowed with the convolution product. I will focus on the combinatorial side of the project.