UC Berkeley

The space of principally regular matrices $\PR_n$ is the space of real symmetric matrices with every principal minor being non-zero. This space plays a vital role in algebraic geometry and algebraic statistics. On one side, it corresponds to the intersection of open cells in the Lagrangian Grassmannian $\text{LGr}(n,2n)$. On the other side, it has applications in conditional independence and determinantal point processes in statistics. Understanding the topology of $\PR_n$ will allow us to explore the new connections between geometry, statistics, and combinatorics. In this work, we discuss the topology of $\PR_n$ for small $n$ where every connected component is realized as an open cell in the complement to conics in $\R^2$ for $n=3$ and quadrics in $\R^3$ for $n=4$. Also, we show that the lower bound of the number of connected components of $\PR_n$ is given as the number of orientations on the uniform Oriented Lagrangian Matroid. Finally, we formulate our main conjecture that every connected component of $\PR_n$ is contractible.