Coxeter and Schubert combinatorics of $\mu$-Involutions

The variety of complete quadrics is the wonderful compactification of $GL_n/O_n$ and admits a cell decomposition into Borel orbits indexed by combinatorial objects called $\mu$-involutions. We study Coxeter-theoretic properties of $\mu$-involutions with results including a combinatorial description for their atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. The corresponding orbit closures can be realized inside the flag variety. In this setting, we study the cohomology representatives of these orbits, which are, up to a scalar, the $\mu$-involution Schubert polynomials. We expand $\mu$-involution Schubert polynomials as a multiplicity-free sum of $\nu$-involution Schubert polynomials when $\nu$ refines $\mu$ and provide recurrences analogous to Monk's rule for Schubert polynomials.