Under construction. Please feel free to contact me.
The cohomology of $C_2$-equivariant Grassmannians
This site is a database of known values of $H^{\ast,\ast}(\text{Gr}_k(\mathbb{R}^{p,q});\underline{\mathbb F}_2)$, expressed as a module over $\mathbb{M}_2:=H^{\ast,\ast}(pt;\underline{\mathbb F}_2)$.
All $RO(C_2)$-grading is motivic, with bidegree $(x,y)$ indicating a $x$-dimensional representation containing $y$ copies of the sign representation (and hence $x-y$ copies of the trivial rep.) The majority of these calculations are produced by the AutoKron, using techniques outlined here.
Note that $\text{Gr}_k(\mathbb{R}^{p,p-q})\cong \text{Gr}_k(\mathbb{R}^{p,q})$ and $\text{Gr}_{p-k}(\mathbb{R}^{p,q})\cong \text{Gr}_k(\mathbb{R}^{p,q})$ so we'll just worry about when $k,q\le \frac p2$.
Because Gonzaga's current system for faculty websites prevents uploading multiple files at once (?!) I've divided the information into four big chunks based on $k$. I will apparently be able to upload a slightly more user-friendly version in June 2026.
For higher $k$, fancier methods will be needed. The AutoKron already gives 16 possiblies for the cohomology of $\text{Gr}_5(\mathbb{R}^{10,1})$.