Linear polarization measurements have been performed for $gamma$-rays in $^{91}$Ru produced with the $^{58}$Ni($^{36}$Ar, $2p1n$$gamma$)$^{91}$Ru reaction at a beam energy of 111 MeV. The EXOGAM Ge clover array has been used to measure the $gamma$-$gamma$ coincidences, $gamma$-ray linear polarization and $gamma$-ray angular distributions. The polarization sensitivity of the EXOGAM clover detectors acting as Compton polarimeters has been determined in the energy range 0.3$-$1.3 MeV. Several transitions have been observed for the first time. Measurements of linear polarization and angular distribution have led to the firm assignments of spin differences and parity of high-spin states in $^{91}$Ru. More specifically, calculations using a semi-empirical shell model were performed to understand the structures of the first and second (21/2$^{+}$) and (17/2$^{+}$) levels. The results are in good agreement with the experimental data, supporting the interpretation of the non yrast (21/2$^{+}$) and (17/2$^{+}$) states in terms of the $J_{rm max}$ and $J_{rm max}-2$ members of the seniority-three $ u(g_{9/2})^{-3}$ multiplet.
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