I have been reading a paper regarding some relations on character degrees. There is a term being used in the proof of a theorem which is not defined. Neither have I been able to find it in any book about representation theory that I have looked at.
I am quoting a part of the proof: "$\chi \in$ Irr($G$) lies over $1_{HN}$ then $\chi$ lies over $1_G$". I am not sure what is meant by 'lies over' in this case
Let $H$ be a subgroup of a group $G$, let $\chi$ be an irreducible character of $G$ and let $\vartheta$ be an irreducible character of $H$. The character $\chi$ is said to lie over $\vartheta$, if $\chi$ is an irreducible constituent of $\vartheta^G$. By the Frobenius reciprocity this is equivalent to say that $\vartheta$ is a constituent of $\chi_N$. In this case, we also say that $\vartheta$ lies under $\chi$.