Let $[H,G]$ be an interval of finite groups.

*Definition*: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.

Let the *fixed-point subspace* $W^{H}:=\{w \in W \ \vert \ kw=w \ , \forall h \in H \}$.

Let the *pointwise stabilizer subgroup* $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

The interval $[H,G]$ is called *linearly primitive* if $\exists V$ irred. complex repr. of $G$ with $G_{(V^H)} = H$.

*Remark*: We recover the usual definition of "linearly primitive" for groups by taking $H =\{ e \}$.

*Definition*: The subset lattice of $\{1, \dots , n \}$ is called the boolean lattice $B_n$ (see $B_3$ below).

**Question**: Is a boolean interval of finite groups linearly primitive?

Every algebraic lattice is isomorphic to an interval in the subgroup lattice of some group.See DOI: 10.1016/0021-8693(89)90171-3, Google, Google Books, Google Scholar. $\endgroup$ – Martin Sleziak Sep 16 '15 at 9:32finitegroups. But I will keep comment, just in case it is useful for other users reading this question. $\endgroup$ – Martin Sleziak Sep 16 '15 at 9:35