Family Algebras and Generalized Exponents for Polyvector Representations of Simple Lie Algebras of Type $B_n$

We give an explicit formula for the exterior powers $\wedge^k\pi_1$ of the defining representation $\pi_1$ of the simple Lie algebra $\mathfrak{so}(2n+1,\mathbb{C})$. We use the technique of family algebras. All representations in question are children of the spinor representation $\sigma$ of $\mathfrak{so}(2n+1,\mathbb{C})$. We also give a survey of main results on family algebras.