On certain submodules of Weyl modules for SO(2n+1,F) with char(F) = 2

Abstract

For $k = 1, 2,...,n-1$ let $V_k = V(\lambda_k)$ be the Weyl module for the special orthogonal group $G = \mathrm{SO}(2n+1,\F)$ with respect to the $k$-th fundamental dominant weight $\lambda_k$ of the root system of type $B_n$ and put $V_n = V(2\lambda_n)$. It is well known that all of these modules are irreducible when $\mathrm{char}(\F) \neq 2$ while when $\mathrm{char}(\F) = 2$ they admit many proper submodules. In this paper, assuming that $\mathrm{char}(\F) = 2$, we prove that $V_k$ admits a chain of submodules $V_k = M_k \supset M_{k-1}\supset ... \supset M_1\supset M_0 \supset M_{-1} = 0$ where $M_i \cong V_i$ for $1,..., k-1$ and $M_0$ is the trivial 1-dimensional module. We also show that for $i = 1, 2,..., k$ the quotient $M_i/M_{i-2}$ is isomorphic to the so called $i$-th Grassmann module for $G$. Resting on this fact we can give a geometric description of $M_{i-1}/M_{i-2}$ as a submodule of the $i$-th Grassmann module. When $\F$ is perfect $G\cong \mathrm{Sp}(2n,\F)$ and $M_i/M_{i-1}$ is isomorphic to the Weyl module for $\mathrm{Sp}(2n,\F)$ relative to the $i$-th fundamental dominant weight of the root system of type $C_n$. All irreducible sections of the latter modules are known. Thus, when $\F$ is perfect, all irreducible sections of $V_k$ are known as well.