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

Abstract

Abstract. For k = 1 , 2 , ... , n - 1 ${k = 1, 2,\ldots ,n-1}$ let V k = V ( λ k ) ${V_k = V(\lambda _k)}$ be the Weyl module for the special orthogonal group G = SO ( 2 n + 1 , 𝔽 ) ${G = \mathrm {SO}(2n+1,\mathbb {F})}$ with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put V n = V ( 2 λ n ) ${V_n = V(2\lambda _n)}$ . It is well known that all of these modules are irreducible when char(𝔽) ≠ 2 while when char(𝔽) = 2 they admit many proper submodules. In this paper, assuming that char(𝔽) = 2, we prove that Vk admits a chain of submodules V k = M k ⊃ M k - 1 ⊃ ⋯ ⊃ M 1 ⊃ M 0 ⊃ M - 1 = 0 ${V_k = M_k \supset M_{k-1}\supset \dots \supset M_1\supset M_0 \supset M_{-1} = 0}$ where Mi ≅ Vi for 1 , ... , k - 1 ${1,\ldots , k-1}$ and M 0 is the trivial 1-dimensional module. We also show that for i = 1 , 2 , ... , k ${i = 1, 2,\ldots , k}$ 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 ≅ Sp(2n,𝔽) and M i / M i-1 is isomorphic to the Weyl module for Sp(2n,𝔽) relative to the i-th fundamental dominant weight of the root system of type Cn . All irreducible sections of the latter modules are known. Thus, when 𝔽 is perfect, all irreducible sections of Vk are known as well.