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Publicly Available Published by De Gruyter November 1, 2013

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

Ilaria Cardinali EMAIL logo and Antonio Pasini
From the journal Journal of Group Theory

Abstract.

For k=1,2,...,n-1 let Vk=V(Ξ»k) be the Weyl module for the special orthogonal group G= SO (2n+1,𝔽) with respect to the k-th fundamental dominant weight Ξ»k of the root system of type Bn and put Vn=V(2Ξ»n). It is well known that all of these modules are irreducible when char(𝔽) β‰  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 Vk=MkβŠƒMk-1βŠƒβ‹―βŠƒM1βŠƒM0βŠƒM-1=0 where Mi β‰… Vi for 1,...,k-1 and M0 is the trivial 1-dimensional module. We also show that for i=1,2,...,k the quotient Mi / Mi-2 is isomorphic to the so-called i-th Grassmann module for G. Resting on this fact we can give a geometric description of Mi-1 / Mi-2 as a submodule of the i-th Grassmann module. When 𝔽 is perfect, G β‰… Sp(2n,𝔽) and Mi / Mi-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.

Received: 2013-5-17
Revised: 2013-9-19
Published Online: 2013-11-1
Published in Print: 2014-7-1

Β© 2014 by Walter de Gruyter Berlin/Boston

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