Let $V_k$ be the Weyl module of dimension ${2n\choose k}-{2n\choose k-2}$ for the group $G = \mathrm{Sp}(2n,\mathbb{F})$ arising from the $k$-th fundamental weight of the Lie algebra of $G$. Thus, $V_k$ affords the grassmann embedding of the $k$-th symplectic polar grassmannian of the building associated to $G$. When $\mathrm{char}(\mathbb{F}) = p > 0$ and $n$ is sufficiently large compared with the difference $n-k$, the $G$-module $V_k$ is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given $h := n-k$. It is known that, for given $h$ and $p$, there exists an integer $n(h,p)$ such that $V_k$ is reducible if and only if $n \geq n(h,p)$. Moreover, let $n \geq n(h,p)$ and $R_k$ the largest proper non-trivial submodule of $V_k$. Then $\mathrm{dim}(R_k) = 1$ if $n = n(h,p)$ while $\mathrm{dim}(R_k) > 1$ if $n > n(h,p)$. In this paper we will show how this result can be obtained by an investigation of a certain chain of $G$-submodules of the exterior power $W_k := \wedge^kV$, where $V = V(2n,\mathbb{F})$

Blok., J., Cardinali, I., Pasini, A. (2011). On natural representations of the symplectic group. BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY SIMON STEVIN, 18, 1-29.

On natural representations of the symplectic group

Abstract

Let $V_k$ be the Weyl module of dimension ${2n\choose k}-{2n\choose k-2}$ for the group $G = \mathrm{Sp}(2n,\mathbb{F})$ arising from the $k$-th fundamental weight of the Lie algebra of $G$. Thus, $V_k$ affords the grassmann embedding of the $k$-th symplectic polar grassmannian of the building associated to $G$. When $\mathrm{char}(\mathbb{F}) = p > 0$ and $n$ is sufficiently large compared with the difference $n-k$, the $G$-module $V_k$ is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given $h := n-k$. It is known that, for given $h$ and $p$, there exists an integer $n(h,p)$ such that $V_k$ is reducible if and only if $n \geq n(h,p)$. Moreover, let $n \geq n(h,p)$ and $R_k$ the largest proper non-trivial submodule of $V_k$. Then $\mathrm{dim}(R_k) = 1$ if $n = n(h,p)$ while $\mathrm{dim}(R_k) > 1$ if $n > n(h,p)$. In this paper we will show how this result can be obtained by an investigation of a certain chain of $G$-submodules of the exterior power $W_k := \wedge^kV$, where $V = V(2n,\mathbb{F})$
Scheda breve Scheda completa Scheda completa (DC)
2011
Blok., J., Cardinali, I., Pasini, A. (2011). On natural representations of the symplectic group. BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY SIMON STEVIN, 18, 1-29.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/35516
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