# Cohomology for Grassmannian

For the Grassmannian $\mathbb{G}(l,V)$ one has for the dual tautological bundle $\mathcal{Q^{\vee}}$: $\mathrm{Hom}(\sum^{\lambda}(\mathcal{Q}^{\vee}),\sum^{\mu}(\mathcal{Q}^{\vee}))=\sum^{(\mu/\lambda)}V^*$. What do we have for $\mathrm{Hom}(\sum^{\lambda}(\mathcal{Q}),\sum^{\mu}(\mathcal{Q}))$?

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$\mathrm{Hom}(\Sigma^\lambda(Q^*), \Sigma^\mu(Q^*))\simeq\mathrm{Hom}(\Sigma^\mu(Q), \Sigma^\lambda(Q))$ as $\Sigma^\lambda(Q^*)\simeq (\Sigma^\lambda(Q))^*$.