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Generally one considers a vector space $k^n$, then $\mathrm{GL}_n(k)$ is realized as the set of automorphisms of $k^n$. We have $k^m\times k^n = k^{m+n}$, however under the natural embedding, $\mathrm{GL}_n\times \mathrm{GL}_m\subsetneq \mathrm{GL}_{n+m}$.

Is there a well defined and interesting product $\star$ on algebraic groups (say), or some enriched version of them, such that $\mathrm{GL}_n \star \mathrm{GL}_m= \mathrm{GL}_{n+m}$ ? In other words can we make algebraic groups "remember" the vector space they naturally act on ?

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    $\begingroup$ Probably such a product should respect split reductive groups, where it would become an operation on Dynkin diagrams such that $\mathsf A_{n - 1} \star \mathsf A_{m - 1}$ equals $\mathsf A_{n + m - 1}$. So perhaps one could look for such an operation? $\endgroup$
    – LSpice
    Commented 10 hours ago

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