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edited Dec 16, 2018 at 16:19

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I'll address $Q1$ and $Q3$. AsAs Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those appearing in the irreducible decomposition of $\mathbb Q [GL_n(\mathbb F_q)/B_n(\mathbb F_q)]$.

Unipotent representations are not closed under the naive induction product, but they are closed under parabolic induction $V*W = {\rm Ind}_{P(n,m)}^{GL_{n+m}} V \otimes W$. This gives $\oplus_n {\rm Rep}^{un}(GL_n(\mathbb F_q))$ the structure of a monoidal category. Instead of being symmetric monoidal, it is now braided monoidal! The Grothendieck ring is a $q$ deformation of the ring of symmetric functions.

Finally, by Morita theory, unipotent representations are equivalent to representations of $\mathcal H_n(q) = {\rm End}_{GL_n}(\mathbb Q GL_n/B_n )$, here $\mathcal H_n(q)$ is the Iwahori-Hecke algebra which $q$-deforms the group ring of $S_n$. It is Schur-Weyl dual to representations of the quantum group $U_q(GL_\infty)$.

I'll address $Q1$ and $Q3$. As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those appearing in the irreducible decomposition of $\mathbb Q [GL_n(\mathbb F_q)/B_n(\mathbb F_q)]$.

Unipotent representations are not closed under the naive induction product, but they are closed under parabolic induction $V*W = {\rm Ind}_{P(n,m)}^{GL_{n+m}} V \otimes W$. This gives $\oplus_n {\rm Rep}^{un}(GL_n(\mathbb F_q))$ the structure of a monoidal category. Instead of being symmetric monoidal, it is now braided monoidal! The Grothendieck ring is a $q$ deformation of the ring of symmetric functions.

Finally, by Morita theory, unipotent representations are equivalent to representations of $\mathcal H_n(q) = {\rm End}_{GL_n}(\mathbb Q GL_n/B_n )$, here $\mathcal H_n(q)$ is the Iwahori-Hecke algebra which $q$-deforms the group ring of $S_n$. It is Schur-Weyl dual to representations of the quantum group $U_q(GL_\infty)$.

As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those appearing in the irreducible decomposition of $\mathbb Q [GL_n(\mathbb F_q)/B_n(\mathbb F_q)]$.

Unipotent representations are not closed under the naive induction product, but they are closed under parabolic induction $V*W = {\rm Ind}_{P(n,m)}^{GL_{n+m}} V \otimes W$. This gives $\oplus_n {\rm Rep}^{un}(GL_n(\mathbb F_q))$ the structure of a monoidal category. Instead of being symmetric monoidal, it is now braided monoidal! The Grothendieck ring is a $q$ deformation of the ring of symmetric functions.

Finally, by Morita theory, unipotent representations are equivalent to representations of $\mathcal H_n(q) = {\rm End}_{GL_n}(\mathbb Q GL_n/B_n )$, here $\mathcal H_n(q)$ is the Iwahori-Hecke algebra which $q$-deforms the group ring of $S_n$. It is Schur-Weyl dual to representations of the quantum group $U_q(GL_\infty)$.

answered Dec 16, 2018 at 16:09

4.6k
1
22
27

I'll address $Q1$ and $Q3$. As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those appearing in the irreducible decomposition of $\mathbb Q [GL_n(\mathbb F_q)/B_n(\mathbb F_q)]$.

Unipotent representations are not closed under the naive induction product, but they are closed under parabolic induction $V*W = {\rm Ind}_{P(n,m)}^{GL_{n+m}} V \otimes W$. This gives $\oplus_n {\rm Rep}^{un}(GL_n(\mathbb F_q))$ the structure of a monoidal category. Instead of being symmetric monoidal, it is now braided monoidal! The Grothendieck ring is a $q$ deformation of the ring of symmetric functions.

Finally, by Morita theory, unipotent representations are equivalent to representations of $\mathcal H_n(q) = {\rm End}_{GL_n}(\mathbb Q GL_n/B_n )$, here $\mathcal H_n(q)$ is the Iwahori-Hecke algebra which $q$-deforms the group ring of $S_n$. It is Schur-Weyl dual to representations of the quantum group $U_q(GL_\infty)$.