# Quantum Linear Groups and Representations of \(GL_n({\mathbb F}_q)\)

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*Jonathan Brundan; Richard Dipper; Alexander Kleshchev*

We give a self-contained account of the results originating in
the work of James and the second author in the 1980s relating the
representation theory of \(GL_n(\mathbb{F}_q)\) over fields of characteristic
coprime to \(q\) to the representation theory of “quantum
\(GL_n\)” at roots of unity.

The new treatment allows us to extend the theory in several
directions. First, we prove a precise functorial connection between the
operations of tensor product in quantum \(GL_n\) and Harish-Chandra
induction in finite \(GL_n\). This allows us to obtain a version of the
recent Morita theorem of Cline, Parshall and Scott valid in addition for
\(p\)-singular classes.

From that we obtain simplified treatments of various basic
known facts, such as the computation of decomposition numbers and blocks of
\(GL_n(\mathbb{F}_q)\) from knowledge of the same for the quantum group,
and the non-defining analogue of Steinberg's tensor product theorem. We also
easily obtain a new double centralizer property between
\(GL_n(\mathbb{F}_q)\) and quantum \(GL_n\), generalizing a
result of Takeuchi.

Finally, we apply the theory to study the affine general linear
group, following ideas of Zelevinsky in characteristic zero. We prove results
that can be regarded as the modular analogues of Zelevinsky's and Thoma's
branching rules. Using these, we obtain a new dimension formula for the
irreducible cross-characteristic representations of
\(GL_n(\mathbb{F}_q)\), expressing their dimensions in terms of the
characters of irreducible modules over the quantum group.