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This work is an account of the results originating in the work of James and the second author in the 1980s relating the representation theory of GL [n(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(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(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.
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Brian Nichols
reviewed Quantum linear groups and representations of GLn (Fq) on May 23, 2016
Quantum linear groups and representations of GLn
In Arnold's poem, "Introit," she offers the image of a river roiling--a river that might be, in the end, aware of its own long history. This particular river broke through a stone barrier to make a new path to the sea. I admit I was lured into the book by this opening image. There is no explanation for why this area of the land would have to be where the river meets the ocean. Arnold was establishing, for me, a logic of natural action, whose logical basis is irrationality. As I read further into the book, I started to see this analogy relating to the sickness that broke into her life. And, at that point, I was invested in what Arnold might do. Unfortunately, by the third section, that urgency surrounding the sickness gave way to a series of unsubtle ruminations about how difficult sickness is. It could be I'm skeptical about the assumed intimacy in confessional poetry, or maybe I'm just worried by what exigence there is in confessionalism.
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