Tensor Products as Induced Representations: The Case of Finite $\mathrm{GL}(3)$

L. Aburto-Hageman, J. Pantoja, J. Soto-Andrade

Pontificia Universidad Católica de Valparaíso

Abstract: We describe the tensor products of two irreducible linear complex representations of the group $G=\mathrm{GL}(3,\mathbb F_q)$ in terms of induced representations by linear characters of maximal tori and also in terms of Gelfand–Graev representations. Our results include MacDonald's conjectures for $G$ and are extensions to $G$ of finite counterparts to classical results on tensor products of principal series as well as holomorphic and antiholomorphic representations of the group $\mathrm{SL}(2,\mathbb R)$; besides, they provide an easy way to decompose these tensor products with the help of Frobenius reciprocity. We also state some conjectures for the general case of $\mathrm{GL}(n,\mathbb F_q)$.

Keywords: tensor products decomposition, irreducible representation of the general linear groups over finite fields, Clebsch–Gordan coefficients, induced representations.

UDC: 512.544.42

Received: 03.05.2010

DOI: 10.4213/mzm8997

 English version: Mathematical Notes, 2012, 91:4, 459–469

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