Commentarii Mathematici Helvetici


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Volume 74, Issue 2, 1999, pp. 280–296
DOI: 10.1007/s000140050089

Published online: 1999-06-30

A character formula for a family of simple modular representations of $ GL_n $

O. Mathieu[1] and Athanase Papadopoulos[2]

(1) Universität Basel, Switzerland
(2) Université de Strasbourg, France

Let K be an algebraically closed field of finite characteristic p, and let $ n\geq 1 $ be an integer. In the paper, we give a character formula for all simple rational representations of $ GL_{n}(K) $ with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight 5 is special if there are integers $ i\leq j $ such that $ \lambda=\sum_{i\leq k\leq j}a_{k}\,\omega_{k} $ and $ \sum_{i\leq k\leq j} a_k\leq {\rm inf}(p-(j-i),p-1) $. Indeed, we compute the character of any simple module whose highest weight 5 can be written as $ \lambda=\lambda_{0}+p\lambda_{1}+...+p^{r}\lambda_{r} $ with all $ \lambda_{i} $ are special. By stabilization, we get a character formula for a family of irreducible rational $ GL_{\infty}(K) $-modules.

Keywords: Tilting modules, modular representations, character formula, polynomial functors, Verlinde's formula

Mathieu O., Papadopoulos Athanase: A character formula for a family of simple modular representations of $ GL_n $. Comment. Math. Helv. 74 (1999), 280-296. doi: 10.1007/s000140050089