JournalscmhVol. 74, No. 2pp. 280–296

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

  • Athanase Papadopoulos

    Université de Strasbourg, France
  • O. Mathieu

    Universität Basel, Switzerland
A character formula for a family of simple modular representations of $ GL_n $ cover
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Abstract

Let K be an algebraically closed field of finite characteristic p, and let n1n\geq 1 be an integer. In the paper, we give a character formula for all simple rational representations of GLn(K)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 iji\leq j such that λ=ikjakωk\lambda=\sum_{i\leq k\leq j}a_{k}\,\omega_{k} and ikjakinf(p(ji),p1)\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 λ=λ0+pλ1+...+prλr\lambda=\lambda_{0}+p\lambda_{1}+...+p^{r}\lambda_{r} with all λi\lambda_{i} are special. By stabilization, we get a character formula for a family of irreducible rational GL(K)GL_{\infty}(K) -modules.

Cite this article

Athanase Papadopoulos, O. Mathieu, A character formula for a family of simple modular representations of GLn GL_n . Comment. Math. Helv. 74 (1999), no. 2, pp. 280–296

DOI 10.1007/S000140050089