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On the degrees and rationality of certain characters of finite Chevalley groups


Authors: C. T. Benson and C. W. Curtis
Journal: Trans. Amer. Math. Soc. 165 (1972), 251-273
MSC: Primary 20C30
DOI: https://doi.org/10.1090/S0002-9947-1972-0304473-1
MathSciNet review: 0304473
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Abstract: Let $ \mathcal{S}$ be a system of finite groups with (B, N)-pairs, with Coxeter system (W, R) and set of characteristic powers $ \{ q\} $ (see [4]). Let A be the generic algebra of the system, over the polynomial ring $ \mathfrak{o} = Q[u]$. Let K be $ Q(u)$, K an algebraic closure of K, and $ {\mathfrak{o}^ \ast }$ the integral closure of $ \mathfrak{o}$ in K. For the specialization $ f:u \to q$ mapping $ \mathfrak{o} \to Q$, let $ {f^ \ast }:{\mathfrak{o}^ \ast } \to \bar Q$ be a fixed extension of f. For each irreducible character $ \chi $ of the algebra $ {A^{\bar K}}$, there exists an irreducible character $ {\zeta _{\chi ,{f^ \ast }}}$ of the group $ G(q)$ in the system corresponding to q, such that $ ({\zeta _{\chi ,{f^ \ast }}},1_{B(q)}^{G(q)}) > 0$, and $ \chi \to {\zeta _{\chi ,{f^ \ast }}}$ is a bijective correspondence between the irreducible characters of $ {A^{\bar K}}$ and the irreducible constituents of $ 1_{B(q)}^{G(q)}$. Assume almost all primes occur among the characteristic powers $ \{ q\} $. The first main result is that, for each $ \chi $, there exists a polynomial $ {d_\chi }(t) \in Q[t]$ such that, for each specialization $ f:u \to q$, the degree $ {\zeta _{\chi ,{f^ \ast }}}(1)$ is given by $ {d_\chi }(q)$. The second result is that, with two possible exceptions in type $ {E_7}$, the characters $ {\zeta _{\chi ,{f^ \ast }}}$ are afforded by rational representations of $ G(q)$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0304473-1
Keywords: Finite group, (B,N)-pair, Coxeter group, generic algebra, irreducible character, generic degree
Article copyright: © Copyright 1972 American Mathematical Society

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