Gauss sums on finite groups (Q447811)

Let \(p > 2\) be a prime number, \(\mathbb F_p\) the prime finite field with \(p\) elements, \(\mathbb F^*_p\) its multiplicative cyclic group of order \(p-1\) and \(i = \sqrt{-1}\). The classical Gauss sum \(g_p\) is given by NEWLINE\[NEWLINE \tau_p= \sum_{x \in \mathbb F^*_p} \left( \frac{x}{p} \right) e^{2 { \pi}i x/p},NEWLINE\]NEWLINE where \( \left( \frac{x}{p} \right)\) is the Legendre symbol. Generalizing the pair \( \left( \mathbb F^*_p, \left( \frac{x}{p} \right) \right)\) to a pair \((G, \chi)\), where \(G\) is a finite group and \( \chi\) its complex character, one can define a Gauss sum \(\tau_{G}( \chi, \psi_{ \rho})\) on \(G\) associated with a modular representation NEWLINE\[NEWLINE \rho : G \rightarrow \mathrm{GL}_n (\mathbb F_q), NEWLINE\]NEWLINE where \(\mathbb F_q\) a finite field with \(q=p^{\nu}\) elements.NEWLINENEWLINENEWLINEAfter short preliminaries, in \(\S 2\) the authors determine explicitly \( \tau_{G}( \chi, \psi_{\rho})\) for the complex reflection group \(G=G(m,1,n)\) and for all all irreducible characters \(\chi\) of \(G\) with an \(n\)-dimensional modular representations \(\rho\). The results include the case of finite symmetric groups and also Weyl groups of type \(D\). In \(\S 3\), applying Clifford's theorem, the authors determine the Gauss sums of the complex reflection groups \(G(m,r,n)\). Finally, in \(\S 4\) the authors treat the case of Weyl groups of finite exceptional type.