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2-weights for classical groups. (English) Zbl 0769.20016

Let \(G\) be a finite symplectic or orthogonal group over a field of odd characteristic. In this paper, the local structures of radical 2- subgroups of \(G\) are described, and a combinatorial way is given to count the number of 2-weights, block by block, for \(G\). Let \(B\) be a block labeled by a semisimple \(2'\)-element \(s\) of the dual group \(G^*\), and let \(m_ \Gamma(s)\) be the multiplicity of an elementary divisor \(\Gamma\) in \(s\). In addition, if \(\Gamma \neq X - 1\), then let \(f_ \Gamma\) be the number of partitions of \(m_ \Gamma(s)\). If \(\Gamma = X - 1\) and \(G\) is symplectic, then let \(f_{\Gamma,\kappa}\) be the number of pairs \((\lambda_ 1,\lambda_ 2)\) of partitions such that \(| \lambda_ 1| + | \lambda_ 2| = {1\over 2}(m_{X-1}(s)-1)- |\kappa|\), and let \(f_ \Gamma = \sum_ \kappa f_{\Gamma,\kappa}\), where \(\kappa\) runs over all 2-cores. If \(\Gamma = X-1\) and \(G\) is orthogonal, then let \(f_{\Gamma,\kappa_ 1,\kappa_ 2,\kappa}\) be the number of 4-tuples \((\lambda_ 1,\lambda_ 2,\lambda_ 3,\lambda_ 4)\) of partitions such that \[ \sum^ 4_{i = 1}|\lambda_ i| = (m_{X-1}(s) - | \kappa_ 1| - |\kappa_ 2| - 2|\kappa|)/4,\quad\text{and let}\quad f_ \Gamma = \sum_{\kappa_ 1,\kappa_ 2,\kappa}f_{\Gamma,\kappa_ 1,\kappa_ 2,\kappa}, \] where \(\kappa\) runs over all 2-cores, \(\kappa_ 1\) and \(\kappa_ 2\) run over all 2- cores with odd and even ranks, respectively when the underlying space of \(G\) is odd-dimensional, and \(\kappa_ 1\) and \(\kappa_ 2\) run over all 2-cores with ranks either both odd or both even such that \(w^{| \kappa_ 1|} = D(V)\) when the underlying space \(V\) of \(G\) is even-dimensional. Here \(D(V)\) is the discriminant of \(V\) and \(w\) is a non-square element of the field. Then the number of \(B\)-weights is \(\prod_ \Gamma f_ \Gamma\).
Reviewer: J.An (Auckland)

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C20 Modular representations and characters
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