On the doubly transitive permutation representations of \(\text{Sp}(2n,\mathbb{F}_2)\).
Let \(G\) denote the finite symplectic group \(\text{Sp}(2n,2)\) of degree \(2n\) over the field \(\mathbb{F}_2\) of order 2. It is well known that \(G\) has two doubly transitive permutation actions on the cosets of the two non-isomorphic orthogonal groups of degree \(2n\). These actions have degree \(2^{n-1}(2^n-1)\) and \(2^{n-1}(2^n+1)\), respectively. The purpose of the paper under review is to investigate the submodule structure of the reduction modulo 2 of the corresponding permutation lattices. Let \(\theta_1\) and \(\theta_2\) be the corresponding permutation characters. As the authors remark, when restricted to elements of odd order, \(\theta_1+\theta_2\) agrees with the permutation character of \(G\) acting on the points of the natural vector space \(V\) of dimension over \(2n\) over \(\mathbb{F}_2\). In turn, one knows that the reduction modulo 2 of the permutation lattice of \(G\) acting on the points of \(V\) has the same Brauer character as that of the exterior algebra \(\bigwedge V\) of \(V\), considered as a module for \(G\). Thus, the composition factors of the reduction modulo 2 of each doubly transitive permutation lattice occur among the composition factors of the exterior algebra. Moreover, on elements of odd order, the generalized character \(\theta_2-\theta_1\) agrees with the Brauer character of the spin module for \(G\). This spin module is irreducible of degree \(2^n\) and it occurs with multiplicity 1 in \(\bigwedge V\), as a composition factor of \(\bigwedge^nV\). With this extra knowledge, we now know that the composition factors of the reduction modulo 2 of the two doubly transitive permutation lattices are almost identical, except that the one of larger dimension contains the spin module, while the other does not. Working over the algebraic closure \(k\) of \(\mathbb{F}_2\), the authors construct filtrations for the modular reductions of the permutation lattices which have subquotients that are modules for the symplectic group \(\text{Sp}(2n,k)\). They relate this filtration to the Weyl modules for \(\text{Sp}(2n,k)\). They do not address the problem of finding the composition factors of these modules, nor their multiplicities, and refer to a paper of \textit{A. M. Adamovich} [Mosc. Univ. Math. Bull. 41, No. 2, 6-9 (1986); translation from Vestn. Mosk. Univ., Ser. I 1986, No. 2, 7-10 (1986; Zbl 0647.20039)] for information in this respect. They might also have mentioned a paper of \textit{A. Kleshchev} and the reviewer [J. Algebra 221, No. 1, 60-89 (1999; Zbl 0943.20013)], which describes the composition factors in the following way. \(G\) contains the symmetric group \(S_{2n+1}\) of degree \(2n+1\) naturally as a subgroup. Any composition factor for \(G\) in its action on \(\bigwedge V\) remains irreducible on restriction to \(S_{2n+1}\) and the corresponding irreducible module of \(S_{2n+1}\) is one labelled by a 2-part partition. Multiplicities can then be determined from properties of the decomposition numbers of \(S_{2n+1}\). In particular, each fundamental module for \(\text{Sp}(2n,k)\) remains irreducible on \(S_{2n+1}\) and it is labelled naturally in the obvious order by one of the 2-part partitions \((2n,1)\), \((2n-1,2),\dots,(n+1,n)\), the last module being the spin module. There is clearly a rich combinatorial structure attached to this exceptional behaviour.
scientific article; zbMATH DE number 1868074