Operation of class sums on permutation modules
The authors present an efficient algorithm to compute the action of a conjugacy class sum of a finite group G on a permutation module V over a finite field F. This may be used to compute the block idempotents \(e_ i\)- knowing the classical character table and the distribution of characters into blocks - and the decomposition of V into blockwise components \(Ve_ i\). Thus, if one wants to know the irreducible constituents of V, the meataxe needs to be applied to smaller modules provided not all direct summands of V belong to the same block. As examples, the authors compute a) a 45-dimensional representation of \(PSL_ 3(4)\) over GF(9) starting with the permutation representation of degree 280 on the cosets of the Sylow 3-normalizer. b) a 27-dimensional representation of the Tits simple group \({}^ 2F_ 4(2)'\) over GF(5) starting with a permutation representation of degree 1755. The algorithm is based on the following key observation. Let \(V=F\Omega\) be a permutation module of G with \(\Omega =\{1,...,n\}\). For \(x\in G\) let K denote the conjugacy class of x in G and let \(\bar K\) denote the class sum of K. Then, the action of \(\bar K\) on V is given by the matrix \(A=(a_{ij})\) with \[ a_{ij}=| K| | \{(i,i^ x)| \quad i\in \Omega \}\cap \{(i^ g,j^ g)| \quad g\in G\}| /| \{i^ g,j^ g)| \quad g\in G\}| \] for \(i,j=1,...,n\). In case the action of G on \(\Omega\) is transitive, only the first row has to be computed.
scientific article; zbMATH DE number 4140019
scientific article