Isomorph Rejection and a Theorem of De Bruijn

Abstract

Let G be a group acting on a finite set S. Let L be a G-stable subset of S and denote by $\Delta $ a transversal for the orbit partition of G acting on L. The problem of devising efficient generating algorithms for $\Delta $ is called the “isomorph rejection” problem. Given a field F of characteristic zero, let $F^S $ denote all functions from S to F. Note that $F^S $ is an algebra under the operations of pointwise addition and multiplication. The problem of generating $\Delta $ or $I_\Delta $ (the Indicator or characteristic function of $\Delta $) may be regarded as a special case of the problem of generating and representing functions $\varphi $ in $F^S $ (in case $\varphi = I_\Delta $). In this context, instead of attempting to generate $I_\Delta $ directly one may instead choose a pair of linear operators $(T_0 ,T_1 )$ and attempt to construct functions $v_1 , \cdots ,v_p $ in $F^S $ such that $T_0 (v_1 + \cdots + v_p ) = T_1 (I_\Delta )$. The construction of $(v_1 , \cdots ,v_p )$ may be regarded as a “weak solution” to the isomorph rejection problem (which becomes an actual solution if $T_0 = T_1 = {\text{identity}}$). In this paper properties of such constructions are considered in the interesting case where $S = R^D $ is itself a finite function set and the operators $T_0 $ and $T_1 $ are constructed from representations of the group G as operators on $F^S $. The extraction of information from such weak solutions is carried out by means of linear functionals on $F^S $. In this setting certain functionals yield information about the cardinality of $\Delta $ in the form of various well-known identities due to deBruijn and Pólya. Some examples of weak solutions to isomorph rejection problems are given.