The sparsity of character tables over finite reductive groups and its additive analogue
GyeongHyeon Nam, Anna Puskás
TL;DR
The paper develops a quantitative, model-free view of the sparsity in character tables of finite reductive groups by leveraging Deligne–Lusztig theory and counts of regular semisimple elements, yielding a concrete asymptotic lower bound for the proportion of zero entries when the underlying group is fixed and q grows. It then shows that if the semisimple rank grows, the proportion of zero entries tends to 1, and extends the analysis to an additive analogue via Fourier transform on finite Lie algebras, aligning the group and Lie-algebra viewpoints. The results are reduced to Weyl-group data through counts of regular semisimple elements in maximal tori and the distribution of conjugacy classes, with explicit dependence on centralizers in the Weyl group. The paper also outlines conjectural additive analogues and discusses polynomiality-on-residue-classes phenomena, highlighting a coherent additive–group parallel that enriches arithmetic statistics of representation-theoretic objects.
Abstract
We consider the proportion of zero entries in the character table of a sequence of reductive groups over a finite field. We prove an asymptotic lower bound when the reductive group is fixed and the size of the finite field increases. Furthermore, we prove that when considering a sequence of reductive groups with increasing semisimple rank, the proportion is asymptotically one. We also establish an additive analogue of this phenomenon in the context of a fixed reductive Lie algebra.