Counting Schur Rings over Cyclic Groups

TL;DR

This paper addresses the complex problem of counting all Schur rings over cyclic groups of prime power order, providing explicit formulas and exploring connections with Higman's PORC conjecture.

Contribution

It introduces a method to count all Schur rings over cyclic groups of prime power order, advancing understanding of their structure and enumeration.

Findings

Formulas for counting Schur rings over cyclic groups of prime power order

Connections established between Schur rings enumeration and Higman's PORC conjecture

Enhanced understanding of Schur ring structure over cyclic groups

Abstract

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for Schur rings over cyclic groups, it is still a difficult problem to count all the partitions. For example, Kovacs, Liskovets, and Poschel determine formulas to count the number of wreath-indecomposable Schur rings. In this paper we solve the problem of counting the number of all Schur rings over cyclic groups of prime power order and draw some parallels with Higman's PORC conjecture.