Row and column detection complexities of character tables

Abstract

Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.