First excited state with moderate rank distribution

TL;DR

This paper demonstrates the existence of infinite sequences of Schur sigma-groups with specific properties, revealing a new 'first excited state' in their transfer kernel types and abelian quotient invariants.

Contribution

It introduces the concept of the first excited state for Schur sigma-groups, expanding understanding of their transfer kernel types and abelian invariants.

Findings

Existence of infinite sequences of Schur sigma-groups with specified properties.

Identification of the first excited state characterized by polarized components.

Distinct logarithmic order differences between ground and excited states.

Abstract

Evidence is provided for the existence of infinite periodic sequences of Schur sigma-groups G with commutator quotient G/G' ~ C(3^e) x C(3), e >= 7, and logarithmic order lo(G) = 10+e. With respect to their maximal subgroups H1,H2,H3;H4, they have moderate rank distribution rho(G) = (rank3(Hi/Hi')) ~ (2,2,3;3) and represent the first excited state of their punctured transfer kernel types kappa(G), which is characterized by a polarized component, e33 or (e+1)32, of the abelian quotient invariants alpha(G) = (Hi/Hi') with lo = 6+e in contrast to the ground state with lo = 4+e.