Groups generated by involutions of diamond-shaped graphs, and deformations of Young's orthogonal form

TL;DR

This paper explores groups generated by involutions on diamond-shaped graphs, focusing on Young graphs, and presents new representations and deformations of Young's orthogonal form, supported by computational experiments and posing asymptotic questions.

Contribution

It introduces a novel construction of groups and representations associated with diamond-shaped graphs, including a deformation of Young's orthogonal form and related computational analysis.

Findings

Identification of subgroup structures in Young graphs

Development of deformed Young's orthogonal form

Results from computer simulations on group representations

Abstract

With an arbitrary finite graph having a special form of 2-intervals (a diamond-shaped graph) we associate a subgroup of a symmetric group and a representation of this subgroup; state a series of problems on such groups and their representations; and present results of some computer simulations. The case we are most interested in is that of the Young graph and subgroups generated by natural involutions of Young tableaux. In particular, the classical Young's orthogonal form can be regarded as a deformation of our construction. We also state asymptotic problems for infinite groups.