Geodesic
Groups generated by involutions of diamond-shaped graphs, and deformations of Young's orthogonal form
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 29-38

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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 orthogonal form can be regarded as a deformation of our construction. We also state asymptotic problems for infinite groups. 
@article{ZNSL_2019_481_a2,
     author = {A. M. Vershik and N. V. Tsilevich},
     title = {Groups generated by involutions of diamond-shaped graphs, and deformations of {Young's} orthogonal form},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {29--38},
     publisher = {mathdoc},
     volume = {481},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a2/}
}
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A. M. Vershik; N. V. Tsilevich. Groups generated by involutions of diamond-shaped graphs, and deformations of Young's orthogonal form. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 29-38. http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a2/