A study of the growth of maximal and typical normalized dimensions of strict Young diagrams

N. N. Vasiliev ; V. S. Duzhin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Tome 437 (2015), pp. 81-99

Résumé

In this paper, we investigate the asymptotics of the normalized dimensions of strict Young diagrams (i.e., the numbers of paths to vertices in the Schur graph). We describe the results of corresponding computer experiments. The strict Young diagrams parametrize the projective representations of the symmetric group $S_n$. So, the asymptotics of the normalized dimensions of diagrams gives us the asymptotics of the dimensions of projective representations as well. Sequences of strict diagrams of high dimension consisting of up to one million cells were built. It was proved by an exhaustive search that the first 250 diagrams of all these sequences have the maximum possible dimensions. Presumably, these sequences contain infinitely many diagrams of maximum dimension, and thus give the correct asymptotics of their growth. Also, we investigate the behavior of the normalized dimensions of typical diagrams with respect to the Plancherel measure on the Schur graph. The calculations strongly agree with A. M. Vershik's hypothesis on the convergence of the normalized dimensions of maximal and Plancherel typical diagrams not only for the standard Young graph, but also for the Schur graph.