Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks
Séminaire lotharingien de combinatoire, Tome 56 (2006-2007)
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We survey a number of powerful recent results concerning diophantine and asymptotic properties of (ordinary) characters of symmetric groups. Apart from their intrinsic interest, these results are motivated by a connection with subgroup growth theory and the theory of random walks. As applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, as well as a finiteness result for the number of Fuchsian presentations of such a group, the latter result solving a long-standing problem of Roger Lyndon's. We also sketch the proof of a well-known conjecture of Roichman's concerning the mixing time of random walks on finite symmetric groups, and of a result describing the parity of the subgroup numbers for a substantial class of one-relator groups.
@article{SLC_2006-2007_56_a0,
author = {Thomas W. M\"uller},
title = {Character {Theory} of {Symmetric} {Groups,} {Analysis} of {Long} {Relators,} and {Random} {Walks}},
journal = {S\'eminaire lotharingien de combinatoire},
publisher = {mathdoc},
volume = {56},
year = {2006-2007},
url = {http://geodesic.mathdoc.fr/item/SLC_2006-2007_56_a0/}
}
Thomas W. Müller. Character Theory of Symmetric Groups, Analysis of Long Relators, and Random Walks. Séminaire lotharingien de combinatoire, Tome 56 (2006-2007). http://geodesic.mathdoc.fr/item/SLC_2006-2007_56_a0/