REPRESENTATIONS OF THE SYMMETRIC GROUPS AND COMBINATORICS OF THE FROBENIUS-YOUNG CORRESPONDENCE By MATTHEW

After introducing the concepts of partitions, Young diagrams, representation theory, and characters of representations, the 2006 paper by A. M. Vershik entitled “A New Approach to the Representation Theory of the Symmetric Group, III: Induced Representations and the Frobenius-Young Correspondence” is discussed. In tracing through Vershik’s line of reasoning, a flaw emerges in his attempt to prove one of the key lemmas. The attempted proof is an induction argument which, if valid, would lead to a purely combinatorial proof of the Frobenius-Young correspondence. Vershik asserts that two statements are equivalent, while in fact the implication is true in only one direction. Counterexamples are then given to Vershik’s combinatorial statement and are then translated into counterexamples in representation theory. Finally, possible directions for further study are noted.