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S.Duzhin
COMBINATORICS OF SYMMETRIC GROUPS
Lecture 4: Representations of the symmetric group
From the combinatorial point of view, the main theorem of the representation theory of
symmetric groups claims the isomorphism of two infinite graphs: the Young graph, built out
of Young diagrams, and the Frobenius graph that comes from irreducible representations
of the symmetric groups.
Definition. The vertices of the Young graph Y are Young diagrams. Two diagrams,
[] and [], are connected by an edge going from [] to [], if jj = jj + 1 and [] can be
obtained from [] by the addition of one cell.
Definition. The vertices of the Frobenius graph F are equivalence classes of the
irreducible representations of groups S n for all n  1. Two representations, U of S n 1
and
V of S n , are connected by an edge, if U appears in the decomposition of the representation
V , restricted to the subgroup S n 1  S n . [We remind that (1) the multiplicity of U in the
restriction can only be 0 or 1, and (2) this multiplicity is equal to the multiplicity of V in
the representation of S n , induced from U .]
Main theorem. Y  = F .
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