Symmetric-group characters and the Frobenius map

Write mi for the number of parts of size i in λ. A permutation commuting with a fixed cycle type can rotate each i-cycle in i ways and permute the mi equal-length cycles in mi factorial ways. Multiplying these independent choices gives zλ. The conjugacy class then has n factorial divided by zλ elements, by orbit–stabilizer for conjugation. This argument explains why repeated fixed points contribute a factorial even though their cycle lengths are all one.

The Frobenius map sums χ(λ)pλ/zλ over partitions of n. It is a linear isomorphism between degree-n class functions and homogeneous degree-n symmetric functions over C. It is not a map from a single group element to its character value, and it is not a formal series over different n unless one explicitly takes the graded direct sum. The familiar factor 1/n factorial appears equivalently when summing over all permutations instead of over cycle types.

Declare power sums orthogonal with squared norm zλ. Over complex coefficients we use the Hermitian extension, conjugate-linear in the second argument, so it matches the character inner product linear in its first argument. Substituting the characteristic formula cancels one centralizer factor and reproduces the class-weighted character sum. In particular, irreducible complex characters map to an orthonormal basis, namely the Schur functions indexed by the same partitions. This identification is a theorem of symmetric-group representation theory, not a consequence of orthogonality alone.

The trivial character maps to hn and the sign character maps to en. For n=2, the power-sum formulas are (p1 squared plus p2)/2 and (p1 squared minus p2)/2. The denominator is compatible with their integral monomial coefficients because the power-sum basis is rational rather than integral. This is a useful connection between the ring issue from the first lesson and the averaging normalization imposed by character theory.

Take an Sm representation V and an Sn representation W. Form their external tensor product on Sm times Sn, embed that subgroup into Sm+n by acting on disjoint blocks, and induce. Its Frobenius characteristic is the ordinary product ch(V)ch(W). Thus LR coefficients describe these induced multiplicities. Restriction gives the corresponding coproduct, splitting one group size into two; this anticipates the Hopf structure of the next part of the module.

Pointwise multiplication of two characters of the same Sn instead describes their internal tensor product. Under the Frobenius map this becomes the internal, or Kronecker, product rather than ordinary symmetric-function multiplication. The degrees distinguish them: ordinary multiplication adds degrees, whereas an internal tensor product stays at degree n. Confusing these operations can produce a formally tidy expression that belongs to the wrong symmetric group, so checking degree is a quick and decisive diagnostic.