The Frobenius formula for product-one triples #
Fix a finite group G and three conjugacy classes C₀, C₁, C∞ of it. The product-one
triples of that data are the triples (x, y, z) with x ∈ C₀, y ∈ C₁, z ∈ C∞ and
z * y * x = 1. They are the finite shadow of a covering of the sphere branched over three points
with prescribed local monodromy, and counting them is the first step in counting such coverings.
Two counts are proved here.
Splitting a triple into its last entry and a factorization
y * x = z⁻¹counts the triples as|C∞|times a structure constant of the class algebra (TauCeti.card_productOneTriples).Feeding that structure constant through the central characters turns the count into a sum over the irreducible characters, the Frobenius formula (
TauCeti.card_productOneTriples_eq_sum_characterTable)#{(x, y, z) | …} = (|C₀| · |C₁| · |C∞| / |G|) · ∑_χ χ(C₀) χ(C₁) χ(C∞) / χ(1).
The route is the class algebra rather than a direct manipulation of characters: the values of a
central character on the class sums are a common left eigenrow of the class-multiplication matrices
(TauCeti.isClassEigenrow_centralCharacterTable), which is exactly the structure-constant identity
∑_C aᵢⱼC ω(K_C) = ω(K_Cᵢ) ω(K_Cⱼ); the second orthogonality relation inverts it, and
TauCeti.centralCharacterTable_eq_div converts between ω and the character table. Characteristic
zero enters only through that conversion, which divides by the degrees χ(1).
What the count is not: it counts triples, not isomorphism classes; it puts no generation condition
on the three entries; and its data are three conjugacy classes of G, not three cycle types. For a
permutation group G the two differ: a single cycle type of the ambient symmetric group can split
into several G-conjugacy classes, and then a count of triples of prescribed cycle types is a sum
of several of these counts.
Main definitions #
TauCeti.productOneTriples: the triples with entries in three prescribed conjugacy classes whose product, in the orderz * y * x, is1.
Main statements #
TauCeti.card_productOneTriples: the count as|C∞|times a structure constant.TauCeti.structureConstant_eq_sum_characterTable: the structure constants of the class algebra, read off the character table.TauCeti.card_productOneTriples_eq_sum_characterTable: the Frobenius formula.
References #
- I. M. Isaacs, Character Theory of Finite Groups (1976), Problem 3.9.
- J.-P. Serre, Topics in Galois Theory, 2nd ed. (2008), §7.2.
- S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications (2004), §5.3, for the use of the formula in counting coverings.
The product-one triples with entries in the conjugacy classes C₀, C₁, C∞: the triples
(x, y, z) with x ∈ C₀, y ∈ C₁, z ∈ C∞ and z * y * x = 1.
The order of the product is the one in which a triple of loops around three branch points concatenates to a nullhomotopic loop.
Equations
- TauCeti.productOneTriples C0 C1 Cinf = {p : G × G × G | ConjClasses.mk p.1 = C0 ∧ ConjClasses.mk p.2.1 = C1 ∧ ConjClasses.mk p.2.2 = Cinf ∧ p.2.2 * p.2.1 * p.1 = 1}
Instances For
The product-one triples fibre over their last entry. Over z ∈ C∞ the fibre consists of the
factorizations y * x = z⁻¹ with y ∈ C₁ and x ∈ C₀, so a structure constant of the class
algebra counts it, independently of z.
The structure constants of the class algebra, read off the character table. The number of
factorizations x * y = g with x ∈ Cᵢ, y ∈ Cⱼ and g a representative of Cₖ is
(|Cᵢ| · |Cⱼ| / |G|) · ∑_χ χ(Cᵢ) χ(Cⱼ) χ(Cₖ⁻¹) / χ(1),
the sum running over the irreducible characters of G.
The Frobenius formula. The number of triples (x, y, z) with x ∈ C₀, y ∈ C₁,
z ∈ C∞ and z * y * x = 1 is
(|C₀| · |C₁| · |C∞| / |G|) · ∑_χ χ(C₀) χ(C₁) χ(C∞) / χ(1),
the sum running over the irreducible characters of G.