The product and the sum of the degrees of a polynomial invariant algebra #
Let a finite group G act linearly on a finite-dimensional vector space M over a field K of
characteristic zero, and suppose that the invariant subalgebra Sym(M)^G has the Hilbert series of
a polynomial algebra on homogeneous generators of degrees d₁, …, d_r:
Hilb(Sym(M)^G, t) = ∏ᵢ (1 - t ^ dᵢ)⁻¹.
This is the case when Sym(M)^G is a polynomial algebra on such generators, for example when G
is generated by pseudoreflections (the Chevalley–Shephard–Todd theorem). If moreover the action is
faithful, then
r = dim M,d₁ d₂ ⋯ d_r = #G,(d₁ - 1) + ⋯ + (d_r - 1) = N, the number of pseudoreflections inG, the elements whose fixed space is a hyperplane. Equivalentlyd₁ + ⋯ + d_r = dim M + N.
For a finite real reflection group the pseudoreflections are the reflections, so these are the
identities ∏ dᵢ = |W| and ∑ dᵢ = r + |Ref(W)|.
Taking M = V* with the contragredient representation gives the formulas for the degrees of the
polynomial functions on a representation V: the contragredient representation is faithful
exactly when ρ is, and it has the same pseudoreflections, so N may be counted on V itself.
The proof compares the expansions at t = 1 of both sides of Molien's formula
Hilb(Sym(M)^G, t) = #G⁻¹ ∑_g det(1 - t g)⁻¹. Write n = dim M. For g ∈ G with fixed space of
dimension m, det(1 - t g) vanishes to order exactly m at t = 1, because g is semisimple
(Module.End.IsFinitelySemisimple.exists_reverse_charpoly_eq_one_sub_X_pow_mul). So the pole of
order n comes only from the kernel of the action, where det(1 - t g) = (1 - t) ^ n, and the
pole of order n - 1 only from the pseudoreflections s, where
det(1 - t s) = (1 - t) ^ (n - 1) (1 - det(s) t)
(Module.End.reverse_charpoly_of_finrank_fixedSubmodule_add_one). On the other side,
1 - t ^ d = (1 - t)(d - d(d - 1)/2 (1 - t) + ⋯). Comparing the orders of the poles gives
r = n, comparing the leading coefficients gives #ker · ∏ dᵢ = #G, and comparing the next
coefficients gives #ker · ∑ (dᵢ - 1) = 2 ∑_s (1 - det s)⁻¹. Pairing s with s⁻¹, whose
determinant is (det s)⁻¹, and using (1 - λ)⁻¹ + (1 - λ⁻¹)⁻¹ = 1 turns the right-hand side into
the number of pseudoreflections.
To compare expansions at t = 1 with formal power series, the identity is cleared of
denominators in K[t] and then mapped to K⟦u⟧ by the substitution t = 1 - u.
Main results #
Representation.card_eq_finrank_of_sum_inv_reverse_charpoly_mul_prod_eq,Representation.natCard_ker_mul_prod_eq_card_of_sum_inv_reverse_charpoly_mul_prod_eq,Representation.natCard_ker_mul_sum_sub_one_eq_card_of_sum_inv_reverse_charpoly_mul_prod_eq: if the Molien series#G⁻¹ ∑_g det(1 - t g)⁻¹is∏ᵢ (1 - t ^ dᵢ)⁻¹, thenr = dim M,#ker ρ · ∏ dᵢ = #Gand#ker ρ · ∑ (dᵢ - 1)is the number of pseudoreflections.Representation.prod_eq_card_of_mk_finrank_mul_prod_eq_one: for a faithful action whose invariant subalgebra has Hilbert series∏ᵢ (1 - t ^ dᵢ)⁻¹,∏ dᵢ = #G.Representation.sum_eq_finrank_add_card_of_mk_finrank_mul_prod_eq_one: for such an action,∑ dᵢ = dim M + N, withNthe number of pseudoreflections.Representation.prod_eq_card_of_bijective_aeval_symmetricAlgebraInvariants,Representation.sum_eq_finrank_add_card_of_bijective_aeval_symmetricAlgebraInvariants: the same for a system of basic invariants, homogeneous invariants on whichSym(M)^Gis a polynomial algebra.Representation.sum_eq_finrank_add_card_of_bijective_aeval_symmetricAlgebraInvariants_dual: the sum formula for the polynomial functionsSym(M*)^G, with the pseudoreflections counted onM.
References #
- J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press (1990), § 3.9.
- R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475–511.
The number of factors of the Molien series. For a finite group acting linearly on a
finite-dimensional vector space M over a field of characteristic zero, if the Molien series
#G⁻¹ ∑_g det(1 - t g)⁻¹ is ∏ᵢ (1 - t ^ dᵢ)⁻¹, then there are dim M factors.
The product of the degrees. For a finite group acting linearly on a finite-dimensional
vector space M over a field of characteristic zero, if the Molien series
#G⁻¹ ∑_g det(1 - t g)⁻¹ is ∏ᵢ (1 - t ^ dᵢ)⁻¹, then #ker ρ * ∏ dᵢ = #G.
The sum of the degrees. For a finite group acting linearly on a finite-dimensional vector
space M over a field of characteristic zero, if the Molien series #G⁻¹ ∑_g det(1 - t g)⁻¹ is
∏ᵢ (1 - t ^ dᵢ)⁻¹, then #ker ρ * ∑ (dᵢ - 1) is the number of pseudoreflections, the elements
whose fixed space is a hyperplane.
The product of the degrees of a polynomial invariant algebra. Let a finite group act
faithfully and linearly on a finite-dimensional vector space M over a field of characteristic
zero. If the invariant subalgebra Sym(M)^G has the Hilbert series ∏ᵢ (1 - t ^ dᵢ)⁻¹ of a
polynomial algebra on homogeneous generators of degrees dᵢ, then ∏ dᵢ = #G.
The sum of the degrees of a polynomial invariant algebra. Let a finite group act faithfully
and linearly on a finite-dimensional vector space M over a field of characteristic zero. If the
invariant subalgebra Sym(M)^G has the Hilbert series ∏ᵢ (1 - t ^ dᵢ)⁻¹ of a polynomial algebra
on homogeneous generators of degrees dᵢ, then ∑ dᵢ = dim M + N, where N is the number of
pseudoreflections, the elements whose fixed space is a hyperplane.
The product of the degrees of basic invariants is the order of the group. Let a finite
group act faithfully and linearly on a finite-dimensional vector space M over a field of
characteristic zero. If the invariant subalgebra Sym(M)^G is a polynomial algebra on invariants
f i homogeneous of degrees d i, then ∏ d i = #G.
The sum of the degrees of basic invariants counts the pseudoreflections. Let a finite group
act faithfully and linearly on a finite-dimensional vector space M over a field of characteristic
zero. If the invariant subalgebra Sym(M)^G is a polynomial algebra on invariants f i
homogeneous of degrees d i, then ∑ d i = dim M + N, where N is the number of
pseudoreflections, the elements whose fixed space is a hyperplane.
For homogeneous basic invariants of a faithful finite action in characteristic zero,
the sum of the degrees minus one is the number of pseudoreflections on M.
The sum of the degrees of basic polynomial functions counts the pseudoreflections. Let a
finite group act faithfully and linearly on a finite-dimensional vector space M over a field of
characteristic zero. If the polynomial functions invariant under the contragredient action,
Sym(M*)^G, form a polynomial algebra on invariants f i homogeneous of degrees d i, then
∑ d i = dim M + N, where N is the number of pseudoreflections of G on M. This is
Representation.sum_eq_finrank_add_card_of_bijective_aeval_symmetricAlgebraInvariants for
ρ.dual, which has the same pseudoreflections as ρ.
For homogeneous basic polynomial functions of a faithful finite action in characteristic
zero, the sum of the degrees minus one is the number of pseudoreflections on M.