Chapter 3. Polynomial invariants of finite reflection groups
FC15 — Humphreys, Reflection Groups and Coxeter Groups.
This chapter contains 9 definitions, 8 theorems, 12 propositions, 6 lemmas, 3 corollaries, 1 example, 3 constructions and 3 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Construction: polynomial functions and invariant ring
For finite-dimensional V over characteristic-zero K, S=S(V) is the graded polynomial algebra on V, with contragredient G-action (g·f)(v)=f(g^{-1}v). The invariant ring is R=S^G.
Proposition: fixed field is the fraction field of the invariant ring
For finite G≤GL(V), the fraction field of R=S^G is exactly the fixed field of G in Frac(S); hence trdeg_K Frac(R)=dim V.
Construction: Reynolds/averaging operator
The averaging map f↦f=(1/|G|)Σ_{g∈G}g·f is a degree-preserving linear projection S→R, fixes R pointwise, and satisfies (pq)=p* q for p∈S,q∈R.
Proposition: finite algebra generation from generators of the Hilbert ideal
Let I=SR_+ be the ideal generated by positive-degree invariants. If homogeneous f_1,…,f_r∈R_+ generate I as an ideal of S, then f_1,…,f_r generate R as a K-algebra.
Lemma: linear divisibility criterion
If a polynomial f vanishes on the zero hyperplane of a nonzero homogeneous linear polynomial l, then l divides f.
Lemma: Chevalley key dependence lemma
For finite reflection W, let I=SR_+. If invariant homogeneous f_1,…,f_r have f_1 outside the ideal of R generated by the others and homogeneous g_i satisfy Σf_i g_i=0, then g_1∈I.
Theorem: Chevalley theorem on polynomial invariants
For essential finite reflection group W on n-dimensional real V, R=S^W is a polynomial algebra on n homogeneous algebraically independent invariants of positive degree.
Definition: basic invariants
A homogeneous algebraically independent generating set f_1,…,f_n of R is a set of basic invariants.
Proposition: module of covariants is free
Viewed as an R-module, S is free of rank |W|; a homogeneous vector-space basis of the coinvariant algebra S/I lifts to an R-module basis of S.
Theorem/recorded result: coinvariant algebra affords the regular representation
The coinvariant algebra S/I has dimension |W| and, as an ungraded W-module, is isomorphic to the regular representation.
Definition: degrees of a finite reflection group
The multiset of degrees d_1,…,d_n of basic invariants is independent of the chosen basic invariants and is called the set of degrees of W.
Proposition: uniqueness of the degrees
If two homogeneous algebraically independent generating sets have degree lists d_i and e_i, then after reordering d_i=e_i for all i.
Comparison/table: degree table for irreducible finite reflection groups
The source tabulates the degrees: A_n:2,3,…,n+1; B_n:2,4,…,2n; D_n:2,4,…,2n−2,n; E_6:2,5,6,8,9,12; E_7:2,6,8,10,12,14,18; E_8:2,8,12,14,18,20,24,30; F_4:2,6,8,12; G_2:2,6; H_3:2,6,10; H_4:2,12,20,30; I_2(m):2,m.
Lemma: dimension of invariants by averaging traces
For any finite-dimensional W-module E over characteristic zero, dim E^W=Tr((1/|W|)Σ_{w∈W}w | E).
Proposition: Molien formula for a finite reflection group
As formal power series, (1/|W|)Σ_{w∈W}1/det(1−tw)=∏_{i=1}n(1−t{d_i})^{-1}.
Theorem: sum and product of degrees
If N is the number of reflections/positive roots, then ∏_i d_i=|W| and Σ_i d_i=N+n, equivalently Σ_i(d_i−1)=N.
Theorem/recorded result: Shephard–Todd fixed-space enumerator
If ∏_i(1+(d_i−1)t)=Σ_k a_k t^k, then a_k is the number of elements of W whose fixed-point space has codimension k.
Definition: Jacobian of n polynomials
For f_1,…,f_n∈K[x_1,…,x_n], J(f_1,…,f_n) is the determinant of the matrix (∂f_i/∂x_j).
Proposition: Jacobian criterion for algebraic independence
Over characteristic-zero K, f_1,…,f_n are algebraically independent iff J(f_1,…,f_n) is not the zero polynomial.
Corollary: degree of the Jacobian of homogeneous independent polynomials
If homogeneous algebraically independent f_i have degrees d_i, then J(f_1,…,f_n) is homogeneous of degree Σ(d_i−1); for basic invariants this degree is N.
Theorem: Shephard–Todd converse theorem
If a finite G≤GL(V) over R has invariant ring generated by n algebraically independent homogeneous polynomials, then G is generated by its reflections.
Definition/result: pseudo-reflections and complex reflection groups
A pseudo-reflection is a finite-order complex linear map with fixed space of codimension one. Finite groups generated by pseudo-reflections (unitary/complex reflection groups) also have polynomial invariant rings with well-defined degrees whose product is the group order; Shephard–Todd classified them.
Proposition: criterion for a proposed set of basic invariants
If homogeneous W-invariants g_1,…,g_n are algebraically independent and the product of their degrees equals |W|, then they form a basic set of invariants.
Examples: explicit basic invariants for An,Bn,Dn
The source gives power-sum-type basic invariants for A_n, even power sums for B_n, and even power sums together with x_1⋯x_n for D_n; their Jacobians verify algebraic independence and recover the degree lists.
Definition: alternating polynomial
A polynomial f∈S is alternating (anti-/skew-invariant) when w·f=det(w)f for every w∈W. The alternating subspace A is graded.
Proposition: factorization of the basic-invariant Jacobian
For basic invariants and J their Jacobian: J=k∏{α∈Φ^+}l_α for nonzero scalar k, where l_α(v)=(α,v); f is alternating iff f=Jr with r∈R; consequently dim A_k=dim R{k−N}.
Definition/construction: induction and restriction of class functions
For finite G and H≤G, restriction carries class functions G→C to H. A class function φ on H induces φG(g)=|H|{-1}Σ_{x:xgx{-1}∈H}φ(xgx{-1}); induced 1_H counts fixed left cosets.
Lemma: basic identities for induced class functions
For class function χ on G, χ·(1_H)G=(χ_H)G; and for class function φ on H, the average of φ^G over G equals the average of φ over H.
Lemma: alternating sum of parabolic invariant dimensions
If R_I is the invariant ring of W_I and A_k the degree-k alternating polynomials, then Σ_{I⊆S}(−1)^{|I|}dim(R_I)_k=dim A_k.
Theorem: factorization of the Poincaré polynomial
The Poincaré polynomial factors as W(t)=Σ_{w∈W}t^{ℓ(w)}=∏_{i=1}n(1+t+⋯+t{d_i−1})=∏_i(t^{d_i}−1)/(t−1).
Definition: Coxeter element
For irreducible W with simple reflections s_1,…,s_n, a product s_1⋯s_n in any order is a Coxeter element.
Proposition: conjugacy of Coxeter elements
All Coxeter elements of an irreducible finite reflection group are conjugate in W.
Definition: Coxeter number and exponents
The common order h of Coxeter elements is the Coxeter number. If ζ is a primitive hth root of unity and the eigenvalues of a Coxeter element are ζ^{m_i}, the integers 0<m_i<h are the exponents, listed increasingly.
Lemma: Coxeter elements have no fixed vectors
A Coxeter element has no eigenvalue 1. The multiset {h−m_i} equals {m_i}, so Σ_i m_i=nh/2.
Construction/result: bipartite factorization and Coxeter plane
The tree Coxeter graph admits a bipartition S=I⊔J into commuting subsets, giving w=yz with involutions y,z. Using the positive Perron eigenvector of the Coxeter form, the source constructs a w-stable Coxeter plane P on which y,z act as reflections and w as rotation through 2π/h.
Proposition: extreme exponents
For irreducible W, the smallest exponent is 1 and the largest is h−1.
Proposition: Coxeter number formula
If N is the number of positive roots/reflections and n=rank W, then h=2N/n.
Corollary: sum identities for exponents
Σ_i m_i=N=Σ_i(d_i−1).
Comparison/table: positive-root counts and Coxeter numbers
The source tabulates N and h for irreducible types; e.g. A_n:N=n(n+1)/2,h=n+1; B_n/C_n:N=n²,h=2n; D_n:N=n(n−1),h=2(n−1); E_6,E_7,E_8:(36,12),(63,18),(120,30); F_4:(24,12); G_2:(6,6); H_3:(15,10); H_4:(60,30); I_2(m):(m,m).
Theorem: degrees are exponents plus one
For irreducible W with exponents m_1,…,m_n, the degrees are m_1+1,…,m_n+1. Hence |W|=∏_i(m_i+1).
Corollary: criterion for −1 to lie in W
The scalar −1 lies in W iff all exponents are odd, equivalently all degrees are even. Then h is even and −1=w^{h/2} for every Coxeter element w.
Proposition: cyclotomic source of Weyl-group exponents
For an irreducible Weyl group with Coxeter number h, every 1≤m≤h−1 relatively prime to h is an exponent.
Definition: root height and height counts
For crystallographic Φ with base Δ, height ht(α) is the sum of simple-root coefficients. Let k_i be the number of positive roots of height i; these form a nonincreasing partition of N.
Theorem: exponents are dual to the root-height partition
For a Weyl group, the decreasing exponent partition m_n≥⋯≥m_1 of N is dual to the partition k_1≥k_2≥⋯, where k_i counts positive roots of height i. In particular the highest root has height h−1.
Identity/result: root-height factorization of the Poincaré polynomial
For a Weyl group, W(t)=∏_{α>0}(t{ht(α)+1}−1)/(t{ht(α)}−1); comparison with the degree factorization yields the dual-partition theorem. The source records a stronger group-algebra identity from which this specializes.