Chapter XI. Representations and Classification of Semisimple Groups
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 7 definitions, 6 theorems, 9 propositions, 3 lemmas, 2 corollaries, 7 constructions and 8 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition/result: weights and multiplicities of a rational representation
For a rational G-module V, a weight is a character λ∈X(T) with V_λ={v | t·v=λ(t)v}; dim V_λ is its multiplicity. W permutes weights and preserves multiplicities.
Definition/result: abstract weight lattice and dominant weights
Every representation weight is an abstract weight: its Cartan integers against all roots are integral. Abstract weights form a lattice Λ containing the root lattice Λ_r with finite index; a base gives fundamental dominant weights and the dominant cone.
Definition: fundamental group, simply connected type and adjoint type
For semisimple G, π(G)=Λ/X(T). G is simply connected (universal type) when X(T)=Λ, and of adjoint type when X(T)=Λ_r.
Definition: maximal vector
A nonzero v in a rational G-module is maximal relative to B when Kv is B-stable, equivalently v is a weight vector fixed by all positive root groups.
Proposition: module generated by a maximal vector
If v is maximal of weight λ and V′=⟨Gv⟩, then every weight is λ minus a nonnegative integral combination of positive roots, λ has multiplicity one, and V′ has a unique maximal proper submodule.
Corollary: highest weight of a maximal-generated module is dominant
The maximal weight λ occurring in the preceding proposition is dominant.
Definition: irreducible rational G-module
A nonzero rational G-module is irreducible when its only G-stable subspaces are 0 and itself.
Theorem/recorded result: geometric reductivity of reductive groups
The source records Haboush’s theorem: every reductive algebraic group is geometrically reductive, in the stated invariant-polynomial sense; this is sufficient for finite generation of invariant rings.
Theorem: highest-weight classification of irreducible modules
Every irreducible rational G-module has a unique B-stable line generated by a maximal vector of dominant highest weight λ with multiplicity one; all other weights are lower by nonnegative positive-root combinations and W preserves multiplicities. Two irreducibles are isomorphic iff their highest weights agree.
Construction/result: addition of highest weights under tensor products
If V,V′ contain maximal vectors of weights λ,μ, then their tensor product contains a maximal vector of weight λ+μ; hence existence for λ and μ implies existence for λ+μ.
Construction: maximal-parabolic representations realizing multiples of fundamental weights
Chevalley’s stabilizer theorem applied to maximal parabolics P_{Δ−{α_i}} gives rational modules with maximal vectors of weights d_iω_i for positive integers d_i.
Construction: canonical highest-weight matrix coefficient
Given highest vector v_λ and dual functional r selecting it, c_λ(g)=r(gv_λ) is a regular matrix coefficient satisfying c_λ(xgy)=λ(x) c_λ(g) λ(y) for x in the opposite Borel and y in B, with the corresponding conventions.
Theorem: existence of irreducible modules of every dominant character weight
For every dominant λ∈X(T), there exists an irreducible rational G-module of highest weight λ.
Construction/result: function-space realization of highest-weight modules
For dominant λ, the semi-invariant function space F_λ⊆K[G] defined by left opposite-Borel covariance is right-translation stable; its canonical maximal vector c_λ generates the irreducible module V(λ), and in characteristic zero F_λ itself is irreducible by complete reducibility.
Definition/result: minimal dominant highest weights
A dominant λ is minimal when no smaller dominant weight exists. Then V(λ) has exactly the W-orbit of λ as its weights, each with multiplicity one, so dim V(λ)=|W·λ|; the source records the fundamental-weight examples by type.
Theorem/result: highest weight of the contragredient
If V(λ) is irreducible and w_0 sends B to the opposite Borel, then V(λ)^* is irreducible of highest weight −w_0(λ). Thus V(λ) is self-dual exactly when −w_0(λ)=λ.
Result: invariant bilinear form on a self-dual irreducible
If V≅V^* is irreducible, Schur’s lemma identifies a one-dimensional space of G-invariant bilinear forms on V, hence such a form is unique up to scalar.
Construction: rank-one and rank-two reductive subgroups used in classification
For simple roots α,β, set Z_α=C_G(T_α) and Z_{αβ}=C_G((T_α∩T_β)°). The latter is reductive of semisimple rank two with root subsystem generated by α,β and is a Levi factor of the corresponding standard parabolic.
Definition: isogeny of algebraic groups
An isogeny G→G′ is a surjective morphism of algebraic groups with finite kernel.
Theorem: classification of simple algebraic groups by root system and fundamental group
Simple algebraic groups with isomorphic root systems and fundamental groups are isomorphic, except that for type D_ℓ with even ℓ≥6 and fundamental group of order 2 there can be two isomorphism types.
Theorem: extension of a based torus isomorphism to semisimple groups
If semisimple G,G′ have maximal tori T,T′ and an isomorphism T→T′ whose pullback identifies their root systems, then it extends to an algebraic-group isomorphism G→G′.
Lemma: morphism criterion for an abstract homomorphism from a semisimple group
If an abstract group homomorphism from semisimple G to an algebraic group H restricts to a morphism on a Borel subgroup, then it is a morphism on G.
Proposition: extension criterion from T to N(T)
After choosing simple-reflection representatives n_α∈N and their rank-two relation elements t_{αβ}, a homomorphism ψ:T→H extends to N with prescribed images h_α exactly when the corresponding Coxeter-power relations among h_α equal ψ(t_{αβ}) for all simple α,β.
Proposition: canonical extension across a rank-one subgroup
An isomorphism T→T′ carrying a root α′ to α extends uniquely to an isomorphism Z_α→Z′_{α′} compatible with chosen admissible root-group parametrizations and simple-reflection representatives.
Corollary: classification of semisimple rank-one groups
Every semisimple algebraic group of rank one is isomorphic to SL(2,K) or PGL(2,K).
Construction: rank-two compatibility conditions (A), (B), (C)
The extension proof is reduced to three rank-two conditions: compatibility of Coxeter relations in N; compatibility of root-group isomorphisms on U_{αβ}; and compatibility of conjugation by chosen simple-reflection representatives.
Proposition: consistent isomorphisms of all root subgroups
Assuming the rank-two conditions, the chosen simple-root isomorphisms extend uniquely and consistently to isomorphisms of every U_γ, equivariant under conjugation by N.
Lemma: Chevalley commutator shape
For roots α,β and admissible parametrizations, the commutator of x_α(u) and x_β(v) is an ordered product over roots rα+sβ with r,s>0, whose parameters are constants times u^r v^s. If no such roots exist, U_α and U_β commute.
Result: extension to a Borel subgroup
Under conditions (A)–(C), the torus and root-subgroup maps combine to an isomorphism of algebraic groups B→B′; rank-two commutator formulas supply multiplicativity.
Lemma: extension of a generically multiplicative morphism
A morphism defined on a nonempty open subset of a connected algebraic group and multiplicative whenever x,y,xy remain in that open subset extends uniquely to an algebraic-group morphism on the whole group.
Proposition: generic multiplicativity of the classification map
Assuming (A)–(C), the set-theoretic bijection built from Bruhat normal forms satisfies φ(xy)=φ(x)φ(y) whenever x,y,xy lie in the big cell.
Result: rank-two reduction completes the semisimple isomorphism theorem
Generic multiplicativity and the open-extension lemma make the Bruhat-defined bijection an algebraic-group isomorphism, reducing Theorem 32.1 entirely to the rank-two conditions checked in §33.
Program/result: rank-two relations determine the global extension data
The source reformulates conditions (A)–(C) as intrinsic relations in each rank-two root subsystem: Coxeter representatives, root-group commutators, and conjugation on root vectors are determined by the root-system type and the initial simple-root parametrizations.
Construction/result: coroot system in the cocharacter lattice
For each simple α, the rank-one subgroup determines a cocharacter α∨ with ⟨α,α∨⟩=2 and t_α=α∨(−1). These coroots form the dual root system Φ∨ in R⊗Y(T), with Weyl action governed by the Cartan integers.
Proposition: rank-two relations of type A₁×A₁
For orthogonal simple roots α,β, the chosen simple-reflection representatives commute up to the stated toral squares, U_α and U_β commute, and conjugation fixes the opposite simple root vectors as prescribed.
Proposition: rank-two relations of type A₂
With the canonical normalization of the α+β root vector, the source gives the Coxeter-representative relation, the commutator x_β(y)x_α(x)=x_α(x)x_β(y)x_{α+β}(xy), and the corresponding simple-reflection actions on the three positive root vectors.
Proposition: rank-two relations of type B₂
For positive roots α,β,α+β,2α+β with α short, the source records the Coxeter representative relation, normalized Chevalley commutator formulas with coefficients 1 and 2, and the simple-reflection actions on the positive root vectors.
Proposition: rank-two relations of type G₂
For the six positive roots of G₂, the source records the Coxeter representative relation, the full normalized Chevalley commutator formulas with coefficients 1,2,3, and the simple-reflection actions on the chosen root vectors.
Result: rank-two verification of the isomorphism theorem
The A₁×A₁, A₂, B₂ and G₂ computations verify all conditions (A)–(C), thereby completing the proof of the semisimple isomorphism theorem.
Recorded results: classical existence models for simple groups
The source records the standard existence models: SL and its quotients for A, symplectic groups for C, orthogonal/projective orthogonal groups for B and D, with spin and half-spin groups supplying the simply connected/intermediate D-types.
Recorded results: exceptional existence models
Groups of types G₂ and F₄ arise as automorphism groups of the Cayley and exceptional Jordan algebras respectively; analogous constructions produce E₆,E₇,E₈.
Recorded construction: Chevalley construction and existence of all semisimple types
Chevalley’s integral-basis construction produces adjoint groups over arbitrary fields; projective-representation or integral highest-weight methods then produce simply connected and intermediate forms, yielding semisimple groups with every allowed fundamental group.