Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms

Depends on

• Linear map between vector spaces over the same field
• Vector space over a field

Used by

• A faithful canonical realization that is not reflection faithful: the affine rank-two system Counterexample
• Hard Lefschetz fails for a non-dominant degree-two element: the rank-one case Counterexample
• Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR Definition
• The local intersection form on Hom(B_y, Bₓ Bₛ) Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• The standard bilinear form ⟨ x,y⟩=∑_i<nxᵢyᵢ on Fⁿ Definition
• The trace form (x,y)↦ Tr_K/F(xy) of a finite extension Definition
• The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction Definition
• A null normal admits no reflection of the displayed form Example
• Reflection matrices in a positive plane, a Lorentzian plane, and a plane with radical Example
• The A-tilde 1 infinity edge, separated from finite dihedral families and from the 4-edge Example
• The form on ℝ² with matrix beginpmatrix0&10&0 endpmatrix is neither symmetric nor alternating Example
• FALSE: ⟨ x,y⟩=∑ᵢxᵢyᵢ makes F₂ⁿ an inner product space False statement
• A nondegenerate symmetric form with a half-dimensional isotropic subspace has zero signature Lemma
• Crystallographic alcove diagrams: the affine list realized by Weyl types A–G Lemma
• Determinant criterion: if the matrix (⟨ vᵢ,wⱼ⟩)_i,j is invertible then both v₁,…,vₘ and w₁,…,wₘ are linearly independent Lemma
• Diagonal criterion: if ⟨ vᵢ,wᵢ⟩≠0 and ⟨ vᵢ,wⱼ⟩=0 for i≠ j, then v₁,…,vₘ are linearly independent Lemma
• Disconnected diagrams, direct products, and comparison of invariant forms Lemma
• Enumeration of the connected positive semidefinite corank-one diagrams Lemma
• Exclusions for positive definite diagrams: trees, valency, labels, chains and arms Lemma
• Gram-map image, full-rank fibers, and rank-stratum dimensions Lemma
• Hard Lefschetz gives the primitive decomposition, orthogonality and the signature formula Lemma
• Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary Lemma
• Invariant forms on an indecomposable Soergel bimodule are unique up to scalar, non-degenerate, and agree with the restricted Bott-Samelson form Lemma
• Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions Lemma
• Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order Lemma
• The harmonic complement of the coinvariant algebra Lemma
• The nilpotent Lefschetz filtration is canonical and self-dual Lemma
• The signature is independent of the diagonalizing basis and unchanged by scalar extension from the rationals to the reals Lemma
• The tensor product of nondegenerate real symmetric forms has multiplicative signature Lemma
• Triangular criterion: if ⟨ vᵢ,wᵢ⟩≠0 and ⟨ vᵢ,wⱼ⟩=0 for j<i, then v₁,…,vₘ are linearly independent Lemma
• Alternating forms are skew-symmetric; the converse holds when charF≠2, while in characteristic 2 alternating forms are symmetric Theorem
• Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^* Theorem
• Classification of finite Coxeter systems, including the H and dihedral families Theorem
• Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even Theorem
• Finiteness criterion: W is finite exactly when the Coxeter form is positive definite Theorem
• If charF≠2, quadratic forms and symmetric bilinear forms correspond by q(v)=B(v,v) and B(u,v)=1/2 b_q(u,v) Theorem
• The interior of the Tits cone, finite parabolic stabilizers, and local finiteness Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• K. Conrad, Bilinear Forms (standard reference, not scraped)
• H. Pinkham, Linear Algebra, Chapter 7 (standard reference, not scraped)