The adjoint representation of the general linear group is matrix conjugation #
The tangent space at the identity of GLₙ is the full matrix algebra
(TauCeti.GeneralLinear.tangentLinearEquivMatrix), and a point of GLₙ acts on it by the
convolution conjugation Ad g d = g ⋆ d ⋆ g⁻¹ (Derivation.adDerivation). This file computes
that action: it is conjugation of the tangent matrix by the invertible matrix of the point,
Ad(g) X = g X g⁻¹.
The proof is the two-fold comultiplication of a generic matrix entry. Evaluating the convolution
product g ⋆ d ⋆ g⁻¹ on Xᵢⱼ runs over comul Xᵢⱼ = ∑ₖ Xᵢₖ ⊗ Xₖⱼ twice, so the (i, j) entry
of the conjugated tangent matrix is the double sum ∑ₖ ∑ₗ gᵢₗ dₗₖ (g⁻¹)ₖⱼ, which is the
(i, j) entry of the matrix product.
Specializing to the diagonal torus computes the expected eigenvalues of the adjoint
representation: conjugation by diag(t) multiplies the (i, j) entry by tᵢ tⱼ⁻¹. Thus the
matrix units have the eigenvalue formulas needed for a future formal weight decomposition, and
the diagonal matrices — the tangent space of the torus itself — are fixed.
Main declarations #
TauCeti.GeneralLinear.counitPointsMulEquiv: the invertible matrix of a point ofGLₙvalued in the counit algebra of the coordinate ring, where the tangent vectors live.TauCeti.GeneralLinear.tangentMatrix_adDerivation: the adjoint action ofGLₙon its tangent space is conjugation of matrices.TauCeti.GeneralLinear.tangentMatrix_adDerivation_apply_of_diagGLandTauCeti.GeneralLinear.tangentMatrix_adDerivation_apply_symm_diagGL: conjugation by a diagonal point scales the(i, j)entry bytᵢ tⱼ⁻¹.TauCeti.GeneralLinear.tangentMatrix_adDerivation_single: the corresponding eigenvalue computation for a matrix unitEᵢⱼ.TauCeti.GeneralLinear.tangentMatrix_adDerivation_diagonal: the tangent space of the diagonal torus is fixed by the adjoint action of the torus.TauCeti.GeneralLinear.tangentMatrix_adDerivation_apply_diagonalTorusPoints: the same computation for a point of the diagonal torus ofGLₙ, indexed by its split-torus coordinates.
References #
- J. S. Milne, Algebraic Groups (2017), §§10.24 and 21.1.
- J. E. Humphreys, Linear Algebraic Groups (1975), §26.3.
- J. C. Jantzen, Representations of Algebraic Groups, I.7.18.
Layer 2 of TauCetiRoadmap/ReductiveGroups/README.md asks for the adjoint action
Ad : G → GL(Lie G), and Layer 7 asks for the root datum of a split pair (G, T), read off the
weight decomposition of Lie G under T. This file supplies the eigenvalue computations needed
for that future decomposition in the worked example GLₙ with its diagonal torus, the roadmap's
prescribed check that the definitions are honest.
The invertible matrix of a point of GLₙ valued in the counit algebra of O(GLₙ).
Tangent vectors at the identity are derivations valued in that counit algebra, so this is the form in which a point conjugating a tangent vector presents itself.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The transported points equivalence is the points equivalence of the transported point.
The matrix of a counit-algebra-valued point evaluates it on the generic matrix entries.
The matrix of the inverse of a counit-algebra-valued point evaluates the inverse point on the generic matrix entries.
The adjoint action of GLₙ on its tangent space is conjugation of matrices.
The tangent vector Ad g d = g ⋆ d ⋆ g⁻¹ has tangent matrix g X g⁻¹, where X is the tangent
matrix of d and g is read as an invertible matrix through
TauCeti.GeneralLinear.counitPointsMulEquiv.
The adjoint action of GLₙ, entrywise: conjugation by g sends the tangent matrix X to
g X g⁻¹.
Conjugation by a diagonal point of GLₙ scales the (i, j) entry of a tangent matrix by
the character tᵢ tⱼ⁻¹ of the diagonal torus.
Conjugation by the diagonal point with coordinates t, with no hypothesis to discharge:
every invertible matrix, in particular every diagonal one, is the matrix of a unique point.
The matrix-unit eigenvalue computation for the adjoint action: conjugation by diag(t)
rescales Eᵢⱼ by tᵢ tⱼ⁻¹.
The tangent space of the diagonal torus is fixed by the adjoint action of the torus: a diagonal tangent matrix is unchanged by conjugation by a diagonal point. This is the fixed-vector computation needed to identify the zero-weight space in a future formal decomposition.
The adjoint action of a point of the diagonal torus of GLₙ: it multiplies the (i, j)
entry of a tangent matrix by the character tᵢ tⱼ⁻¹ read off the split-torus coordinates of
the point.