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Induction-restriction adjunction

读物

• https://ncatlab.org/nlab/show/induced+representation
• https://www.uni-math.gwdg.de/tammo/rep.pdf Chapter 4 Induced Representations

Definitions of adjunction

• The universal mapping property definition of adjunction relates the equivariant maps that I am trying to understand.

Internal structure vs. external relationships

• The induction-restriction adjunction is based on breaking up the action into an internal coding (within the vector space) and an external coding (amongst the cosets, the copies of the vector space) so that it can be done and undone.
• Given R-modules (where R=CH) and S-modules (where S=CG) we get two adjoints that turn an H-module into a G-module. The right adjoint, the produced representation HomR(S,W) lets you add S homomorphism extensions externally. It takes W to the fixed points under G. If G is the trivial group, then derived functors are called cohomology of H. The left adjoint, the induced representation S XR W lets you multiply by S on the left, so that works internally, in the mathematical structure. It takes W to the quotient W/(1-g)W. If G is the trivial group, then derived functors are called homology of H. Left adjoints are right exact and right adjoints are left exact, so if the left adjoint equals the right adjoint, then they are exact, they preserve exact sequences. Exact sequences express divisions of everything internally, and adjoint strings express them externally.
• So the equivalence of the left-adjoint and right-adjoint for turning an H-module into a G-module is the equivalence of external relations and internal structure. And this is the point of Schur-Weyl duality and it is reflected in the construction of the representation theory of the symmetric group and of the general linear group.

Inner product - order out of chaos

• An inner product like <x,x'> from V and its dual V' divides up space into two subsystems. Adjoint functors coordinate the division.
• Chaos is natural, such as the ugly variety of possible bases for a vector space. Order is not natural. Unitary matrix or operator implies orthogonality of basis which is orderly and not natural. It is also a sign of there being an observer. Adjunctions break down the orderly components needed to generate a coordinate system and impose order, to relate two subsystems, and so on.
• Math consists of a small set (twelve?) of ways of adding a node (variable, label, etc.) Consider the kinds of variables. And in each case removing that variable is trivial because the information is already explicit (forgetting it), whereas the adjoint functor of adding the variable is quite involved (a free construction). And each adjoint functor is in a different branch of math, thus they distinguish the branches of math. And in that branch of math you have to spend a month mastering the absolute basics to understand that trivial adjunction. But if you knew all of those trivial adjunctions then you could build up everything, all of math. You could also explain what is involved in building up, for example, coordinate systems, order out of chaos.
• Given a vector subspace, its complement is not unique. Likewise, bases are not unique. However, there is a unique "best basis" which is orthonormal. (In what sense is that a universal property?) And with regard to that basis, the representation is in terms of unitary matrices.
• The induced representation involves an ambiguous understanding of w+0+0 in W+W+W as w in W.

Adjunctions

• Adjunctions are the basis for recurring chemical processes, biological, neurological processes.

Adjoint string

• Adjoint string - divisions, adjunction - topology - one perspective in a division, one variable added.