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Andrius Kulikauskas
 m a t h 4 w i s d o m  g m a i l
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Inductionrestriction adjunction
Definition of induction
 What is the definition of the induced functor in terms of what it does to morphisms? Consider the tensor product definition of induction and what that suggests.
Example
 For the equivariant map in the case of representations of {$S_2$}, solve for all possible values of the representation, and then show how that relates to the equivariant map. And consider also the action of the equivariant map, permuting rows and recalculating, by which it yields the needed answer.
Equivariance
 How does equivariance {$\alpha\circ\psi=\theta\circ\alpha$} relate to adjunction (left,_){$\cong$}(_,right) ?
Yoneda lemma
 Relate the restrictioninduction adjunction to Yoneda's lemma by way of Cauchy's theorem.
Axiom of choice
 Does induction restriction adjunction use the axiom of choice? Do you need to choose coset representatives?
 Can you calculate for all elements without choosing any of them? What about other adjunctions?
Order of composition
 How is the relation between composition by matrix multiplication and by permutation the foundation for SchurWeyl duality?
 Is the composition of permutations contravariant in that it proceeds by insertion?
Internal and external structure
 Are internal structure (exact sequences) and external relationships (adjoint strings) two representations of divisions of everything, in terms of increasing slack and decreasing slack?
Order out of chaos
 Consider how entropy relates to the transformation of chaos to order by way of adjunctions.
Inducedrestricted
 Suppose G/H=3. What is the induced representation? Show how it gets expressed for a subgroup {$S_2$} of {$S_3$}.
 Inductionrestrictioncoinduction etc. may be thought of as defining the action +2. Similarly, the threecycle of derived functors may define +3. What would define the action +1?
 Inductionrestriction adjunction is built with the symmetry group. How does it relate to Cayley's theorem? and the Yoneda lemma?
 In what sense does a propagator describe an adjoint relationship?
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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 inductionrestriction 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 Rmodules (where R=CH) and Smodules (where S=CG) we get two adjoints that turn an Hmodule into a Gmodule. 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/(1g)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 leftadjoint and rightadjoint for turning an Hmodule into a Gmodule is the equivalence of external relations and internal structure. And this is the point of SchurWeyl 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.
