Canonical induction, Green functors, lefschetz invariant of monomial G-posets
Author(s)
Advisor
Barker, Laurence JohnDate
2019-06Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
Green functors are a kind of group functor, rather like Mackey functors, but
with a further multiplicative structure. They are defined on a category whose
objects are finite groups and whose morphisms are generated by maps such as
induction, restriction, inflation, deflation. The aim of this thesis is general formulation for canonical induction, suitable for Green functors, optionally equipped
with inflations.
Let p be a prime number. In Section 3, we apply the Boltje’s theory of canonical
induction [1] to p-permutation modules and give a restriction-preserving Z[1/p]-
linear canonical induction formula from the inflations of projective modules.
In Section 4, we give a general formulation of canonical induction theory for
Green biset functors equipped with induction, restriction, inflation maps.
Let G be a finite group and C be an abelian group. In Section 5, motivated in
part by a search for connection with Peter Symonds’ proof [2] of the integrality
of a canonical induction formula, we introduce a Lefschetz invariant for the Cmonomial Burnside ring. These invariants let us to construct generalize tensor
induction functors associated to any C-monomial (G, H)-biset from the category
of C-monomial G-posets to the category of C-monomial H-posets. We will show
that these functors induce well-defined tensor induction maps from BC(G) to
BC(H), which in turn gives a group homomorphism BC(G)
× → BC(H)
× between
the unit groups of C-monomial Burnside rings.
Keywords
Green functorsP-permutation modules
Canonical induction formula
Burnside ring
Monomial Burnside ring
Tensor induction
Lefschetz invariant