Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018

    am starting something. Just saving for the moment, not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018
    • (edited Sep 27th 2018)

    okay, I have recorded the basic definition/statement

    I tried to bring out the nice picture as in tom Dieck 09, 6.2, in particular I focused, for the moment, on field extensions from \mathbb{Q}.

    I have taken the liberty of calling the direct sum of a k^\widehat k-irrep with all its distinct Galois translates its Galois group averaging (here) since I find that saying this helps to see at a glance what’s really going on here.

    I have also taken the liberty of giving an alternative characterization of this “Galois group averaged representation”, namely as the smallest rep that is in the kernel of Ψ nid\Psi^n - id for nn coprime to |G|{\vert G\vert}. Because this makes the relation to the Adams conjecture manifest. Hope I got this right.

    But representation theorists please feel invited to criticize.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018
    • (edited Sep 27th 2018)

    I have also added a remark (here) that highlights the close similarity of the construction to the J-homomorphism and the Adams conjecture.

    So I highlighted in particular the (open?) question (raised in another thread here) of whether indeed the Schur index construction is the incarnation of the equivariant J-homomorphism over the point, and if the “Galois group averaging” involved is the equivariant Adams conjecture-statement on the point – at least for those groups GG for which A(G)βR (G)A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G) is surjective – because it looks directly analogous.

    This may be straightforward to check by chasing through Segal’s proof of 𝕊 G 0(*)A(G)\mathbb{S}^0_G(\ast) \simeq A(G). If this question turns out to be really open, I’ll sit down and do that. But I’d much rather just cite it from somewhere. If it is indeed true.