geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
In representation theory, a Schur index is a measure for how much an irreducible representation $V$ over some field $k$ becomes the direct sum of several irreducible representations as one passes from the ground field $k$ to some field extension $\widehat k$ of $k$.
There are two different-looking but equivalent perspectives on Schur indices, one via Galois theory, the other via the lambda-ring-structure on representation rings $R_{\widehat k}(G)$.
Throughout, $G$ is a finite group with order denoted ${\vert G\vert} \in \mathbb{N}$.
Given a field $k$, we write $R_K(G)$ for the representation ring of $G$ over $k$.
(representation ring is a lambda-ring)
Let $k$ be a field of characteristic zero.
The representation ring $R_k(G)$ is canonically a lambda-ring.
Hence, in particular, for each $n \in \mathbb{N}$, there exists the corresponding Adams operation
Notice that for $k = \mathbb{C}$ the complex numbers, the representation ring is identified with the $G$-equivariant K-theory of the point
and under this identification the Adams operations here are those familiar from K-theory.
(Adams operation-action on characters)
Let $k$ be a field of characteristic zero.
Then the $n$th Adams operation on the representation ring (Remark )
has the following simple explicit description in terms of the characters $\chi_V$ of representations $V \in R_{k}(G)$:
hence for all $g \in G$
(Adams operations represent Galois action)
Now let $\widehat k$ be a splitting field for $G$, for instance the complex numbers $\mathbb{C}$, but containing at least the cyclotomic field $\mathbb{Q}\left[ \exp(2 \pi i/e(G) \right]$, where $e(G) \in \mathbb{N}$ is the exponent of $G$.
Then the action of the Adams operations (Remark )
for $n$ coprime to the order of the group,
equals the canonical action of the Galois group of $\mathbb{Q}\left[ e^{2 \pi i/e(G)} \right]$ over $\mathbb{Q}$ on $R_{\widehat k}(G)$ by field automorphisms, which in turns is isomorphic to the multiplicative group of integers modulo e(G):
One hence also says that $\psi^n V$ is a Galois translate of $V$.
Moreover, if $V \in R_{\widehat k}(G)$ is an irreducible representation, then also its Galois translate $\psi^n V$ is an irreducible representation, for $n$ coprime to ${\vert G \vert}$.
(Galois group averaging on representations)
For $V \in R_{\widehat k}(G)$ an irreducible representation, say that its group averaging with respect to the Galois group/Adams operations from Prop. is the smallest representation $V_{avg}$ containing $V$ as a subrepresentation such that
for all $n$ coprime to ${\vert G\vert}$. (See also at Adams conjecture.)
By Prop. this is equivalently the direct sum
of $V$ with all of its distinct Galois translates.
(Schur index for complex-to-rational)
Let $k = \mathbb{Q}$ be the rational numbers and let $\widehat k$ be the complex numbers or at least the cyclotomic field $\mathbb{Q}\left[e^{2 \pi i/e(G)}\right]$ for $e(G) \in \mathbb{Z}$ the exponent of $G$.
Then for
a $\widehat k$-linear representation, its Schur index
is the smallest natural number such that there exists a rational irreducible representation $W \in R_{\mathbb{Q}}(G)$ whose extension of scalars (e.g. complexification) is $s$ times the Galois group averaging $V_{avg}$ (Def. ) of $V$:
Such irreducible $W$ exists uniquely.
(e.g. tom Dieck 09, theorem 6.2.1)
The uniqueness of the rational irrep $W$ in Prop. means that the Schur index construction there provides a linear map (not necessarily a ring homomorphism)
For some finite groups $G$ the permutation representation-assigning map
from the Burnside ring to the rational representation ring is surjective (this Prop.) and admits a canonical section; for $G = C_n$ a cyclic group it is actually an isomorphism. Hence in these cases we may regard the Schur index construction of Prop. as a linear map of the form
hence going from the equivariant K-theory of the point to equivariant stable cohomotopy of the point
Question:
Is this the $G$-equivariant J-homomorphism over the point?
Notice that, by the construction in Prop. this map satisfies
for all $n$ coprime to ${\vert G \vert}$.
Question:
Is this the $G$-equivariant Adams conjecture-statement over the point?
Lecture notes include
Textbook accounts include
Tammo tom Dieck, Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979
Bertram Huppert, chapter 38 of Character theory of finite groups, de Gruyter 1998