nLab Schur index

Idea

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 $R_{\widehat k}(G)$.

Definition

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 $k$.

Galois action and Adams operations

Remark

(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

$$\psi^n \;\colon\; R_k(G) \longrightarrow R_k(G) \,.$$

Notice that for $k = \mathbb{C}$ the complex numbers, the representation ring is identified with the $G$-equivariant K-theory of the point

$$R_{\mathbb{C}}(G) \;\simeq\; KU^0_G(\ast)$$

and under this identification the Adams operations here are those familiar from K-theory.

(tom Dieck 79, section 3.5)

Proposition

(Adams operation-action on characters)

Let $k$ be a field of characteristic zero.

Then the $n$th Adams operation on the representation ring (Remark )

$$\psi^n \;\colon\; R_k(G) \longrightarrow R_k(G) \,.$$

has the following simple explicit description in terms of the characters $\chi_V$ of representations $V \in R_{k}(G)$:

$$\chi_{\psi^n V} = \chi_{V}\left( (-)^n \right)$$

hence for all $g \in G$

$$\chi_{\psi^n(V)}(g) \;=\; \chi_V( g^n )$$

tom Dieck 79, Prop. 3.5.1

Proposition

(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 )

$$\psi^n \;\colon\; R_{\widehat k}(G) \longrightarrow R_{\widehat k}(G) \,.$$

for $n$ coprime to the order of the group,

$$(n, {\vert G \vert}) = 1 \,,$$

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):

$$Gal\left( \mathbb{Q}\left[ e^{2 \pi i/e(G)} \right] \;:\; \mathbb{Q} \right) \;\simeq\; \left( \mathbb{Z}/{e(G)} \right)^{\times} \;\simeq\; \big\{ \psi^{n} \;\vert\; (n,{\vert G\vert}) = 1 \big\}$$

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 ${\vert G \vert}$.

(tom Dieck 79, Prop. 3.5.2)

The Schur index

Definition

(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

$$\Psi^n\left(V_{avg}\right) - V_{avg} \;=\; 0 \;\;\;\;\; \in R_{\widehat{k}}(G)$$

for all $n$ coprime to ${\vert G\vert}$. (See also at Adams conjecture.)

By Prop. this is equivalently the direct sum

$$V + \Psi^{n_1}\left(V\right) + \psi^{n_2}\left(V\right) + \cdots$$

of $V$ with all of its distinct Galois translates.

Proposition

(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

$$V \in R_{\widehat k}(G)$$

a $\widehat k$-linear representation, its Schur index

$$s_V \in \mathbb{N}$$

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 $V$:

$$W \otimes_{\mathbb{Q}} \mathbb{C} \;\simeq\; s_V \cdot V_{avg}$$

Such irreducible $W$ exists uniquely.

(e.g. tom Dieck 09, theorem 6.2.1)

Remark

(equivariant J-homomorphism?)

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)

$$\array{ R_{\widehat k}(G) \; &\overset{J_{\mathbb{Q}}}{\longrightarrow}& \; R_{\mathbb{Q}}(G) \\ V &\mapsto& s_V \cdot \big(V + \Psi^{n_1}(V) + \cdots \big) } \,.$$

For some finite groups $G$ the permutation representation-assigning map

$$A(G) \overset{\beta}{\longrightarrow} R_{\mathbb{Q}}(G)$$

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

$$J \;\colon\; R_{\mathbb{C}}(G) \longrightarrow A(G)$$

hence going from the equivariant K-theory of the point to equivariant stable cohomotopy of the point

$$J \;\colon\; KU^0_G(\ast) \longrightarrow \mathbb{S}_G^0(\ast) \,.$$

Question:

Is this the $G$-equivariant J-homomorphism over the point?

Notice that, by the construction in Prop. this map satisfies

$$J \big( \Psi^n(V) - V \big) \;=\; 0$$

for all $n$ coprime to ${\vert G \vert}$.

Question:

Is this the $G$-equivariant Adams conjecture-statement over the point?

References

Lecture notes include

• Tammo tom Dieck, section 6.2 of Representation theory, 2009 (pdf)

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