Schur index



In representation theory, a Schur index is a measure for how much an irreducible representation VV over some field kk becomes the direct sum of several irreducible representations as one passes from the ground field kk to some field extension k^\widehat k of kk.

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


Throughout, GG is a finite group with order denoted |G|{\vert G\vert} \in \mathbb{N}.

Given a field kk, we write R K(G)R_K(G) for the representation ring of GG over kk.

Galois action and Adams operations


(representation ring is a lambda-ring)

Let kk be a field of characteristic zero.

The representation ring R k(G)R_k(G) is canonically a lambda-ring.

Hence, in particular, for each nn \in \mathbb{N}, there exists the corresponding Adams operation

ψ n:R k(G)R k(G). \psi^n \;\colon\; R_k(G) \longrightarrow R_k(G) \,.

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

R (G)KU G 0(*) 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)


(Adams operation-action on characters)

Let kk be a field of characteristic zero.

Then the nnth Adams operation on the representation ring (Remark )

ψ n:R k(G)R k(G). \psi^n \;\colon\; R_k(G) \longrightarrow R_k(G) \,.

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

χ ψ nV=χ V(() n) \chi_{\psi^n V} = \chi_{V}\left( (-)^n \right)

hence for all gGg \in G

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

tom Dieck 79, Prop. 3.5.1


(Adams operations represent Galois action)

Now let k^\widehat k be a splitting field for GG, for instance the complex numbers \mathbb{C}, but containing at least the cyclotomic field [exp(2πi/e(G)]\mathbb{Q}\left[ \exp(2 \pi i/e(G) \right], where e(G)e(G) \in \mathbb{N} is the exponent of GG.

Then the action of the Adams operations (Remark )

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

for nn coprime to the order of the group,

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

equals the canonical action of the Galois group of [e 2πi/e(G)]\mathbb{Q}\left[ e^{2 \pi i/e(G)} \right] over \mathbb{Q} on R k^(G)R_{\widehat k}(G) by field automorphisms, which in turns is isomorphic to the multiplicative group of integers modulo e(G):

Gal([e 2πi/e(G)]:)(/e(G)) ×{ψ n|(n,|G|)=1} 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 ψ nV\psi^n V is a Galois translate of VV.

Moreover, if VR k^(G)V \in R_{\widehat k}(G) is an irreducible representation, then also its Galois translate ψ nV\psi^n V is an irreducible representation, for nn coprime to |G|{\vert G \vert}.

(tom Dieck 79, Prop. 3.5.2)

The Schur index


(Galois group averaging on representations)

For VR k^(G)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 avgV_{avg} containing VV as a subrepresentation such that

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

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

By Prop. this is equivalently the direct sum

V+Ψ n 1(V)+ψ n 2(V)+ V + \Psi^{n_1}\left(V\right) + \psi^{n_2}\left(V\right) + \cdots

of VV with all of its distinct Galois translates.


(Schur index for complex-to-rational)

Let k=k = \mathbb{Q} be the rational numbers and let k^\widehat k be the complex numbers or at least the cyclotomic field [e 2πi/e(G)]\mathbb{Q}\left[e^{2 \pi i/e(G)}\right] for e(G)e(G) \in \mathbb{Z} the exponent of GG.

Then for

VR k^(G) V \in R_{\widehat k}(G)

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

s V s_V \in \mathbb{N}

is the smallest natural number such that there exists a rational irreducible representation WR (G)W \in R_{\mathbb{Q}}(G) whose extension of scalars (e.g. complexification) is ss times the Galois group averaging V avgV_{avg} (Def. ) of VV:

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

Such irreducible WW exists uniquely.

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


(equivariant J-homomorphism?)

The uniqueness of the rational irrep WW in Prop. means that the Schur index construction there provides a linear map (not necessarily a ring homomorphism)

R k^(G) J R (G) V s V(V+Ψ n 1(V)+). \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 GG the permutation representation-assigning map

A(G)βR (G) 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 nG = 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:R (G)A(G) 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:KU G 0(*)𝕊 G 0(*). J \;\colon\; KU^0_G(\ast) \longrightarrow \mathbb{S}_G^0(\ast) \,.


Is this the GG-equivariant J-homomorphism over the point?

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

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

for all nn coprime to |G|{\vert G \vert}.


Is this the GG-equivariant Adams conjecture-statement over the point?


Lecture notes include

Textbook accounts include