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Stable Representation Theory
Daniel A. RamrasDepartment of Mathematical Sciences
New Mexico State University
September 10, 2010
Daniel A. Ramras Stable Representation Theory

Representation Theory
Given a (finitely generated, discrete group) G, we want todescribe the homomorphisms
ρ : G −→ U(n)
and understand what these representations say about G.
Can one find an algebraic or combinatorial description of allrepresentations?
Difficult or impossible in general.
Today: Topological approachUsing the natural topology of U(n) ⊂ Mn×n(C), we’ll considerthe space Hom(G, U(n)) of representations.
Daniel A. Ramras Stable Representation Theory

Spaces of Representations
If G is generated by k elements, then
Hom(G, U(n)) ⊂ U(n)k .
We’ll also consider isomorphism classes of representations:
ρ ∼= AρA−1
for any A ∈ U(n).
The space of isomorphism classes is then
Hom(G, U(n))/U(n) = Hom(G, U(n))/(ρ ∼ AρA−1)
Daniel A. Ramras Stable Representation Theory

Example: Finite Groups
If G is finite, then a representation ρ : G→ U(n) is determinedup to isomorphism by its character,
Trace(ρ) : G→ C
The traceis a continuous invarianttakes on only finitely many values.
Conclusion: Hom(G, U(n))/U(n) is a finite, discrete space!
However, there is interesting topology lurking here...
Daniel A. Ramras Stable Representation Theory

The Atiyah–Hirzebruch–Segal Theorem
The Representation Ring:
R(G) = {[ρ]− [ψ] : ρ, ψ are unitary representations of G}
Consists of formal differences between isomorphismclasses of representationsBlock sum, tensor product give addition and multiplication.
Theorem (AHS)For any finite group G, there is a spectral sequence
H∗(G) =⇒ R(G).
Algebraic procedure for constructing R(G) out of thecohomology of G.
Daniel A. Ramras Stable Representation Theory

Cohomology and Classifying Spaces
The cohomology groups Hk (G) can be definedAlgebraically: Hochschild cohomologyTopologically: for a space X , the group Hk (X ) measuresk–dimensional “combinatorial holes” in X .
DefinitionA classifying space for a group G is a quotient space X/G,where X is a contractible simplicial complex on which G actsfreely.
FactAll classifying spaces for G have the same cohomology,denoted H∗(G).
Daniel A. Ramras Stable Representation Theory

Representations and Cohomology
Lots of mathematics has grown out of attempts to betterunderstand the Atiyah–Hirzebruch–Segal Theorem.
Geometric Rigidity: Are all classifying spaces for a givengroup G homeomorphic to one another?
Borel Conjecture:YES, if these classifying spaces are highdimensionalmanifolds (of the same dimension).
Studied via obstruction theory, and depends on relatingmodules over Z[G] (“representations”) to a modifiedversion of the homology of G.
Daniel A. Ramras Stable Representation Theory

Representations and Cohomology
Novikov Conjecture: asks whether certain differentialinvariants of manifolds are actually homotopy invariant.
Studied by relating certain Banach algebras built from G(“representations”) to another modified homology theory.
Quillen–Lichtenbaum Conjecture: relates algebraicK –theory to Galois cohomology theory.
Today:We’ll examine representations and cohomology of infinite,discrete groups, where the connection is much more explicitand geometric.
Goal: Shed light on some of these more abstract areas.
Daniel A. Ramras Stable Representation Theory

Representations and cohomology of infinite discretegroups
Example (G = Z)Hom(Z, U(n)) = U(n)Hom(Z, U(n))/U(n) = U(n)/(A ∼ PAP−1)
Spectral Theorem:Every A ∈ U(n) has n eigenvalues, when counted withmultiplicity.
This yields an eigenvalue map
Hom(Z, U(n))/U(n) −→ (S1 × · · ·× S1︸ ︷︷ ︸n
)/Σn =: Symn(S1).
This map is a homeomorphism by Rouché’s Theorem.
Daniel A. Ramras Stable Representation Theory

Stabilization
Something interesting happens when we stabilize: let n →∞:⋃
nHom(Z, U(n))/U(n) ∼=
⋃
nSymn(S1) = Sym∞(S1)
Theorem (Dold–Thom)For any (based) simplicial complex X, the homotopy groups ofSym∞(X ) are isomorphic to the homology groups of X .
The homotopy groups πk (Z ) measure k–dimensional“spherical holes” in Z :
πk (Z ) = {f : Sk −→ Z}/homotopy.
Conclusion: π∗Hom(Z, U)/U ∼= H∗(S1) = H∗(Z),S1 is a classifying space for Z, because R/Z = S1.
Daniel A. Ramras Stable Representation Theory

Stable representation theory of Zn
By similar methods, we can analyze the free abelian group Zk :
Rk/Zk = (R/Z)k = (S1)k , so the k–dimensional torus is aclassifying space for Zk .Commuting unitary matrices are simultaneouslydiagonalizable, so to every representation of Zk we canassociate its eigenvalues:
Hom(Zk , U(n))/U(n) ∼= Symn((S1)k ).
Once again, the Dold–Thom Theorem tells us that
π∗Hom(Zk , U)/U ∼= H∗(Zk ).
Daniel A. Ramras Stable Representation Theory

Surface Groups
Any Riemann surface S is a classifying space for itsfundamental group: if S has genus g,
π1S = 〈a1, b1, . . . , ag , bg [a1, b1] · · · [ag , bg] = 1〉.
Theorem (R.)For each Riemann surface S, there is a homotopy equivalenceHom(π1S, U)/U # Sym∞(S), and hence
πk Hom(π1S, U)/U ∼= Hk (π1S).
Relies on Morse Theory for the Yang–Mills functional andstable homotopy theory.
Heuristic: Representations ! Vector bundles over S !Characteristic (co)homology classes
Daniel A. Ramras Stable Representation Theory

Crystallographic Groups
DefinitionA crystallographic group is a discrete subgroupΓ < Isom(Rk ) = Rk ! O(k), such that Rk/Γ is compact.
FactIf Γ is a torsionfree crystallographic group, then Γ acts freely onRk , and Rk/Γ is a classifying space for Γ.
The translations in Γ form a free abelian subgroup T ∼= Zkof finite index.
Consequence: The irreducible representations of Γ havedimension at most Γ/T , making computations feasible.
Daniel A. Ramras Stable Representation Theory

A Crystallographic Family
Example
The group Γk = 〈t1, . . . , tk , a  [ti , tj ] = 1, atia−1 = t−1i 〉 iscrystallographic.
T = 〈t1, . . . , tk , a2〉 ∼= Zk+1,Γk/T = 〈a〉/〈a2〉 ∼= Z/2Z.
Γk acts on Rk+1 by the isometries
v ti$→ v + ei
v a$→
−1. . .
−11
v +
12
ek+1
Daniel A. Ramras Stable Representation Theory

The classifying space
The classifying space Rk+1/Γk has the form(Rk+1/Zk+1)/ (Z/2Z)
= (S1)k+1/(z1, . . . , zk , a) ∼ (z−11 , . . . , z−1k ,−a).
Example (The Klein Bottle)When k = 1, this quotient space is the usual model for theKlein Bottle!
Daniel A. Ramras Stable Representation Theory

The irreducible representations of Γk
Irreducible representations of Γk have dimension ≤ 2.
Recall: ρ is irreducible if ρ ![
ψ 00 τ
].
We will analyze the closure of the 2–dimensionalirreducible representations.
Fact
Irr2(Γk ) ={([
τi 00 τ−1i
],
[0 α1 0
]): τi , α ∈ S1
}/ ∼,
where the equivalence relation ∼ is given by:([
τi 00 τ−1i
],
[0 α1 0
])∼
([τ−1i 0
0 τi
],
[0 α1 0
]).
Daniel A. Ramras Stable Representation Theory

Comparing Representations and Cohomology
Irr2(Γk ) = (S1)k+1/ (Z/2Z), with action
(τ1, . . . , τk , α) !→ (τ−11 , . . . , τ−1k , α).
We saw that Rk+1/Γk = (S1)k+1/ (Z/2Z), with action
(z1, . . . , zk , a) ∼ (z−11 , . . . , z−1k ,−a).
The actions are homotopic to one another, and the cohomologygroups of the quotients have the same rank (Grothendieck).
Theorem (R.)
For each group Γk = 〈t1, . . . , tk , a  [ti , tj ] = 1, atia−1 = t−1i 〉,
Rank (πqHom(Γk , U)/U) = Rank Hq(Γk )
(except possibly when q = 1, 2).
Daniel A. Ramras Stable Representation Theory

Stabilization versus Group Completion
In order to extend these ideas to all crystallographic groups, weneed to analyze the idea of stabilization.
In the A–H–S Theorem, we formed R(G) by adding formalinverses to each representation of G:
This is called Group Completion.
For infinite discrete groups, we stabilized by letting thedimensions of the representations tend to infinity.
Question:How can we relate group completion and stabilization?
ExampleThe integers Z can be built from the natural numbers N byeither process.
Daniel A. Ramras Stable Representation Theory

The Grothendieck Group and Stabilization
The group completion, or Grothendieck Group, of a (niceenough) abelian topological monoid A is the group
Gr(A) = A× A/ ∼
where (a1, a2) ∼ (a′1, a′2) if a1 + a′2 + a = a′1 + a2 + a for some a.Analogous to the construction of fraction fields.
PropositionIf A is a discrete abelian monoid and a0 ∈ A satisfies:
∀a ∈ A, ∃ ā ∈ A such that a + ā ∈ 〈a0〉,
thenGr(A) ∼= lim
→(A +a0−→ A +a0−→ A +a0−→ · · · ).
Example: every vector bundle V → X has an orthogonalcomplement: V ⊕ V⊥ ∼= X × RN ∈ 〈X × R〉.
Daniel A. Ramras Stable Representation Theory

Group completion versus stabilization forrepresentation spaces
Definition
The deformation representation ring Rdef(G) is the groupcompletion of
∐n Hom(G, U(n))/U(n).
The groups we’ve focused on today satisfy:
For all ρ : G → U(n), there exists k such that
ρ⊕ ρk # I,
where I is the trivial representation.
This implies that for all these groups G,
Rdef(G) # lim⊕1−→
(∐
nHom(G, U(n))/U(n)
)
Daniel A. Ramras Stable Representation Theory

General results for crystallographic groups
Theorem (R.)
For any crystallographic group Γ < Isom(Rk ), the homotopygroups of the deformation representation ring Rdef(Γ) vanishabove dimension k.
Conjecture:
The nontrivial homotopy groups of Rdef(Γ) agree with H∗(Γ),up to torsion.
The discrepancy in the torsion is meaningful: it has to do withthe question of which torsion classes in the cohomology can berealized as characteristic classes of (families of)representations.
Daniel A. Ramras Stable Representation Theory

Concluding Remarks
For infinite discrete groups, the relationship betweenrepresentations and cohomology can be made
• explicit • geometric
Implications for other areas?
Each representation ρ : G → U(n) gives rise to a vector bundle
Eρ = (X × Cn)/G
!!X/G.
The geometry of bundles helps explain the relations betweenrepresentations and cohomology for infinite discrete groups,and may offer explanations for other related phenomena.
Daniel A. Ramras Stable Representation Theory