bicmr.pku.edu.cnbicmr.pku.edu.cn/conference/isga/download/Beijing-invthy2-2013.pdf · Plan 1. The 3 formulations of invariant theory 2. Classical and quantum invariant theory for

The Universityof Sydney

Invariant theory:classical, quantum and super

Gus Lehrer

University of SydneyNSW 2006Australia

Journal of Algebra 50thPeking University

Beijing, June 10, 2013

Plan

1. The 3 formulations of invariant theory2. Classical and quantum invariant theory for GLn

3. Classical and quantum invariant theory for On and Spn

4. The problem which Brauer left5. The second fundamental theorem–solution of Brauer’sproblem in the classical cases.6. From classical to quantum-the BMW algebra7. Positive characteristic and roots of unity–the non-semisimplecase.8. Invariant theory for supergroups.

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The three formulations of invariant theory

Let G be a connected reductive group over C, and V arepresentation of G.

First: Describe EndG(V⊗r ) for each r . [Non-commutativealgebra]

Second: Describe ((⊗r V ∗)⊗ (⊗sV ))G. [Multilinear algebra]

Third: Let W = ⊕`V ⊕m V ∗. Give a presentation of C[W ]G

Geometrically: Describe W//G. [Commutative algebra, GIT]

In each context, these are all equivalent, by similar, but variousarguments




































History

In each case the problem divides into:(i) Find generators (FFT)(ii) Give all relations among these(SFT)

There are very few pairs (G, V ) for which satisfactory answersare known

The modern version of the subject might be said to havestarted with Gauss’ Disquisitiones Arithmeticae in 1802.

He analysed binary quadratic forms.

Most modern literature is in the context of Schur-Weyl duality,started by Schur in his thesis of 1901.

History






History






History






History






History






History






Classical theory-type A

Let V = Cn, G = GL(V ), and consider first formulation:

Determine EndG(V⊗r ) ' ((EndV )⊗r )G, whereg.v1 ⊗ . . .⊗ vr := gv1 ⊗ . . .⊗ gvr (g ∈ GL(V ), vi ∈ V ).

For π ∈ Symr , µr (π) : v1 ⊗ . . .⊗ vr 7→ vπ−1(1) ⊗ . . .⊗ vπ−1(r)

lies in EndG(V⊗r )

Hence we have µr : CSymr → EndG(V⊗r ).

FFT(Schur): µr is surjective ∀r .




































For any r , let a(r) = ∑π∈Symrε(π)π (Symr -alternator)

Since a(n + 1)V⊗n+1 ⊆ Λn+1(V ) = 0, clearlya(n + 1) ∈ Ker(µr ).

SFT(Schur) Ker(µr ) = 〈a(n + 1)〉. µr is an isomorpism ifr < n + 1.

Both FFT and SFT are most easily proved using semisimplicity,but are valid much more generally.

These statements may be easily translated into theirequivalents in the multilinear and commutative algebraformulations.





















Quantum theory-type ALet K = C(q), q an indeterminate,Uq = Uq(gln), a K -Hopf algebra.

C: the category of f.d. gln(C)-modules.If Cq is the category of f.d. Uq-modules of type (1, 1, ..., 1),we have a weight-preserving equivalence C ∼→ Cq,where v ∈ Wq ∈ Cq has weight λ if Kiv = q〈αi ,λ〉v .

There is a universal R-matrix R ∈ ˜(Uq ⊗Uq) such that for anyVq ∈ Cq:(i) R := PR ∈ EndUq (Vq ⊗ Vq), where P(v ⊗w) = w ⊗ v .(ii) If Ri = R acting on the (i , i + 1) factors of V⊗r

q ,then RiRj = RjRi if |i − j | ≥ 2, and RiRi+1Ri = Ri+1RiRi+1

Hence we have µr ,q : KBr → EndUq (V⊗rq ).

for any Vq ∈ Cq, where Br is the r -string braid group.











































FFT: If Vq is the ‘natural module’ for Uq(gln), µr ,q is surjective.This is easily proved directly from the classical case usingintegral forms of Uq and Vq.

Next, observe that if ε1, ...,εn are the standard weights,Vq = Lε1,q, and Vq ⊗ Vq ' L2ε1,q ⊕ Lε1+ε2,q.

Known: R acts on L2ε1,q and Lε1+ε2,q as the scalar q,−q−1

respectively.Hence µr ,q factors throughνr : KBr/〈(R1 − q)(R1 + q−1)〉 = Hr (q)→ EndUq (V⊗r

q ).

For any r , let aq(r) = ∑w∈Symr(−q)−`(w)Tw ∈ Hr (q) (The

Hr (q)-alternator). Tw =standard basis element of Hr (q)

SFT: Ker(νr ) = 〈aq(n + 1)〉. νr is an isomorpism if r < n + 1.





q ).








q ).








q ).








q ).








q ).








q ).




Classical theory-orthogonal and symplectic casesLet V = Cn, (−,−) a non-degenerate symmetric or skewsymmetric form on V .G = O(V ) or Sp(V ), the isometry group of V , (−,−).

Let (bi), (b′i ) be dual bases of V , so (b′i , bj) = δij .Define c0 := ∑i bi ⊗ b′i . Then c0 is independent of basis,and ∈ (V ⊗ V )G.

Define e ∈ EndG(V⊗2) by e(v ⊗w) = (v , w)c0, andei ∈ EndG(V⊗r ) as e, acting on factors (i , i + 1) of V⊗r .

Note that e2i = εnei , where ε = 1 if G = O(V ),

and ε = −1 if G = Sp(V ).

We also still have the permutation endomorphisms of type A,which are generated by the interchanges ri of the i , i + 1 factorsin EndG(V⊗r ).





































Brauer showed (1937) that if the diagrams si and fi are asshown:

...

i − 1

...

,

...

i − 1

...

.

Figure : si( 7→ εri), fi( 7→ ei)

then the endomorphisms they represent satisfy the samecomposition laws as the diagrams under concatenation,provided free circles are replaced by εn.

Brauer showed that these diagrams generate an algebraBr (εn), which has rank ∏1≤i≤n(2i − 1), with basis the ‘Brauerdiagrams from n to n’.


...

i − 1

...

,

...

i − 1

...

.





...

i − 1

...

,

...

i − 1

...

.





...

i − 1

...

,

...

i − 1

...

.





...

i − 1

...

,

...

i − 1

...

.




This shows that there is a homomorphismηr : Br (εn)→ EndG(V⊗r )Brauer used this to prove in 1937 the following form of the FFT.

FFT (Brauer 1937): ηr : Br (εn)→ EndG(V⊗r ) is surjective

This may be proved using the FFT for type A, together with adensity argument, based on the fact that G is birationallyequivalent to Adim G. (cf. Atiyah-Bott-Patodi 1972.)

But Brauer proved no version of the SFT-i.e. the question ofKer(ηr ) was left open.

This question is complicated by the fact that Br (εn) is notusually semisimple.In fact: Rui and Si have shown that: Br (εn) is semisimple iffr ≤ d + 1 where d = n if G = O(V ) and d = n

2 is G = Sp(V ).Hermann Weyl called Br (δ) ‘enigmatic’.

















































Brauer’s open question.Determine the kernel of ηr : Br (εn)→ EndG(V⊗r ).

Note that there have existed versions of the SFT in formulation2 (multilinear); but no structural version. The following providesa solution, and we will see applies in non-semisimple situationsas well.

Theorem (L-Ruibin Zhang)(SFT): There is a quasi-idempotentΨ ∈ Bd+1(εn), explicitly described in terms of diagrams, suchthat Ker(ηr ) is the ideal of Br (εn) generated by Ψ. (Recalld = n if G = O(V ) and d = n

2 is G = Sp(V ).)

To describe Ψ we will need the element Σε(r) ∈ Br (εn).

Σε(r) ∈ Br (εn) := ∑w∈Symr(−ε)`(w)w .

Σε(r) ∈ Br (εn) will be denoted by a rectangle diagrammatically.




2 is G = Sp(V ).)







2 is G = Sp(V ).)







2 is G = Sp(V ).)







2 is G = Sp(V ).)







2 is G = Sp(V ).)







2 is G = Sp(V ).)







2 is G = Sp(V ).)




The symplectic case

The answer is easist to describe when G = Spn = Sp2d .

Let Ψ = ∑ALL Brauer diagrams D∈Bd+1(−2d) D

Then Ψ has the following properties (L-Zhang, arXiv):

I Ψ2 = (d + 1)!Ψ

I Ψfi = fiΨ = 0 ∀i , 1 ≤ i ≤ dI Ψπ = πΨ ∀π ∈ Symd+1 Thus 1

(d+1)! Ψ is the centralidempotent in Bd+1(−2d) corresponding to the trivialrepresentation.

I Ψ generates Ker(ηr ) ∀r

The proof is not direct–it involves other expressions for Ψ, interms of Σ−(d + 1)

The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





The symplectic case




I Ψ2 = (d + 1)!Ψ





Specifically, we have

Ψ =[ d+1

2 ]

∑k=0

ak Ξk where ak =1

(2kk !)2(d + 1− 2k)!.

and Ξk (k = 0, 1, . . . , [d+12 ]) is given by

...

Σ−(d + 1)

... ...k...

Σ−(d + 1)

....

To prove all the properties, other expressions for Ψ are needed.


Ψ =[ d+1

2 ]

∑k=0

ak Ξk where ak =1

(2kk !)2(d + 1− 2k)!.


...

Σ−(d + 1)

... ...k...

Σ−(d + 1)

....



Ψ =[ d+1

2 ]

∑k=0

ak Ξk where ak =1

(2kk !)2(d + 1− 2k)!.


...

Σ−(d + 1)

... ...k...

Σ−(d + 1)

....



Ψ =[ d+1

2 ]

∑k=0

ak Ξk where ak =1

(2kk !)2(d + 1− 2k)!.


...

Σ−(d + 1)

... ...k...

Σ−(d + 1)

....


The orthogonal caseLet G = O(V ) = On(C).

Let Ψ = E[ n+12 ], where, for p = 0, 1, 2, . . . , n + 1, Ep is defined

diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p

Properties of Ψ (L-Zhang, arXiv, Ann Math 2012):

I Ψ2 = ([n+12 ])!(n + 1− [n+1

2 ])!ΨI Ψfi = fiΨ = 0 ∀i , 1 ≤ i ≤ nI Ψ generates Ker(ηr ) ∀r

The proof again requires other characterisations of Ψ, andcomputations in the Brauer algebra.



diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1





diagramatically as

En+1−p =

...

...p

Σ+(n + 1)

...

...p


I Ψ2 = ([n+12 ])!(n + 1− [n+1



The quantum case-types B, C and D.

Let K = C(q), Uq = Uq(g) where g is the Lie algebra ofrelevant type; A is the subring of K consiting of functions withno pole at 1.

If Vq ' K n is the ‘natural’ Uq-module, then (in the usualnotation for weights) Vq = Lε1 and

Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.

The eigenvalues of R on these 3 summands are respectivelyq,−q−1,εqε−n.Hence in the homomorphism µr ,q : KBr → EndUq (V⊗r

q ), theimages σ of the generators of the braid group satisfy the cubicrelation (σ − q)(σ + q−1)(σ −εqε−n) = 0.




Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.






Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.






Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.






Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.






Vq ⊗ Vq = L2ε ⊕ Lε1+ε2 ⊕ L0.



One deduces easily that µr ,q factors through the BMW algebrawith appropriate parameters; i.e. we have

νr ,q : BMWr (εqε−n, q − q−1)→ EndUq (V⊗rq ).

Theorem (FFT): νr ,q is surjective.

This is proved using an A-form of all structures involved toreduce to the classical case by taking limq→1.

What of the SFT? The key to using limq→1 is the cellularstructure of both BMWr (εqε−n, q − q−1) and Br (εn).

Let BMWεr (q) be the BMW algebra over A with parameters

εqε−n, q − q−1. Then:




































I BMWεr (q) and Br (εn) have cellular structures with the

same cell datum, and writing limq→1(−) = −⊗A C, wehave limq→1 BMWε

r (q) = Br (εn).I All cell modules W (λ) of Br (εn) are of the form

W (λ) = limq→1 Wq(λ), where Wq(λ) is the correspondingcell module for BMWε

r (q).

This leads to the following situation.

0 → Ker(νq,r ) → BMWεr (K )

νq,r→ EndUq (V⊗rq ) → 0

↑−⊗AK ↑−⊗AK ↑−⊗AK

0 → Ker(νA,r ) → BMWεr (q)

νA,r→ EndUA(V⊗rA ) → 0

↓limq→1 ↓limq→1 ↓limq→1

0 → Ker(νr ) → Br (εn)νr→ EndG(V⊗r ) → 0





r (q).




↑−⊗AK ↑−⊗AK ↑−⊗AK









r (q).




↑−⊗AK ↑−⊗AK ↑−⊗AK









r (q).




↑−⊗AK ↑−⊗AK ↑−⊗AK









r (q).




↑−⊗AK ↑−⊗AK ↑−⊗AK





From this one deduces:Theorem (LZ): Let Ψ ∈ Br (εn) be such that Kerνr = 〈Ψ〉.Assume that Ψq ∈ BMWε

r (q) is such that• Ψ2

q = f (q)Ψq where f (1) 6= 0• limq→1 Ψq = cΨ, with c 6= 0.Then Ker(νr ,q) = 〈Ψq〉.

Cor: Ker(νr ,q) = 〈Ψq〉 for some quasi-idempotentΨq ∈ BMWε

d+1(q) in both the symplectic and orthogonal cases.

In the symplectic case, we may take Ψq to be the idempotentcorresponding to the ‘trivial representation’ of BMWε

d+1(q).In the orthogonal case we use an argument about liftingidempotents.Remark: Jun Hu and Xsiao have proved that the abovestatements may be generalised to all characteristics, and haveproved a linear version of the quantum statement.
































































SuperalgebrasLet V = V0 ⊕ V1 be a Z2-graded C-vector space.

If dim V0 = m, dim V1 = n, say that sdim V = (m|n).

Suppose V has an even non-degenerate bilinear form (−,−)which is symmetric on V0, skew symmetric on V1, and satisfies(V0, V1) = (V1, V0) = 0. So sdim V = (m|2n). This is anorthosymplectic superspace.

If V , W are Z2-graded , so are V ∗ andHomC(V , W ) ' W ⊗C V ∗. In particular, so is EndC(V ).• If sdim (V ) = (m|n) the general linear supergroupgl(V ) = gl(m|n) is the Z2-graded Lie algebra EndC(V ), with Lieproduct [X , Y ] = XY − (−1)[X ][Y ]YX .• The orthosymplectic Lie algebra osp(m|2n) is the Z2-gradedsubalgebra of gl(m|2n) defined by{X ∈ gl(m|2n) | (Xv , w) + (−1)[X ][v ](v ; Xw) = 0}.

























The Lie superalgebra gl(V ) acts on V⊗r via

X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .

The subalgebra osp(m|2n) acts correspondingly on V⊗r .Further the group G := O(V0)× Sp(V1) also acts on V⊗r ,compatibly with osp.We have the endomorphisms τ , e ∈ Endosp,G(V ⊗ V ):

τ(v ⊗w) = (−1)[v ][w ]w ⊗ ve is defined in a similar way to the classical orthogonal andsymplectic cases, using dual homogeneous bases of V .This shows that: the Brauer algebra Br (m− 2n) acts on V⊗r .This action commutes with that of osp and of G.


X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .




X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .




X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .




X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .




X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .




X .v1⊗ . . .⊗ vr =r

∑i=1

(−1)[X ]([v1]+···+[vi−1])v1⊗ . . .⊗Xvi ⊗ . . .⊗ vr .



This leads to:

Theorem (LZ 2013, see also Serge’ev) The mapBr (m− 2n)→ Endosp,G(V⊗r ) is surjective.

The proof is by converting to an equivalent statement for theorthosymplectic supergroup over an infinite dimensionalGrassmann algebra, and using the geometric method of Atiyahet al.

This case is more complicated than the usual classical ones,because we do not have complete reducibility, even for V ⊗ V .

This leads to:




This leads to:




This leads to:




Further questions

Integral versions of all cases; analysis at roots of unity;tilting modules

For which pairs g, V do we have ABr → EndUA(g)(V⊗r )surjective? And for which subrings A of K ?

When does the above map factor through a cellularalgebra? (cf. ALZ)

Further questions




Further questions




Further questions




Selected BibliographyH. H. Andersen, G. I. Lehrer and R. B. Zhang, arXiv:1303.0984.Richard Brauer, “On algebras which are connected with thesemisimple continuous groups”, Ann. of Math. (2) 38 (1937),857–872.Jie Du, B. Parshall and L. Scott, “Quantum Weyl reciprocity andtilting modules." Commun. Math. Phys. 195 (1998), 321–352.J. J. Graham and G. I. Lehrer, “Cellular algebras”, InventionesMath. 123 (1996), 1–34.Hu, Jun and Xiao, Zhankui, “On tensor spaces forBirman-Murakami-Wenzl algebras." J. Algebra 324 (2010)2893–2922.Hebing Rui and Mei Si, “A criterion on the semisimple Braueralgebras. II.” J. Combin. Theory Ser. A 113 (2006), 1199–1203.

G. I. Lehrer and R. B. Zhang, arXiv:1207.5889G. I. Lehrer and R. B. Zhang, “The second fundamentaltheorem of invariant theory for the orthogonal group", Ann. ofMath. (2) 176 (2012), no. 3, 2031–2054.G. I. Lehrer and R. B. Zhang, “Strongly multiplicity free modulesfor Lie algebras and quantum groups”, J. of Alg. 306 (2006),138–174.R. Goodman and N.R. Wallach, “Representations andInvariants of the Classical Groups." Cambridge UniversityPress, third corrected printing, 2003.A. Sergeev, “An analog of the classical invariant theory for Liesuperalgebras. II", Michigan Math. J. 49 (2001), Issue 1,147-168.Hermann Weyl, “The Classical Groups. Their Invariants andRepresentations." Princeton University Press, Princeton, N.J.,1939.

Documents

bicmr.pku.edu.cnbicmr.pku.edu.cn/conference/isga/download/Beijing-invthy2-2013.pdf · Plan 1. The 3 formulations of invariant theory 2. Classical and quantum invariant theory for