multiplicative invariant theoryzhang/seattlenit2012/...multiplicative invariant theory seattle...

Multiplicative InvariantTheory

Workshop “Noncommutative Invariant Theory”U Washington, Seattle 05/27/2012

Jessie HammTemple University, Philadelphia

Special Thanks to Dr. Martin Lorenz

OverviewPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property

Multiplicative Invariant Theory Seattle 05/27/2012 – slide 2

■ Introductionto multiplicative invariants: definitions, examples, . . .




■ Regularity: reflection groups and semigroup algebras




■ Regularity: reflection groups and semigroup algebras

■ TheCohen-Macaulay property: reminders on CM rings, some resultson multiplicative invariants, and some problems

Part I: Introduction

Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property

Multiplicative invariantsPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


■ Given: a groupG and aG-latticeL ∼= Z

n; so

G → GL(L) ∼= GLn(Z)

anintegral representationof G

k

k k k

k k

k k




n; so


■ Choose a base ringk and form thegroup algebra

k[L] =⊕

m∈L

kxm ∼= k[x±1

1 , . . . , x±1n ] , x

mxm′

= xm+m′

TheG-action onL extends uniquely to a“multiplicative” or“exponential”action on thek-algebrak[L]: g(xm) = x

g(m)

k k




n; so


■ Choose a base ringk and form thegroup algebra

k[L] =⊕

m∈L

kxm ∼= k[x±1

1 , . . . , x±1n ] , x

mxm′

= xm+m′

TheG-action onL extends uniquely to a“multiplicative” or“exponential”action on thek-algebrak[L]: g(xm) = x

g(m)

■Problem: k[L]G = {f ∈ k[L] | g(f) = f ∀g ∈ G} = ?

Example #1Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Multiplicative inversion in rank 2:(k = Z)

G = 〈g | g2 = 1〉L = Ze1 ⊕ Ze2action:g(ei) = −ei





Puttingxi = xei we have: Z[L] = Z[x±1

1 , x±12 ] with g(xi) = x−1

i





Puttingxi = xei we have: Z[L] = Z[x±1

1 , x±12 ] with g(xi) = x−1

i

Straightforward calculation

Z[L]G = Z[ξ1, ξ2]⊕ ηZ[ξ1, ξ2]

ξi = xi + x−1i

η = x1x2 + x−11 x−1

2

ηξ1ξ2 = η2 + ξ21 + ξ22 − 4





Hence: Z[L]G ∼= Z[x, y, z]/(x2 + y2 + z2 − xyz − 4)

Example #1′: linear analogPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Linear inversion in rank 2: G = 〈g | g2 = 1〉L = Ze1 ⊕ Ze2action:g(ei) = −ei




Now: S(L) = Z[x1, x2] with g(xi) = −xi




Now: S(L) = Z[x1, x2] with g(xi) = −xi

One obtains:

S(L)G = Z[ξ1, ξ2]⊕ ηZ[ξ1, ξ2]

ξi = x2i

η = x1x2

η2 = ξ1ξ2




Hence: Z[L]G ∼= Z[x, y, z]/(z2 − xy)

Some Special FeaturesPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Back to general multiplicative actions:

L aG-lattice

k a commutative base ring

k[L] the group algebra



Multiplicative invariants have a Z-structure:

ak-basis ofk[L]G is given by the distinct orbit sums

orb(m) :=∑

m′∈G(m)

xm′

(m ∈ L)

⇓

k[L]G = k⊗

Z

Z[L]G



It suffices to considerfinite groups:

eachorb(m) is supported onLfin = {m ∈ L | [G : Gm] < ∞}

stabilizer ofm ∈ L

G acts onLfin through the finite quotientG = G/KerG(Lfin). Thus:

k[L]G = k[Lfin]G



In particular,k[L]G is always affine/k

(Hilbert # 14 ok).

Finite Linear GroupsPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Jordan (1880):GLn(Z) has only finitely many finite subgroups up toconjugacy.



Jordan (1880):GLn(Z) has only finitely many finite subgroups up toconjugacy.

there are onlyfinitely manymultiplicative invariant algebras

k[L]G (up to∼=) with rankL bounded



n # fin. G ≤ GLn(Z) # max’l G(up to conj.) (up to conj.)

1 2 1

2 13 2

3 73 4

4 710 9

5 6079 17

6 85311 39

Pioneers of MITPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


■ Bourbaki : “Invariants exponentiels”(Chap. VI§ 3 of Groupes etalgebres de Lie, 1968)

R(g) ∼= Z[Λ]W ∼= Z[x1, . . . , xrank g]

whereR(g) = representation ring of a semisimple Lie algebrag, Λ =weight lattice ofg, andW = Weyl group.




R(g) ∼= Z[Λ]W ∼= Z[x1, . . . , xrank g]


■ Steinberg, Richardson(1970s)




R(g) ∼= Z[Λ]W ∼= Z[x1, . . . , xrank g]


■ Steinberg, Richardson(1970s)

■ “∆-methods” for group rings:Passman, Zalesskiı, Roseblade, DanFarkas “multiplicative invariants”

(mid 1980s)

Part II: Regularity

Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property

Regularity at 1Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Notations: G a finite groupL ∼= Z

n a faithfulG-lattice

k = k a field withchar k ∤ |G|

Will explain the following result . . .



Theorem 1TFAE (1) k[L]G is regular atπ(1)(2) G acts as a reflection group onL(3)k[L]G = k[M ] is a semigroup

algebra withε(M) ⊆ k

∗





∗

Here,

X = Spec k[L]π

−→ X/G = Spec k[L]G

∈

1 = Ker ε

whereε : k[L] −→ k is the counit:ε(xm) = 1 for all m ∈ L





∗

(1)⇒ (2) useslinearization: PutE = Ker ε. Then

L

k

= L⊗ k

∼→ E/E2

m⊗ 1 7→ xm − 1 + E2

leads to S(L

k

)Gπ(0)

∼=

k[L]Gπ(1). Now use the S-T-C Theorem.





∗

(2)⇒ (3) usesroot systems: ∃ root systemΦ so that

ZΦ ⊆ L ⊆ Λ(Φ) with G = W(Φ)

UseBourbaki’s Thm: Z[Λ(Φ)]W(Φ) is a polynomial algebra.





∗

(3)⇒ (1) usestorus actions:

(3) ⇔X/G = Spec k[L]G is an affine toric variety so thatπ(1)belongs to the open torus orbit

This implies (1).

Example #1 RevisitedPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Recall:multiplicative inversion(rank2)


k[L]G ∼= k[x, y, z]/(x2 + y2 + z2 − xyz − 4)

k[L]G is not a semigroupalgebra:

Ex #2: Un and the root latticeAn−1Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1

Sn-action: σ(ei) = eσ(i) (σ ∈ Sn)



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


Note: Sn acts as a reflection group;transpositions are reflections



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


Putxi = xei ∈ k[Un]; soσ(xi) = xσ(i) for σ ∈ Sn. Then

k[Un] = k[x±11 , . . . , x±1

n ] = k[x1, . . . , xn][s−1n ] ,

wheresn =∏n

1 xi is thenth elementary symmetric polynomial.



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


∴ k[Un]Sn = k[x1, . . . , xn][s

−1n ]Sn

= k[x1, . . . , xn]Sn [s−1

n ]

= k[s1, . . . , sn−1, s±1n ]

∼= k[Zn−1+ ⊕ Z]

elem. symmetric poly’s



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


Now,

k[An−1] = k[Un]0 ,

the degree0-component for the (Sn-stable) “total degree” grading of

k[Un] = k[x±11 , . . . , x±1

n ].



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


Get

k[An−1]Sn ∼= k[M ]

withM =

{

(t1, . . . , tn−1) ∈ Z

n−1+ |

∑

iti ∈ nZ

}



Notation: Un =⊕n

1 Zei∼= Z

n

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Z

n−1


C[An−1]Sn is not regular:

(n > 2; picture forn = 3)

RegularityPart I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property


Here is the global version of Theorem 1(same notations and hypotheses)



Theorem 1′ TFAE (1) k[L]G is regular(2)G acts as a reflection group onL

andH1(G/D, LD) = 0(3) k[L]G ∼= k[Zr

+ ⊕ Z

s](4)∃ root systemΦ s.t.L/LG ∼= Λ(Φ)

andG = W(Φ)

Here,D is the subgroup ofG that is generated by the “diagonalizable”reflections, conjugate inGL(L) to

d =

(

−11

...1

)



n

# finiteG ≤ GLn(Z)(up to conjugacy)

# reflection groupsG(up to conjugacy)

# cases withk[L]G regular

2 13 9 7

3 73 29 18

4 710 102 51

Part III: The Cohen-MacaulayProperty

Part I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property

Reminder: CM RingsPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


■ Hypotheses: R a comm. noetherian ringa an ideal ofR




■ Always:

height a ≥ depth a = inf{i | H ia(R) 6= 0}




■ Always:


(Zariski) topologydimension theory

(homological) algebra




■ Always:


(Zariski) topologydimension theory

(homological) algebra

■ Def: R is Cohen-Macaulayiff equality holds forall (maximal) idealsa

Some Examples of CM RingsPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


■ Standard example:R an affine domain/PIDk, finite / somepolynomial subalgebraP = k[x1, . . . , xn]. Then:

R CM ⇔ R is freeoverP

Some Examples of CM RingsPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


■ Standard example:R an affine domain/PIDk, finite / somepolynomial subalgebraP = k[x1, . . . , xn]. Then:

R CM ⇔ R is freeoverP

■ Hierarchy: catenary

regular +3 complete⋂ +3 Gorenstein +3 CM

KS

dim 0

6>✉✉✉✉✉✉✉✉✉

✉✉✉✉✉✉✉✉✉ dim 1reduced

KS

dim 2normal

^f❋❋❋❋❋❋❋❋

❋❋❋❋❋❋❋❋

Invariant RingsPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


Hypotheses: R a CM ringG a finite group acting onR




If the trace mapR → RG , r 7→∑

Gg(r), is epi (“non-modular case”)

thenRG is CM; otherwiseusually not.




If the trace mapR → RG , r 7→∑

Gg(r), is epi (“non-modular case”)

thenRG is CM; otherwiseusually not.

Here is anecessary condition. . .



Hypotheses: R a CM ringG a finite group acting onRRk = { k-reflectionsonR }AssumeR noetherian/RG

automorphisms belonging tothe inertia group of someprime of height≤ k

Proposition(L. - Pathak)

RG CM & H i(G, R) = 0 (0 < i < k)

⇒ res : Hk(G, R) →∏

H⊆Rk+1

Hk(H, R)



Hypotheses: R a CM ringG a finite group acting onRRk = {k-reflectionsonR }AssumeR noetherian/RG

Proposition(L. - Pathak)

RG CM & H i(G, R) = 0 (0 < i < k)

⇒ res : Hk(G, R) →∏

H⊆Rk+1

Hk(H, R)

Note: The (H i = 0)-condn

is vacuous fork = 1 bireflections.

Mult. Invariants: CM-propertyPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


Notations: G is a finite group6= 1L aG-lattice, WLOG faithful




SoG → GL(L), g 7→ gL. In this setting,

g ∈ G is ak-reflectiononk[L]

⇐⇒ rank(gL − IdL) ≤ k

”g is ak-reflection onL” — or onL⊗

Z

Q




Theorem 2(L, TAMS ’06)

If Z[L]G is CM then allGm/R2(Gm) for m ∈ L are

perfect groups, but notall Gm are.

subgroup gen. bybireflections onL







Corollary (“3-copies conjecture”) Z[L⊕r]G is never CMfor r ≥ 3.







Note that the conclusions of Theorem 2 only refer to therationaltype

of L. In fact . . .







Proposition If k[L]G is CM then so isk[L′]G for any G-latticeL′ so thatL′ ⊗ Q

∼= L⊗ Q.

Example: Sn-latticesPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


What are theSn-latticesL such thatZ[L]Sn is CM ?



We know:

■ only the structure ofL

Q

= L⊗

Z

Q matters (Proposition)

■ Sn must act as a bireflection group onL (Theorem 2), and hence onall simple constituents ofL

Q



Classification results of irreducible finite linear groups containing abireflection (Huffman and Wales, 70s) imply,for n ≥ 7:

L

Q

∼= Q

r ⊕ (Q−)s⊕ (An−1)

t

Q

(s+ t ≤ 2)

sign representation ofSn



In all cases,Z[L]Sn is indeed CM, with the possibleexception of

L = A2n−1

This case reduces to

Problem(open forp ≤ n/2)

Are the “vector invariants”

Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?

SummaryPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


LetL be aG-lattice, whereG is a finite group.

G is generated byreflectionsonL

Bourbaki, Farkas

L.

*2

Z[L]G is asemigroup algebra

?

jr

Hochster

��

G is generated bybireflectionsonL

Z[L]G isCohen-Macaulay+3

?

ks

ThanksPart I: Introduction Part II: RegularityPart III: The Cohen-Macaulay Property


Thanks for your attention!

multiplicative invariant theoryzhang/seattlenit2012/...multiplicative invariant theory seattle...

Documents