Multiplicative Invariant Theory Workshop Noncommutative Invariant - PowerPoint PPT Presentation

Multiplicative Invariant Theory Workshop “Noncommutative Invariant Theory” U Washington, Seattle 05/27/2012 Jessie Hamm Temple University, Philadelphia Special Thanks to Dr. Martin Lorenz

Overview Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property ■ Introduction to multiplicative invariants: definitions, examples, . . . Multiplicative Invariant Theory Seattle 05/27/2012 – slide 2

Overview Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property ■ Introduction to multiplicative invariants: definitions, examples, . . . ■ Regularity: reflection groups and semigroup algebras Multiplicative Invariant Theory Seattle 05/27/2012 – slide 2

Overview Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property ■ Introduction to multiplicative invariants: definitions, examples, . . . ■ Regularity: reflection groups and semigroup algebras ■ The Cohen-Macaulay property: reminders on CM rings, some results on multiplicative invariants, and some problems Multiplicative Invariant Theory Seattle 05/27/2012 – slide 2

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

k k k k k k k k Multiplicative invariants Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Given : a group G and a G -lattice L ∼ Z n ; so = ■ G → GL( L ) ∼ = GL n ( Z ) an integral representation of G Multiplicative Invariant Theory Seattle 05/27/2012 – slide 4

k k Multiplicative invariants Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Given : a group G and a G -lattice L ∼ Z n ; so = ■ G → GL( L ) ∼ = GL n ( Z ) Choose a base ring k and form the group algebra ■ x m x m ′ = x m + m ′ k x m ∼ � k [ x ± 1 1 , . . . , x ± 1 k [ L ] = n ] , = m ∈ L The G -action on L extends uniquely to a “multiplicative” or g ( x m ) = x g ( m ) k [ L ] : “exponential” action on the k -algebra Multiplicative Invariant Theory Seattle 05/27/2012 – slide 4

Multiplicative invariants Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Given : a group G and a G -lattice L ∼ Z n ; so = ■ G → GL( L ) ∼ = GL n ( Z ) Choose a base ring k and form the group algebra ■ x m x m ′ = x m + m ′ k x m ∼ � k [ x ± 1 1 , . . . , x ± 1 k [ L ] = n ] , = m ∈ L The G -action on L extends uniquely to a “multiplicative” or g ( x m ) = x g ( m ) k [ L ] : “exponential” action on the k -algebra ■ k [ L ] G = { f ∈ k [ L ] | g ( f ) = f ∀ g ∈ G } = ? Problem: Multiplicative Invariant Theory Seattle 05/27/2012 – slide 4

Example #1 Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Multiplicative inversion in rank 2 : k = L = Z e 1 ⊕ Z e 2 ( Z ) action: g ( e i ) = − e i Multiplicative Invariant Theory Seattle 05/27/2012 – slide 5

Example #1 Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Multiplicative inversion in rank 2 : k = L = Z e 1 ⊕ Z e 2 ( Z ) action: g ( e i ) = − e i Putting x i = x e i we have: Z [ x ± 1 1 , x ± 1 with g ( x i ) = x − 1 Z [ L ] = 2 ] i Multiplicative Invariant Theory Seattle 05/27/2012 – slide 5

Example #1 Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Multiplicative inversion in rank 2 : k = L = Z e 1 ⊕ Z e 2 ( Z ) action: g ( e i ) = − e i Putting x i = x e i we have: Z [ x ± 1 1 , x ± 1 with g ( x i ) = x − 1 Z [ L ] = 2 ] i Straightforward calculation � Z [ L ] G = Z [ ξ 1 , ξ 2 ] ⊕ η Z [ ξ 1 , ξ 2 ] ξ i = x i + x − 1 i η = x 1 x 2 + x − 1 1 x − 1 2 ηξ 1 ξ 2 = η 2 + ξ 2 1 + ξ 2 2 − 4 Multiplicative Invariant Theory Seattle 05/27/2012 – slide 5

Example #1 Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Multiplicative inversion in rank 2 : k = L = Z e 1 ⊕ Z e 2 ( Z ) action: g ( e i ) = − e i Z [ L ] G ∼ Z [ x, y, z ] / ( x 2 + y 2 + z 2 − xyz − 4) = Hence: Multiplicative Invariant Theory Seattle 05/27/2012 – slide 5

Example #1 ′ : linear analog Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Linear inversion in rank 2 : L = Z e 1 ⊕ Z e 2 action: g ( e i ) = − e i Multiplicative Invariant Theory Seattle 05/27/2012 – slide 6

Example #1 ′ : linear analog Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Linear inversion in rank 2 : L = Z e 1 ⊕ Z e 2 action: g ( e i ) = − e i S ( L ) = Z [ x 1 , x 2 ] with g ( x i ) = − x i Now: Multiplicative Invariant Theory Seattle 05/27/2012 – slide 6

Example #1 ′ : linear analog Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Linear inversion in rank 2 : L = Z e 1 ⊕ Z e 2 action: g ( e i ) = − e i S ( L ) = Z [ x 1 , x 2 ] with g ( x i ) = − x i Now: One obtains: S ( L ) G = Z [ ξ 1 , ξ 2 ] ⊕ η Z [ ξ 1 , ξ 2 ] ξ i = x 2 i η = x 1 x 2 η 2 = ξ 1 ξ 2 Multiplicative Invariant Theory Seattle 05/27/2012 – slide 6

Example #1 ′ : linear analog Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property G = � g | g 2 = 1 � Linear inversion in rank 2 : L = Z e 1 ⊕ Z e 2 action: g ( e i ) = − e i Z [ L ] G ∼ Z [ x, y, z ] / ( z 2 − xy ) = Hence: Multiplicative Invariant Theory Seattle 05/27/2012 – slide 6

Some Special Features Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Back to general multiplicative actions: L a G -lattice a commutative base ring k k [ L ] the group algebra Multiplicative Invariant Theory Seattle 05/27/2012 – slide 7

Some Special Features Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Multiplicative invariants have a Z -structure: k [ L ] G is given by the distinct orbit sums a k -basis of x m ′ � orb ( m ) := ( m ∈ L ) m ′ ∈ G ( m ) ⇓ k [ L ] G = Z [ L ] G k ⊗ Z Multiplicative Invariant Theory Seattle 05/27/2012 – slide 7

Some Special Features Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property It suffices to consider finite groups : each orb ( m ) is supported on L fin = { m ∈ L | [ G : G m ] < ∞} stabilizer of m ∈ L G acts on L fin through the finite quotient G = G/ Ker G ( L fin ) . Thus: k [ L ] G = k [ L fin ] G Multiplicative Invariant Theory Seattle 05/27/2012 – slide 7

Some Special Features Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property k [ L ] G is always affine/ In particular, k (Hilbert # 14 ok). Multiplicative Invariant Theory Seattle 05/27/2012 – slide 7

Finite Linear Groups Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Jordan (1880): GL n ( Z ) has only finitely many finite subgroups up to conjugacy. Multiplicative Invariant Theory Seattle 05/27/2012 – slide 8

Finite Linear Groups Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Jordan (1880): GL n ( Z ) has only finitely many finite subgroups up to conjugacy. there are only finitely many multiplicative invariant algebras � k [ L ] G (up to ∼ = ) with rank L bounded Multiplicative Invariant Theory Seattle 05/27/2012 – slide 8

Finite Linear Groups Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property n # fin. G ≤ GL n ( 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 Multiplicative Invariant Theory Seattle 05/27/2012 – slide 8

Pioneers of MIT Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Bourbaki : “Invariants exponentiels” (Chap. VI § 3 of Groupes et ■ alg` ebres de Lie , 1968) Z [Λ] W ∼ R ( g ) ∼ Z [ x 1 , . . . , x rank g ] = = where R ( g ) = representation ring of a semisimple Lie algebra g , Λ = weight lattice of g , and W = Weyl group. Multiplicative Invariant Theory Seattle 05/27/2012 – slide 9

Pioneers of MIT Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Bourbaki : “Invariants exponentiels” (Chap. VI § 3 of Groupes et ■ alg` ebres de Lie , 1968) Z [Λ] W ∼ R ( g ) ∼ Z [ x 1 , . . . , x rank g ] = = where R ( g ) = representation ring of a semisimple Lie algebra g , Λ = weight lattice of g , and W = Weyl group. ■ Steinberg, Richardson (1970s) Multiplicative Invariant Theory Seattle 05/27/2012 – slide 9

Pioneers of MIT Part I: Introduction Part II: Regularity Part III: The Cohen-Macaulay Property Bourbaki : “Invariants exponentiels” (Chap. VI § 3 of Groupes et ■ alg` ebres de Lie , 1968) Z [Λ] W ∼ R ( g ) ∼ Z [ x 1 , . . . , x rank g ] = = where R ( g ) = representation ring of a semisimple Lie algebra g , Λ = weight lattice of g , and W = Weyl group. ■ Steinberg, Richardson (1970s) “ ∆ -methods” for group rings: Passman , Zalesski˘ ■ ı , Roseblade , Dan Farkas � “multiplicative invariants” (mid 1980s) Multiplicative Invariant Theory Seattle 05/27/2012 – slide 9

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