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Modular Invariant Theory of the Cyclic Group via Classical Invariant Theory David Wehlau June 5, 2010 Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 1 Outline Modular Representation Theory of Cp Modular RepresentationTheory of Cp . Representation Theory of Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the Conjecture Some Consequences SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 2 Modular Representation Theory of Cp The Modular Group of Prime Order Vn Known Modular Invariants of Cp Representation Theory of SL2 (C) The Connection Modular Representation Theory of Cp Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 3 The Modular Group of Prime Order Modular Representation Theory of Cp The Modular Group of Prime Order Let F be any field of characteristic p > 0 and let Cp denote the cyclic group of order p. We fix a generator σ of Cp . Vn Known Modular Invariants of Cp Representation Theory of SL2 (C) Suppose V is an indecomposable representation of Cp defined over F. Consider the Jordan Normal Form of the matrix of σ in GL(V ). This is the matrix: The Connection 1 1 0 0 1 1 0 0 1 . . . .. .. .. 0 0 0 0 0 0 Shank’s Conjecture Proof of the conjecture Some Consequences where 1 ≤ n ≤ p. Fredericton, New Brunswick, 5 June 2010 ··· ··· .. . .. . ··· ··· 0 0 0 0 0 0 .. .. . . 1 1 0 1 n×n CMS Summer Meeting – slide 4 Vn 1. We get p inequivalent indecomposable representations: V1 ( V2 ( · · · ( Vp C 2. Vn p ∼ = V1 for all n. 3. ∆ := σ − 1 Pp−1 i p−1 · f = ∆p−1 (f ) where ∆ = σ − 1. 4. Tr(f ) = i=0 σ · f = (σ − 1) Cp Thus a non-zero invariant h is a transfer if and only if h ∈ Vp . Qp−1 i 5. Norm(f ) = i=0 σ · f . Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 5 Known Modular Invariants of Cp Modular Representation Theory of Cp The Modular Group of Prime Order 1. Fp [⊕m V2 ]Cp 2. Fp [V3 ]Cp 3. Fp [V4 ]Cp 4. Fp [V5 ]Cp 5. Fp [V2 ⊕ V3 ]Cp 6. Fp [V3 ⊕ V3 ]Cp 7. Fp [V2 ⊕ V2 ⊕ V3 ]Cp Vn Known Modular Invariants of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 6 Modular Representation Theory of Cp Representation Theory of SL2 (C) Representations of SL2 (C) The Connection Shank’s Conjecture Representation Theory of SL2 (C) Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 7 Representations of SL2 (C) Modular Representation Theory of Cp Representation Theory of 1. R1 denotes the defining two dimensional representation of SL2 (C) with fixed basis {X, Y }. SL2 (C) Representations of SL2 (C) 2. Rd := Sd (R1 ) = span{X d , X d−1 Y, . . . , Y d }. The Connection Shank’s Conjecture Proof of the conjecture 3. Every irreducible SL2 (C) representation is isomorphic to Rd for some d. Some Consequences 4. Classical invariant theorists studied not just the ring of invariants C[W ]SL2 (C) of a representation W but also C[R1 ⊕ W ]SL2 (C) , the ring of covariants of W . Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 8 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Robert’s Isomorphism (1861) The Integers Notation The Connection From SL2 (C) to Cp Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 9 Robert’s Isomorphism (1861) Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection C[R1 ⊕ W ]SL2 (C) −→ C[W ]U f (·, ·) 7→ f (X, ·) Robert’s Isomorphism (1861) The Integers where Notation From SL2 (C) to Cp Shank’s Conjecture Proof of the conjecture Some Consequences U = {τ ∈ SL2 (C) | τ · X = X} 1 u = |u∈C . 0 1 Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 10 Robert’s Isomorphism (1861) Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection C[R1 ⊕ W ]SL2 (C) −→ C[W ]U f (·, ·) 7→ f (X, ·) ∼ Robert’s Isomorphism (1861) The Integers where Notation From SL2 (C) to Cp Shank’s Conjecture Proof of the conjecture Some Consequences U = {τ ∈ SL2 (C) | τ · X = X} 1 u = |u∈C . 0 1 Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 11 The Integers Modular Representation Theory of Cp U contains the subgroup Representation Theory of SL2 (C) The Connection Robert’s Isomorphism (1861) U⊃ 1 k 0 1 |k∈Z The Integers Notation From SL2 (C) to Cp Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 12 The Integers Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Robert’s Isomorphism (1861) The Integers Notation From SL2 (C) to Cp Shank’s Conjecture U contains the subgroup 1 k U⊃ |k∈Z =Z 0 1 The fact that Z is a dense subgroup of U in the Zariski topology implies C[W ]U = C[W ]Z . Proof of the conjecture Some Consequences Note that the action of Z is generated by the element Fredericton, New Brunswick, 5 June 2010 1 1 0 1 CMS Summer Meeting – slide 13 Notation Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Robert’s Isomorphism (1861) The Integers Define • Mn := Rn−1 (Q) • Ln := Rn−1 (Z) = Mn (Z) • If W is an SL2 (C)-representation, M = W (Q) and L = W (Z) If W = ⊕ti=1 Rni −1 then M = ⊕ti=1 Mni and L = ⊕ti=1 Lni . Notation From SL2 (C) to Cp The element 1 ∈ Z ⊂ U acts on Rn−1 , Mn and Ln via the matrix Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 1 1 0 0 1 1 0 0 1 . . . .. .. .. 0 0 0 0 0 0 ··· ··· .. . .. . ··· ··· 0 0 0 0 0 0 .. .. . . 1 1 0 1 n×n CMS Summer Meeting – slide 14 From SL2 (C) to Cp Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Robert’s Isomorphism (1861) The Integers Notation From SL2 (C) to Cp Shank’s Conjecture C[R1 ⊕ W ]SL2 (C) ∼ = C[W ]U = C[W ]Z ∼ = Q[M ]Z ⊗Q C ∼ = (Z[L]Z ⊗Z Q) ⊗Q C ∼ = Z[L]Z ⊗Z C S • (L∗ ) = Z[L] ⊃ Z[L]Z Proof of the conjecture ↓ (mod p) Some Consequences S • (V ∗ ) = Fp [V ] ↓ (mod p) ⊃ Fp [V ]Cp Elements of Fp [V ]Cp which lie in the image of Z[L]Z under reduction modulo p are called integral invariants. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 15 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Shank’s Conjecture (1996) Shank’s Conjecture Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 16 Shank’s Conjecture (1996) Modular Representation Theory of Cp F[V ]Cp is generated by Representation Theory of 1. Certain specific norms (one for each summand of V ). 2. Certain transfers (finitely many). 3. A finite set of integral invariants. SL2 (C) The Connection Shank’s Conjecture Shank’s Conjecture (1996) Proof of the conjecture Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 17 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Proof of the conjecture Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 18 The Tensor Algebra Fp [Vn ]d = S d (Vn∗ ) is a summand of ⊗d V ∗ . We consider ⊗d Vn . (⊗d Mn ) ⊂ ⊂ S d (Ln ) ⊂ (⊗d Ln ) Ln1 ⊕ Ln2 ⊕ · · · ⊕ Lns ↓ (mod p) S d (Vn ) ∼ = ⊂ ⊂ S d (Mn ) ⊂ ↓ (mod p) (⊗d Vn ) ∼ = Mn1 ⊕ Mn2 ⊕ · · · ⊕ Mns ↓ (mod p) Vn 1 ⊕ Vn 2 ⊕ · · · ⊕ Vn s We need to show that the co-kernel of the map on invariants is spanned by transfers for d < p. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 19 Example (p = 7) 4 S (L5 ) ⊂ ↓ (mod p) S 4 (V5 ) ⊗ M5 ∼ = 5L1 ⊕ 12L3 ⊕ 16L5 ⊕ 17L7 ⊕15L9 ⊕ 10L11 ⊕ 6L13 ⊕ 3L15 ⊕ L17 4 ⊗ L5 ↓ (mod p) ⊂ ⊗4 V5 Fredericton, New Brunswick, 5 June 2010 5M1 ⊕ 12M3 ⊕ 16M5 ⊕ 17M7 ⊕15M9 ⊕ 10M11 ⊕ 6M13 ⊕ 3M15 ⊕ M17 ⊂ ⊂ ⊂ S (M5 ) 4 ⊂ 4 ∼ = ↓ (mod p) 2V1 ⊕ 3V3 ⊕ V5 ⊕ 87V7 CMS Summer Meeting – slide 20 Three Representation Rings Modular Representation Theory of Cp Let m ≤ n. We compare: Representation Theory of Rm−1 ⊗ Rn−1 Mm ⊗ Mn Vm ⊗ Vn SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 21 Clebsch-Gordan Rule Modular Representation Theory of Cp Rm ⊗ Rn ∼ = Rn−m ⊕ Rn−m+2 ⊕ · · · ⊕ Rm+n Representation Theory of SL2 (C) The Connection Shank’s Conjecture Rm−1 ⊗ Rn−1 ∼ = Rn−m+1 ⊕ Rn−m+3 ⊕ · · · ⊕ Rm+n−1 Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 22 Jordan Normal Form Modular Representation Theory of Cp Mm ⊗ Mn ∼ = Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1 Representation Theory of SL2 (C) The Connection Shank’s Conjecture Roth (1934) Aiken (1934) (Littelwood 1936) Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Markus and Robinson (1975) Brualdi (1985) Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 23 Representation ring of Cp If 1 ≤ m ≤ n ≤ p, Vm ⊗ Vn ∼ = ( Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ Vm+n−1 , if m + n ≤ p + 1; Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ V2p−m−n−1 ⊕ (⊕m+n−p Vp ), if m + n ≥ p. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 24 Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1 ⊂ ∼ = ⊂ Mm ⊗ Mn Lm ⊗ Ln Ln−m+1 ⊕ Ln−m+3 ⊕ · · · ⊕ Lm+n−1 ↓ (mod p) Vm ⊗ Vn ∼ = ↓ (mod p) Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ Vn+m−1 if m + n ≤ p + 1. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 25 ∼ = ⊂ Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1 ⊂ Mm ⊗ Mn Lm ⊗ Ln Ln−m+1 ⊕ Ln−m+3 ⊕ · · · ⊕ Lm+n−1 ↓ (mod p) Vm ⊗ Vn ∼ = ↓ (mod p) Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ V2p−m−n−1 ⊕ (⊕m+n−p Vp ) if m + n ≥ p. It suffices to prove that Lr surjects onto Vr here for all r ≤ 2p − m − n − 1 when m ≤ n ≤ p. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 26 Modular Representation Theory of Cp Consider m = 3 and n = 4. Representation Theory of M3 ⊗ M4 SL2 (C) ∼ = M2 ⊕ M4 ⊕ M6 ⊂ Shank’s Conjecture ⊂ The Connection L3 ⊗ L4 L2 ⊕ L4 ⊕ L6 Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings ↓ (mod p) Clebsch-Gordan Rule ↓ (mod p) Jordan Normal Form Representation ring of Cp V3 ⊗ V4 Explicit Decompositions ∼ = V2 ⊕ V4 ⊕ V6 Some Consequences if p ≥ 7. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 27 Modular Representation Theory of Cp If m = 3, n = 4 and p = 5. Representation Theory of SL2 (C) M3 ⊗ M4 ∼ = M2 ⊕ M4 ⊕ M6 ⊂ Shank’s Conjecture ⊂ The Connection L3 ⊗ L4 L2 ⊕ L4 ⊕ L6 Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form ↓ (mod p) ↓ (mod p) Representation ring of Cp Explicit Decompositions V3 ⊗ V4 ∼ = V2 ⊕ V5 ⊕ V5 Some Consequences Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 28 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences M3 ⊗ M4 ∼ = Q[s]/(s3 ) ⊗ Q[t]/t4 ∼ = Q[s, t]/(s3 , t4 ). M3 ⊗ M4 ∼ = M2 ⊕ M4 ⊕ M6 . ∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st. 0 1 1 s t 2 s2 st t2 3 s2 t st2 t3 4 s2 t2 st3 5 s2 t3 0 • 1 • • 2 • • • 3 • • • 4 • • 5 • Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 29 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences M3 ⊗ M4 ∼ = Q[s]/(s3 ) ⊗ Q[t]/t4 ∼ = Q[s, t]/(s3 , t4 ). M3 ⊗ M4 ∼ = M2 ⊕ M4 ⊕ M6 . ∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st. 0 1 1 s t 2 s2 st t2 3 s2 t st2 t3 4 s2 t2 st3 5 s2 t3 • 0 1 • • 2 • • • 3 • • • 4 • • 5 • Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 30 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences M3 ⊗ M4 ∼ = Q[s]/(s3 ) ⊗ Q[t]/t4 ∼ = Q[s, t]/(s3 , t4 ). M3 ⊗ M4 ∼ = M2 ⊕ M4 ⊕ M6 . ∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st. 0 1 1 s t 2 s2 st t2 3 s2 t st2 t3 4 s2 t2 st3 5 s2 t3 • 0 1 • • 2 • • • 3 • • • 4 • • 5 • Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 31 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences M3 ⊗ M4 ∼ = Q[s]/(s3 ) ⊗ Q[t]/t4 ∼ = Q[s, t]/(s3 , t4 ). M3 ⊗ M4 ∼ = M2 ⊕ M4 ⊕ M6 . ∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st. 0 1 1 s t 2 s2 st t2 3 s2 t st2 t3 4 s2 t2 st3 5 s2 t3 • 0 1 • • 2 • • • 3 • • • 4 • • 5 • Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 32 Modular Representation Theory of Cp M3 ⊗ M4 ∼ = Q[s, t]/(s3 , t4 ) ∼ = M2 ⊕ M4 ⊕ M6 . Representation Theory of SL2 (C) The Connection Shank’s Conjecture If p ≥ 7: V3 ⊗ V4 ∼ = Fp [s, t]/(s3 , t4 ) ∼ = V2 ⊕ V4 ⊕ V6 . Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 Deg M3 ⊗ M4 0 1 2 3 4 5 • • • • • • • • • • • • V3 ⊗ V4 • • • • • • • • • • • • CMS Summer Meeting – slide 33 Modular Representation Theory of Cp M3 ⊗ M4 ∼ = Q[s, t]/(s3 , t4 ) ∼ = M2 ⊕ M4 ⊕ M6 . Representation Theory of SL2 (C) The Connection Shank’s Conjecture If p = 5: V3 ⊗ V4 ∼ = Fp [s, t]/(s3 , t4 ) ∼ = V2 ⊕ V5 ⊕ V5 . Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 Deg M3 ⊗ M4 0 1 2 3 4 5 • • • • • • • • • • • • CMS Summer Meeting – slide 34 Modular Representation Theory of Cp M3 ⊗ M4 ∼ = Q[s, t]/(s3 , t4 ) ∼ = M2 ⊕ M4 ⊕ M6 . Representation Theory of SL2 (C) The Connection Shank’s Conjecture If p = 5: V3 ⊗ V4 ∼ = Fp [s, t]/(s3 , t4 ) ∼ = V2 ⊕ V5 ⊕ V5 . Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 Deg M3 ⊗ M4 0 1 2 3 4 5 • • • • • • • • • • • • V3 ⊗ V4 • • • • • • • • • • • • CMS Summer Meeting – slide 35 Modular Representation Theory of Cp M3 ⊗ M4 ∼ = Q[s, t]/(s3 , t4 ) ∼ = M2 ⊕ M4 ⊕ M6 . Representation Theory of SL2 (C) The Connection Shank’s Conjecture If p = 5: V3 ⊗ V4 ∼ = Fp [s, t]/(s3 , t4 ) ∼ = V2 ⊕ V5 ⊕ V5 . Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 Deg M3 ⊗ M4 0 1 2 3 4 5 • • • • • • • • • • • • V3 ⊗ V4 • • • • • • • • • • • • CMS Summer Meeting – slide 36 Modular Representation Theory of Cp M3 ⊗ M4 ∼ = Q[s, t]/(s3 , t4 ) ∼ = M2 ⊕ M4 ⊕ M6 . Representation Theory of SL2 (C) The Connection Shank’s Conjecture If p = 5: V3 ⊗ V4 ∼ = Fp [s, t]/(s3 , t4 ) ∼ = V2 ⊕ V5 ⊕ V5 . Proof of the conjecture The Tensor Algebra Example (p = 7) Three Representation Rings Clebsch-Gordan Rule Jordan Normal Form Representation ring of Cp Explicit Decompositions Some Consequences Fredericton, New Brunswick, 5 June 2010 Deg M3 ⊗ M4 0 1 2 3 4 5 • • • • • • • • • • • • V3 ⊗ V4 • • • • • • • • • • • • CMS Summer Meeting – slide 37 Explicit Decompositions If p ≥ 7 M3 ⊗ M4 M6 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2 7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3 7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0 M4 3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2 7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3 7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0 M2 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0 V6 V3 ⊗ V4 V4 V2 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2 7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3 7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0 3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2 7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3 7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0 Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 38 Explicit Decompositions If p = 5 M3 ⊗ M4 M6 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2 7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3 7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0 M4 3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2 7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3 7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0 M2 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0 V5 V3 ⊗ V4 V5 V2 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2 7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3 7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 0= 10s2 t3 s 7→ s2 + st + s2 t 7→ 2s2 t + st2 + 2s2 t2 7→ 3s2 t2 + st3 + 3s2 t3 7→ 4s2 t3 7→ 0 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2st3 + 3s2 t2 7→ 0 Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 39 Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture Some Consequences Some Consequences Some New Modular Invariants Selected Bibliography Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 40 Some New Modular Invariants 1. F[V2 ⊕ V4 ]Cp Covariants of R1 ⊕ R3 were computed classically. 2. F[V3 ⊕ V4 ]Cp Covariants of R2 ⊕ R3 were computed classically. 3. F[⊕m V3 ]Cp Covariants of ⊕m R2 were computed classically. 4. F[⊕m V4 ]Cp Covariants of ⊕m R3 were computed by Schwarz(1987). 5. F[V6 ]Cp Covariants of the quintic were computed by Gordan in 1869. 6. F[V7 ]Cp Covariants of the sextic were computed by Gordan in 1869. 7. F[V8 ]Cp Covariants of the septic were computed by Bedratyuk(2009). 8. F[V9 ]Cp Covariants of the octic were computed by von Gall in 1880. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 41 Selected Bibliography Modular Representation Theory of Cp Representation Theory of SL2 (C) The Connection Shank’s Conjecture Proof of the conjecture Some Consequences Some New Modular Invariants Selected Bibliography [1] Gert Almkvist, Representations of Z/pZ in characteristic p and reciprocity theorems, J. Algebra 68 (1981), no. 1, 1–27. [2] G. Almkvist and R. Fossum, Decompositions of exterior and symmetric powers of indecomposable Z/pZ -modules in characteristic p, Lecture Notes in Math. 641, 1–114, Springer-Verlag, 1978. [3] Marvin Marcus and Herbert Robinson, Elementary divisors of tensor products, Comm. ACM 18 (1975) 36–39. [4] Michael Roberts, On the Covariants of a Binary Quantic of the nth Degree, The Quarterly Journal of Pure and Applied Mathematics 4 (1861) 168178. [5] R.J. Shank, S.A.G.B.I. bases for rings of formal modular seminvariants, Commentarii Mathematici Helvetici, 73 (1998) no. 4, 548–565. Fredericton, New Brunswick, 5 June 2010 CMS Summer Meeting – slide 42