Cohomological Invariants An invariant a is called normalized if it sends trivial torsor to zero. We denote the subgroup of normalized invariants by Inv d ( G , d − 1) norm . A normalized invariant a is called decomposable if it is given by a cup-product with an invariant of degree 2 (class in the Brauer group). We denote the subgroup of decomposable invariants by Inv 3 ( G , 2) dec . The factor group Inv 3 ( G , 2) norm / Inv 3 ( G , 2) dec is denoted by Inv 3 ( G , 2) ind and is called the group of indecomposable invariants. 8 / 31
Cohomological Invariants The group Inv 3 ( G , 2) ind has been studied by Garibaldi, Kahn, Levine, Rost, Serre and others in the simply-connected case and is closely related to the Rost invariant. Recently, Merkurjev showed how to compute this group in general using new results on motivic cohomology. In particular, it was computed by him for all adjoint split groups and by Bermudez and Ruozzi for all split simple groups. 9 / 31
Cohomological Invariants The group Inv 3 ( G , 2) ind has been studied by Garibaldi, Kahn, Levine, Rost, Serre and others in the simply-connected case and is closely related to the Rost invariant. Recently, Merkurjev showed how to compute this group in general using new results on motivic cohomology. In particular, it was computed by him for all adjoint split groups and by Bermudez and Ruozzi for all split simple groups. 9 / 31
Representation theory Recall that a classical character map identifies the representation ring of G with the subring Z [ T ∗ ] W of W -invariant elements of the integral group ring Z [ T ∗ ], where W is the Weyl group which acts naturally on the group of characters T ∗ of a split maximal torus T of G . In particular, the ideal ( � I W ) generated by augmented W -invariant elements in Z [Λ], where Λ is the respective weight lattice, can be identified with the ideal generated by classes of augmented (i.e. virtual of dimension 0) representations of the simply-connected cover of G . 10 / 31
Representation theory Recall that a classical character map identifies the representation ring of G with the subring Z [ T ∗ ] W of W -invariant elements of the integral group ring Z [ T ∗ ], where W is the Weyl group which acts naturally on the group of characters T ∗ of a split maximal torus T of G . In particular, the ideal ( � I W ) generated by augmented W -invariant elements in Z [Λ], where Λ is the respective weight lattice, can be identified with the ideal generated by classes of augmented (i.e. virtual of dimension 0) representations of the simply-connected cover of G . 10 / 31
Representation theory Recall that a classical character map identifies the representation ring of G with the subring Z [ T ∗ ] W of W -invariant elements of the integral group ring Z [ T ∗ ], where W is the Weyl group which acts naturally on the group of characters T ∗ of a split maximal torus T of G . In particular, the ideal ( � I W ) generated by augmented W -invariant elements in Z [Λ], where Λ is the respective weight lattice, can be identified with the ideal generated by classes of augmented (i.e. virtual of dimension 0) representations of the simply-connected cover of G . 10 / 31
Representation theory Recall that a classical character map identifies the representation ring of G with the subring Z [ T ∗ ] W of W -invariant elements of the integral group ring Z [ T ∗ ], where W is the Weyl group which acts naturally on the group of characters T ∗ of a split maximal torus T of G . In particular, the ideal ( � I W ) generated by augmented W -invariant elements in Z [Λ], where Λ is the respective weight lattice, can be identified with the ideal generated by classes of augmented (i.e. virtual of dimension 0) representations of the simply-connected cover of G . 10 / 31
Semidecomposable Invariants We introduce a subgroup of semi-decomposable invariants Inv 3 ( G , 2) sdec which consists of invariants a ∈ Inv 3 ( G , 2) norm such that for every field extension L / F and a G -torsor Y over L � φ i ∪ b i ( Y ) for some φ i ∈ L × and b i ∈ Inv 2 ( G , 1) norm . a ( Y ) = i finite Observe that by definition we have Inv 3 ( G , 2) dec ⊆ Inv 3 ( G , 2) sdec ⊆ Inv 3 ( G , 2) norm . Roughly speaking, Semi-decomposable = Locally decomposable. 11 / 31
Semidecomposable Invariants We introduce a subgroup of semi-decomposable invariants Inv 3 ( G , 2) sdec which consists of invariants a ∈ Inv 3 ( G , 2) norm such that for every field extension L / F and a G -torsor Y over L � φ i ∪ b i ( Y ) for some φ i ∈ L × and b i ∈ Inv 2 ( G , 1) norm . a ( Y ) = i finite Observe that by definition we have Inv 3 ( G , 2) dec ⊆ Inv 3 ( G , 2) sdec ⊆ Inv 3 ( G , 2) norm . Roughly speaking, Semi-decomposable = Locally decomposable. 11 / 31
Semidecomposable Invariants We introduce a subgroup of semi-decomposable invariants Inv 3 ( G , 2) sdec which consists of invariants a ∈ Inv 3 ( G , 2) norm such that for every field extension L / F and a G -torsor Y over L � φ i ∪ b i ( Y ) for some φ i ∈ L × and b i ∈ Inv 2 ( G , 1) norm . a ( Y ) = i finite Observe that by definition we have Inv 3 ( G , 2) dec ⊆ Inv 3 ( G , 2) sdec ⊆ Inv 3 ( G , 2) norm . Roughly speaking, Semi-decomposable = Locally decomposable. 11 / 31
Semidecomposable Invariants We introduce a subgroup of semi-decomposable invariants Inv 3 ( G , 2) sdec which consists of invariants a ∈ Inv 3 ( G , 2) norm such that for every field extension L / F and a G -torsor Y over L � φ i ∪ b i ( Y ) for some φ i ∈ L × and b i ∈ Inv 2 ( G , 1) norm . a ( Y ) = i finite Observe that by definition we have Inv 3 ( G , 2) dec ⊆ Inv 3 ( G , 2) sdec ⊆ Inv 3 ( G , 2) norm . Roughly speaking, Semi-decomposable = Locally decomposable. 11 / 31
Main Theorem. Let G be a split semisimple linear algebraic group over a field F and let X gen denote the associated versal flag. There is a short exact sequence 0 → Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec → Inv 3 ( G , 2) ind → CH 2 ( X gen ) tors → 0 , together with a group isomorphism I W ) ∩ Z [ T ∗ ]) Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec ≃ c 2 (( � , c 2 ( Z [ T ∗ ] W ) where c 2 is the second Chern class map. In addition, if G is simple , then Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec , so there is an isomorphism Inv 3 ( G , 2) ind ≃ CH 2 ( X gen ) tors . 12 / 31
Main Theorem. Let G be a split semisimple linear algebraic group over a field F and let X gen denote the associated versal flag. There is a short exact sequence 0 → Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec → Inv 3 ( G , 2) ind → CH 2 ( X gen ) tors → 0 , together with a group isomorphism I W ) ∩ Z [ T ∗ ]) Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec ≃ c 2 (( � , c 2 ( Z [ T ∗ ] W ) where c 2 is the second Chern class map. In addition, if G is simple , then Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec , so there is an isomorphism Inv 3 ( G , 2) ind ≃ CH 2 ( X gen ) tors . 12 / 31
Main Theorem. Let G be a split semisimple linear algebraic group over a field F and let X gen denote the associated versal flag. There is a short exact sequence 0 → Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec → Inv 3 ( G , 2) ind → CH 2 ( X gen ) tors → 0 , together with a group isomorphism I W ) ∩ Z [ T ∗ ]) Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec ≃ c 2 (( � , c 2 ( Z [ T ∗ ] W ) where c 2 is the second Chern class map. In addition, if G is simple , then Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec , so there is an isomorphism Inv 3 ( G , 2) ind ≃ CH 2 ( X gen ) tors . 12 / 31
Main Theorem. Let G be a split semisimple linear algebraic group over a field F and let X gen denote the associated versal flag. There is a short exact sequence 0 → Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec → Inv 3 ( G , 2) ind → CH 2 ( X gen ) tors → 0 , together with a group isomorphism I W ) ∩ Z [ T ∗ ]) Inv 3 ( G , 2) sdec Inv 3 ( G , 2) dec ≃ c 2 (( � , c 2 ( Z [ T ∗ ] W ) where c 2 is the second Chern class map. In addition, if G is simple , then Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec , so there is an isomorphism Inv 3 ( G , 2) ind ≃ CH 2 ( X gen ) tors . 12 / 31
Example If G is not simple, then Inv 3 ( G , 2) sdec does not necessarily coincide with Inv 3 ( G , 2) dec : Consider a quadratic form q of degree 4 with trivial discriminant (it corresponds to a SO 4 -torsor). There is an invariant given by q �→ α ∪ β ∪ γ , where α is represented by q and � � β, γ � � = � α � q is the 2-Pfister form [Garibaldi-Merkurjev-Serre Ex. 20.3]. This invariant is semi-decomposable but not decomposable. 13 / 31
Example If G is not simple, then Inv 3 ( G , 2) sdec does not necessarily coincide with Inv 3 ( G , 2) dec : Consider a quadratic form q of degree 4 with trivial discriminant (it corresponds to a SO 4 -torsor). There is an invariant given by q �→ α ∪ β ∪ γ , where α is represented by q and � � β, γ � � = � α � q is the 2-Pfister form [Garibaldi-Merkurjev-Serre Ex. 20.3]. This invariant is semi-decomposable but not decomposable. 13 / 31
Example If G is not simple, then Inv 3 ( G , 2) sdec does not necessarily coincide with Inv 3 ( G , 2) dec : Consider a quadratic form q of degree 4 with trivial discriminant (it corresponds to a SO 4 -torsor). There is an invariant given by q �→ α ∪ β ∪ γ , where α is represented by q and � � β, γ � � = � α � q is the 2-Pfister form [Garibaldi-Merkurjev-Serre Ex. 20.3]. This invariant is semi-decomposable but not decomposable. 13 / 31
Invariants vs. Geometry Let p be a prime integer and G = SL p s / µ p r for some integers s ≥ r > 0. If p is odd, we set k = min { r , s − r } and if p = 2 we assume that s ≥ r + 1 and set k = min { r , s − r − 1 } . It is shown by Bermudez-Ruozzi (2013) that the group Inv 3 ( G , 2) ind is cyclic of order p k . By Karpenko (1998) if X is the Severi-Brauer variety of a generic algebra A gen , then CH 2 ( X ) tors is also a cyclic group of order p k (to show this Karpenko uses the Grothendieck γ -filtration). The canonical morphism X gen → X is an iterated projective bundle, hence, CH 2 ( X gen ) tors ≃ CH 2 ( X ) tors is a cyclic group of order p k . The exact sequence of the theorem implies that Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec 14 / 31
Invariants vs. Geometry Let p be a prime integer and G = SL p s / µ p r for some integers s ≥ r > 0. If p is odd, we set k = min { r , s − r } and if p = 2 we assume that s ≥ r + 1 and set k = min { r , s − r − 1 } . It is shown by Bermudez-Ruozzi (2013) that the group Inv 3 ( G , 2) ind is cyclic of order p k . By Karpenko (1998) if X is the Severi-Brauer variety of a generic algebra A gen , then CH 2 ( X ) tors is also a cyclic group of order p k (to show this Karpenko uses the Grothendieck γ -filtration). The canonical morphism X gen → X is an iterated projective bundle, hence, CH 2 ( X gen ) tors ≃ CH 2 ( X ) tors is a cyclic group of order p k . The exact sequence of the theorem implies that Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec 14 / 31
Invariants vs. Geometry Let p be a prime integer and G = SL p s / µ p r for some integers s ≥ r > 0. If p is odd, we set k = min { r , s − r } and if p = 2 we assume that s ≥ r + 1 and set k = min { r , s − r − 1 } . It is shown by Bermudez-Ruozzi (2013) that the group Inv 3 ( G , 2) ind is cyclic of order p k . By Karpenko (1998) if X is the Severi-Brauer variety of a generic algebra A gen , then CH 2 ( X ) tors is also a cyclic group of order p k (to show this Karpenko uses the Grothendieck γ -filtration). The canonical morphism X gen → X is an iterated projective bundle, hence, CH 2 ( X gen ) tors ≃ CH 2 ( X ) tors is a cyclic group of order p k . The exact sequence of the theorem implies that Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec 14 / 31
Invariants vs. Geometry Let p be a prime integer and G = SL p s / µ p r for some integers s ≥ r > 0. If p is odd, we set k = min { r , s − r } and if p = 2 we assume that s ≥ r + 1 and set k = min { r , s − r − 1 } . It is shown by Bermudez-Ruozzi (2013) that the group Inv 3 ( G , 2) ind is cyclic of order p k . By Karpenko (1998) if X is the Severi-Brauer variety of a generic algebra A gen , then CH 2 ( X ) tors is also a cyclic group of order p k (to show this Karpenko uses the Grothendieck γ -filtration). The canonical morphism X gen → X is an iterated projective bundle, hence, CH 2 ( X gen ) tors ≃ CH 2 ( X ) tors is a cyclic group of order p k . The exact sequence of the theorem implies that Inv 3 ( G , 2) sdec = Inv 3 ( G , 2) dec 14 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry Consider K 0 ( X gen ). It can be shown that the Grothendieck γ -filtration on X gen coincides with the topological filtration. So that for simple groups γ 2 / 3 ( X gen ) ≃ τ 2 / 3 ( X gen ) ≃ Tors CH 2 ( X gen ) ≃ Inv 3 ( G , 2) ind The group γ 2 / 3 ( X gen ) can be computed using Panin’s theorem (this involves the Steinberg basis and indices of Tits algebras). This gives a way to describe invariants via algebraic cycles (characteristic classes of bundles over X gen ). How to construct non-trivial torsion elements in γ 2 / 3 ( X gen ) directly ? for strongly inner and some inner G (Garibaldi-Z., 2012) is open for a general G . 15 / 31
Invariants vs. Geometry What about non-versal case? Namely, what is the relation between γ 2 / 3 ( X ) and Inv 3 ( G , 2) ind , where X is a twisted form of G / B by means of an arbitrary G -torsor ? Does γ 2 / 3 ( X ) correspond to a group of ’conditional invariants’ ? Yes, for some PGO 4 n -torsors (Junkins, 2013) Here the non-trivial torsion elements of γ 2 / 3 ( X gen ) can be constructed using the twisted γ -filtration (Z., 2012). In particular, a nontrivial torsion element constructed by Junkins can be viewed as an invariant of algebras with orthogonal involutions satisfying some restrictions on indices of the Tits algebras. 16 / 31
Invariants vs. Geometry What about non-versal case? Namely, what is the relation between γ 2 / 3 ( X ) and Inv 3 ( G , 2) ind , where X is a twisted form of G / B by means of an arbitrary G -torsor ? Does γ 2 / 3 ( X ) correspond to a group of ’conditional invariants’ ? Yes, for some PGO 4 n -torsors (Junkins, 2013) Here the non-trivial torsion elements of γ 2 / 3 ( X gen ) can be constructed using the twisted γ -filtration (Z., 2012). In particular, a nontrivial torsion element constructed by Junkins can be viewed as an invariant of algebras with orthogonal involutions satisfying some restrictions on indices of the Tits algebras. 16 / 31
Invariants vs. Geometry What about non-versal case? Namely, what is the relation between γ 2 / 3 ( X ) and Inv 3 ( G , 2) ind , where X is a twisted form of G / B by means of an arbitrary G -torsor ? Does γ 2 / 3 ( X ) correspond to a group of ’conditional invariants’ ? Yes, for some PGO 4 n -torsors (Junkins, 2013) Here the non-trivial torsion elements of γ 2 / 3 ( X gen ) can be constructed using the twisted γ -filtration (Z., 2012). In particular, a nontrivial torsion element constructed by Junkins can be viewed as an invariant of algebras with orthogonal involutions satisfying some restrictions on indices of the Tits algebras. 16 / 31
Invariants vs. Geometry What about non-versal case? Namely, what is the relation between γ 2 / 3 ( X ) and Inv 3 ( G , 2) ind , where X is a twisted form of G / B by means of an arbitrary G -torsor ? Does γ 2 / 3 ( X ) correspond to a group of ’conditional invariants’ ? Yes, for some PGO 4 n -torsors (Junkins, 2013) Here the non-trivial torsion elements of γ 2 / 3 ( X gen ) can be constructed using the twisted γ -filtration (Z., 2012). In particular, a nontrivial torsion element constructed by Junkins can be viewed as an invariant of algebras with orthogonal involutions satisfying some restrictions on indices of the Tits algebras. 16 / 31
Invariants vs. Geometry What about non-versal case? Namely, what is the relation between γ 2 / 3 ( X ) and Inv 3 ( G , 2) ind , where X is a twisted form of G / B by means of an arbitrary G -torsor ? Does γ 2 / 3 ( X ) correspond to a group of ’conditional invariants’ ? Yes, for some PGO 4 n -torsors (Junkins, 2013) Here the non-trivial torsion elements of γ 2 / 3 ( X gen ) can be constructed using the twisted γ -filtration (Z., 2012). In particular, a nontrivial torsion element constructed by Junkins can be viewed as an invariant of algebras with orthogonal involutions satisfying some restrictions on indices of the Tits algebras. 16 / 31
Geometry vs. Invariants As Inv 3 ( G , 2) ind has been computed for all simple groups, we immediately compute Tors CH 2 for all versal flags for twisted flag varieties that share the same upper-motive as the versal flag (one uses here various motivic decomposition results), e.g. for (generic) maximal orthogonal Grassmannian, for some generalized Severi-Brauer varieties... 17 / 31
Geometry vs. Invariants As Inv 3 ( G , 2) ind has been computed for all simple groups, we immediately compute Tors CH 2 for all versal flags for twisted flag varieties that share the same upper-motive as the versal flag (one uses here various motivic decomposition results), e.g. for (generic) maximal orthogonal Grassmannian, for some generalized Severi-Brauer varieties... 17 / 31
Non-triviality of invariants. Case C n Let G = PGSp 2 n be the split projective symplectic group. For a field extension L / F , the set H 1 ( L , G ) is identified with the set of isomorphism classes of central simple L -algebras A of degree 2 n with a symplectic involution σ . A decomposable invariant of G then takes an algebra with involution ( A , σ ) to the cup-product φ ∪ [ A ] for a fixed element φ ∈ F × . In particular, decomposable invariants of G are independent of the involution. 18 / 31
Non-triviality of invariants. Case C n Let G = PGSp 2 n be the split projective symplectic group. For a field extension L / F , the set H 1 ( L , G ) is identified with the set of isomorphism classes of central simple L -algebras A of degree 2 n with a symplectic involution σ . A decomposable invariant of G then takes an algebra with involution ( A , σ ) to the cup-product φ ∪ [ A ] for a fixed element φ ∈ F × . In particular, decomposable invariants of G are independent of the involution. 18 / 31
Non-triviality of invariants. Case C n Suppose that 4 | n . It is known that the group of indecomposable invariants Inv 3 ( G , 2) ind is cyclic of order 2. If char( F ) � = 2, Garibaldi, Parimala and Tignol constructed a degree 3 cohomological invariant ∆ 2 n of the group G with coefficients in Z / 2 Z . They showed that if a ∈ A is a σ -symmetric element of A × and σ ′ = Int( a ) ◦ σ , then ∆ 2 n ( A , σ ′ ) = ∆ 2 n ( A , σ ) + Nrp( a ) ∪ [ A ] , (1) where Nrp is the pfaffian norm. In particular, ∆ 2 n does depend on the involution and, therefore, the invariant ∆ 2 n is not decomposable. Hence the class of ∆ 2 n in Inv 3 ( G , 2) ind is nontrivial. 19 / 31
Non-triviality of invariants. Case C n Suppose that 4 | n . It is known that the group of indecomposable invariants Inv 3 ( G , 2) ind is cyclic of order 2. If char( F ) � = 2, Garibaldi, Parimala and Tignol constructed a degree 3 cohomological invariant ∆ 2 n of the group G with coefficients in Z / 2 Z . They showed that if a ∈ A is a σ -symmetric element of A × and σ ′ = Int( a ) ◦ σ , then ∆ 2 n ( A , σ ′ ) = ∆ 2 n ( A , σ ) + Nrp( a ) ∪ [ A ] , (1) where Nrp is the pfaffian norm. In particular, ∆ 2 n does depend on the involution and, therefore, the invariant ∆ 2 n is not decomposable. Hence the class of ∆ 2 n in Inv 3 ( G , 2) ind is nontrivial. 19 / 31
Non-triviality of invariants. Case C n Suppose that 4 | n . It is known that the group of indecomposable invariants Inv 3 ( G , 2) ind is cyclic of order 2. If char( F ) � = 2, Garibaldi, Parimala and Tignol constructed a degree 3 cohomological invariant ∆ 2 n of the group G with coefficients in Z / 2 Z . They showed that if a ∈ A is a σ -symmetric element of A × and σ ′ = Int( a ) ◦ σ , then ∆ 2 n ( A , σ ′ ) = ∆ 2 n ( A , σ ) + Nrp( a ) ∪ [ A ] , (1) where Nrp is the pfaffian norm. In particular, ∆ 2 n does depend on the involution and, therefore, the invariant ∆ 2 n is not decomposable. Hence the class of ∆ 2 n in Inv 3 ( G , 2) ind is nontrivial. 19 / 31
Non-triviality of invariants. Case C n Suppose that 4 | n . It is known that the group of indecomposable invariants Inv 3 ( G , 2) ind is cyclic of order 2. If char( F ) � = 2, Garibaldi, Parimala and Tignol constructed a degree 3 cohomological invariant ∆ 2 n of the group G with coefficients in Z / 2 Z . They showed that if a ∈ A is a σ -symmetric element of A × and σ ′ = Int( a ) ◦ σ , then ∆ 2 n ( A , σ ′ ) = ∆ 2 n ( A , σ ) + Nrp( a ) ∪ [ A ] , (1) where Nrp is the pfaffian norm. In particular, ∆ 2 n does depend on the involution and, therefore, the invariant ∆ 2 n is not decomposable. Hence the class of ∆ 2 n in Inv 3 ( G , 2) ind is nontrivial. 19 / 31
Nontriviality of invariants. Case C n So the class ∆ 2 n ( A ) ∈ H 3 ( L , Z / 2 Z ) of ∆ 2 n ( A , σ ) depends only on the L × ∪ [ A ] L -algebra A of degree 2 n and exponent 2 but not on the involution. Since ∆ 2 n ( A , σ ) is not decomposable, it is not semi-decomposable by our main theorem. The latter implies that ∆ 2 n ( A ) is nontrivial generically , i.e. there is a central simple algebra A of degree 2 n over a field extension of F with exponent 2 such that ∆ 2 n ( A ) � = 0. 20 / 31
Nontriviality of invariants. Case C n So the class ∆ 2 n ( A ) ∈ H 3 ( L , Z / 2 Z ) of ∆ 2 n ( A , σ ) depends only on the L × ∪ [ A ] L -algebra A of degree 2 n and exponent 2 but not on the involution. Since ∆ 2 n ( A , σ ) is not decomposable, it is not semi-decomposable by our main theorem. The latter implies that ∆ 2 n ( A ) is nontrivial generically , i.e. there is a central simple algebra A of degree 2 n over a field extension of F with exponent 2 such that ∆ 2 n ( A ) � = 0. 20 / 31
Nontriviality of invariants. Case C n So the class ∆ 2 n ( A ) ∈ H 3 ( L , Z / 2 Z ) of ∆ 2 n ( A , σ ) depends only on the L × ∪ [ A ] L -algebra A of degree 2 n and exponent 2 but not on the involution. Since ∆ 2 n ( A , σ ) is not decomposable, it is not semi-decomposable by our main theorem. The latter implies that ∆ 2 n ( A ) is nontrivial generically , i.e. there is a central simple algebra A of degree 2 n over a field extension of F with exponent 2 such that ∆ 2 n ( A ) � = 0. 20 / 31
Nontriviality of invariants. Case A n Let G = SL n / µ m , where n and m are positive integers such that n and m have the same prime divisors and m | n . Given a field extension L / F the natural surjection G → PGL n yields a map α : H 1 ( L , G ) → H 1 ( L , PGL n ) ⊂ Br( L ) taking a G -torsor Y over L to the class of a central simple algebra A ( Y ) of degree n and exponent dividing m . By definition, a decomposable invariant of G is of the form Y �→ φ ∪ [ A ( Y )] for a fixed φ ∈ F × . 21 / 31
Nontriviality of invariants. Case A n Let G = SL n / µ m , where n and m are positive integers such that n and m have the same prime divisors and m | n . Given a field extension L / F the natural surjection G → PGL n yields a map α : H 1 ( L , G ) → H 1 ( L , PGL n ) ⊂ Br( L ) taking a G -torsor Y over L to the class of a central simple algebra A ( Y ) of degree n and exponent dividing m . By definition, a decomposable invariant of G is of the form Y �→ φ ∪ [ A ( Y )] for a fixed φ ∈ F × . 21 / 31
Nontriviality of invariants. Case A n Let G = SL n / µ m , where n and m are positive integers such that n and m have the same prime divisors and m | n . Given a field extension L / F the natural surjection G → PGL n yields a map α : H 1 ( L , G ) → H 1 ( L , PGL n ) ⊂ Br( L ) taking a G -torsor Y over L to the class of a central simple algebra A ( Y ) of degree n and exponent dividing m . By definition, a decomposable invariant of G is of the form Y �→ φ ∪ [ A ( Y )] for a fixed φ ∈ F × . 21 / 31
Nontriviality of invariants. Case A n The map SL m → SL n taking a matrix M to the tensor product M ⊗ I n / m with the identity matrix, gives rise to a group homomorphism PGL m → G . The induced homomorphism ϕ : Inv 3 ( G , 2) norm → Inv 3 ( PGL m , 2) norm = F × / F × m is a splitting of the inclusion homomorphism F × / F × m = Inv 3 ( G , 2) dec ֒ → Inv 3 ( G , 2) norm . 22 / 31
Nontriviality of invariants. Case A n The map SL m → SL n taking a matrix M to the tensor product M ⊗ I n / m with the identity matrix, gives rise to a group homomorphism PGL m → G . The induced homomorphism ϕ : Inv 3 ( G , 2) norm → Inv 3 ( PGL m , 2) norm = F × / F × m is a splitting of the inclusion homomorphism F × / F × m = Inv 3 ( G , 2) dec ֒ → Inv 3 ( G , 2) norm . 22 / 31
Nontriviality of invariants. Case A n Collecting descriptions of p -primary components of Inv 3 ( G , 2) ind we get � gcd( n if n m , m ) , m is odd; Inv 3 ( G , 2) ind ≃ m k Z q / m Z q , where k = gcd( n if n 2 m , m ) , m is even. (2) Let ∆ n , m be a (unique) invariant in Inv 3 ( G , 2) norm such that its class in Inv 3 ( G , 2) ind corresponds to m k q + m Z q and ϕ (∆ n , m ) = 0. Note that the order of ∆ n , m in Inv 3 ( G , 2) norm is equal to k . Therefore, ∆ n , m takes values in H 3 ( − , Z / k Z (2)) ⊂ H 3 ( − , 2). 23 / 31
Nontriviality of invariants. Case A n Collecting descriptions of p -primary components of Inv 3 ( G , 2) ind we get � gcd( n if n m , m ) , m is odd; Inv 3 ( G , 2) ind ≃ m k Z q / m Z q , where k = gcd( n if n 2 m , m ) , m is even. (2) Let ∆ n , m be a (unique) invariant in Inv 3 ( G , 2) norm such that its class in Inv 3 ( G , 2) ind corresponds to m k q + m Z q and ϕ (∆ n , m ) = 0. Note that the order of ∆ n , m in Inv 3 ( G , 2) norm is equal to k . Therefore, ∆ n , m takes values in H 3 ( − , Z / k Z (2)) ⊂ H 3 ( − , 2). 23 / 31
Nontriviality of invariants. Case A n Collecting descriptions of p -primary components of Inv 3 ( G , 2) ind we get � gcd( n if n m , m ) , m is odd; Inv 3 ( G , 2) ind ≃ m k Z q / m Z q , where k = gcd( n if n 2 m , m ) , m is even. (2) Let ∆ n , m be a (unique) invariant in Inv 3 ( G , 2) norm such that its class in Inv 3 ( G , 2) ind corresponds to m k q + m Z q and ϕ (∆ n , m ) = 0. Note that the order of ∆ n , m in Inv 3 ( G , 2) norm is equal to k . Therefore, ∆ n , m takes values in H 3 ( − , Z / k Z (2)) ⊂ H 3 ( − , 2). 23 / 31
Nontriviality of invariants. Case A n Fix a G -torsor Y over F and consider the twists Y G and SL 1 ( A ( Y )) by Y of the groups G and SL n respectively. By (2) the image of ∆ n , m under the natural composition Inv 3 ( G , 2) norm ≃ Inv 3 ( Y G , 2) norm − → Inv 3 ( SL 1 ( A ( Y )) , 2) norm is a m k -multiple of the Rost invariant. Recall that the Rost invariant takes the class of φ in F × / Nrd( A ( Y ) × ) = H 1 ( F , SL 1 ( A ( Y ))) to the cup-product φ ∪ [ A ( Y )] ∈ H 3 ( F , 2). So we get ∆ n , m ( φ Y ) − ∆ n , m ( Y ) ∈ F × ∪ m k [ A ( Y )] . (3) (Here the group F × acts transitively on the fiber over A ( Y ) of the map α . If φ ∈ F × , we write φ Y for the corresponding element in the fiber.) 24 / 31
Nontriviality of invariants. Case A n Fix a G -torsor Y over F and consider the twists Y G and SL 1 ( A ( Y )) by Y of the groups G and SL n respectively. By (2) the image of ∆ n , m under the natural composition Inv 3 ( G , 2) norm ≃ Inv 3 ( Y G , 2) norm − → Inv 3 ( SL 1 ( A ( Y )) , 2) norm is a m k -multiple of the Rost invariant. Recall that the Rost invariant takes the class of φ in F × / Nrd( A ( Y ) × ) = H 1 ( F , SL 1 ( A ( Y ))) to the cup-product φ ∪ [ A ( Y )] ∈ H 3 ( F , 2). So we get ∆ n , m ( φ Y ) − ∆ n , m ( Y ) ∈ F × ∪ m k [ A ( Y )] . (3) (Here the group F × acts transitively on the fiber over A ( Y ) of the map α . If φ ∈ F × , we write φ Y for the corresponding element in the fiber.) 24 / 31
Nontriviality of invariants. Case A n Fix a G -torsor Y over F and consider the twists Y G and SL 1 ( A ( Y )) by Y of the groups G and SL n respectively. By (2) the image of ∆ n , m under the natural composition Inv 3 ( G , 2) norm ≃ Inv 3 ( Y G , 2) norm − → Inv 3 ( SL 1 ( A ( Y )) , 2) norm is a m k -multiple of the Rost invariant. Recall that the Rost invariant takes the class of φ in F × / Nrd( A ( Y ) × ) = H 1 ( F , SL 1 ( A ( Y ))) to the cup-product φ ∪ [ A ( Y )] ∈ H 3 ( F , 2). So we get ∆ n , m ( φ Y ) − ∆ n , m ( Y ) ∈ F × ∪ m k [ A ( Y )] . (3) (Here the group F × acts transitively on the fiber over A ( Y ) of the map α . If φ ∈ F × , we write φ Y for the corresponding element in the fiber.) 24 / 31
Nontriviality of invariants. Case A n Fix a G -torsor Y over F and consider the twists Y G and SL 1 ( A ( Y )) by Y of the groups G and SL n respectively. By (2) the image of ∆ n , m under the natural composition Inv 3 ( G , 2) norm ≃ Inv 3 ( Y G , 2) norm − → Inv 3 ( SL 1 ( A ( Y )) , 2) norm is a m k -multiple of the Rost invariant. Recall that the Rost invariant takes the class of φ in F × / Nrd( A ( Y ) × ) = H 1 ( F , SL 1 ( A ( Y ))) to the cup-product φ ∪ [ A ( Y )] ∈ H 3 ( F , 2). So we get ∆ n , m ( φ Y ) − ∆ n , m ( Y ) ∈ F × ∪ m k [ A ( Y )] . (3) (Here the group F × acts transitively on the fiber over A ( Y ) of the map α . If φ ∈ F × , we write φ Y for the corresponding element in the fiber.) 24 / 31
Nontriviality of invariants. Case A n Given a central simple L -algebra A of degree n and exponent dividing m , we define an element ∆ n , m ( A ) ∈ H 3 ( L , Z / k Z (2)) L × ∪ m k [ A ] as follows. Choose a G -torsor Y over L with A ( Y ) ≃ A and set ∆ n , m ( A ) to be the class of ∆ n , m ( Y ) in the factor group. It follows from (3) that ∆ n , m ( A ) is independent of the choice of Y . 25 / 31
Nontriviality of invariants. Case A n Given a central simple L -algebra A of degree n and exponent dividing m , we define an element ∆ n , m ( A ) ∈ H 3 ( L , Z / k Z (2)) L × ∪ m k [ A ] as follows. Choose a G -torsor Y over L with A ( Y ) ≃ A and set ∆ n , m ( A ) to be the class of ∆ n , m ( Y ) in the factor group. It follows from (3) that ∆ n , m ( A ) is independent of the choice of Y . 25 / 31
Nontriviality of invariants. Case A n Given a central simple L -algebra A of degree n and exponent dividing m , we define an element ∆ n , m ( A ) ∈ H 3 ( L , Z / k Z (2)) L × ∪ m k [ A ] as follows. Choose a G -torsor Y over L with A ( Y ) ≃ A and set ∆ n , m ( A ) to be the class of ∆ n , m ( Y ) in the factor group. It follows from (3) that ∆ n , m ( A ) is independent of the choice of Y . 25 / 31
Nontriviality of invariants. Case A n Proposition. Let A be a central simple L -algebra of degree n and exponent dividing m . Then the order of ∆ n , m ( A ) divides k . If A is a generic algebra, then the order of ∆ n , m ( A ) is equal to k . Proof: If k ′ is a proper divisor of k , then the multiple k ′ ∆ n , m is not decomposable. By our theorem k ′ ∆ n , m is not semi-decomposable and, hence, k ′ ∆ n , m ( A ) � = 0. 26 / 31
Nontriviality of invariants. Case A n Proposition. Let A be a central simple L -algebra of degree n and exponent dividing m . Then the order of ∆ n , m ( A ) divides k . If A is a generic algebra, then the order of ∆ n , m ( A ) is equal to k . Proof: If k ′ is a proper divisor of k , then the multiple k ′ ∆ n , m is not decomposable. By our theorem k ′ ∆ n , m is not semi-decomposable and, hence, k ′ ∆ n , m ( A ) � = 0. 26 / 31
Nontriviality of invariants. Case A n Proposition. Let A be a central simple L -algebra of degree n and exponent dividing m . Then the order of ∆ n , m ( A ) divides k . If A is a generic algebra, then the order of ∆ n , m ( A ) is equal to k . Proof: If k ′ is a proper divisor of k , then the multiple k ′ ∆ n , m is not decomposable. By our theorem k ′ ∆ n , m is not semi-decomposable and, hence, k ′ ∆ n , m ( A ) � = 0. 26 / 31
Nontriviality of invariants. Case A n Example. Let A be a central simple F -algebra of degree 2 n divisible by 8 and exponent 2. Choose a symplectic involution σ on A . The group PGSp 2 n is a subgroup of SL 2 n / µ 2 , hence, if char( F ) � = 2, the restriction of the invariant ∆ 2 n , 2 on PGSp 2 n is the invariant ∆ 2 n ( A , σ ) considered before. It follows that ∆ 2 n , 2 ( A ) = ∆ 2 n ( A ) in the group H 3 ( F , Z / 2 Z ) / ( F × ∪ [ A ]). 27 / 31
Nontriviality of invariants. Case A n Example. Let A be a central simple F -algebra of degree 2 n divisible by 8 and exponent 2. Choose a symplectic involution σ on A . The group PGSp 2 n is a subgroup of SL 2 n / µ 2 , hence, if char( F ) � = 2, the restriction of the invariant ∆ 2 n , 2 on PGSp 2 n is the invariant ∆ 2 n ( A , σ ) considered before. It follows that ∆ 2 n , 2 ( A ) = ∆ 2 n ( A ) in the group H 3 ( F , Z / 2 Z ) / ( F × ∪ [ A ]). 27 / 31
Nontriviality of invariants. Case A n Example. Let A be a central simple F -algebra of degree 2 n divisible by 8 and exponent 2. Choose a symplectic involution σ on A . The group PGSp 2 n is a subgroup of SL 2 n / µ 2 , hence, if char( F ) � = 2, the restriction of the invariant ∆ 2 n , 2 on PGSp 2 n is the invariant ∆ 2 n ( A , σ ) considered before. It follows that ∆ 2 n , 2 ( A ) = ∆ 2 n ( A ) in the group H 3 ( F , Z / 2 Z ) / ( F × ∪ [ A ]). 27 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Nontriviality of invariants. Case A n The class ∆ n , m is trivial on decomposable algebras: Proposition. Let n 1 , n 2 , m be positive integers such that m divides n 1 and n 2 . Let A 1 and A 2 be two central simple algebras over F of degree n 1 and n 2 respectively and of exponent dividing m . Then ∆ n 1 n 2 , m ( A 1 ⊗ F A 2 ) = 0. Proof: The tensor product homomorphism SL n 1 × SL n 2 → SL n 1 n 2 yields a homomorphism Sym 2 ( T ∗ n 1 n 2 ) → Sym 2 ( T ∗ n 1 ) ⊕ Sym 2 ( T ∗ n 2 ) , where T n 1 , T n 2 and T n 1 n 2 are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q n 1 n 2 of Sym 2 ( T ∗ n 1 n 2 ) is equal to n 2 q n 1 + n 1 q n 2 . Since n 1 and n 2 are divisible by m , the pull-back of the invariant ∆ n 1 n 2 , m under the homomorphism ( SL n 1 / µ m ) × ( SL n 2 / µ m ) → SL n 1 n 2 / µ m is trivial. 28 / 31
Invariants vs. Representation Theory How to show that Inv 3 ( G , 2) dec = Inv 3 ( G , 2) sdec for simple groups of type C 4 n ? We want to show that c 2 ( x ) ∈ 2 Z q for every element I W ) ∩ Z [ T ∗ ]. x ∈ ( � Given a weight χ ∈ Λ we denote by W ( χ ) its W -orbit and we e χ := � define � λ ∈ W ( χ ) (1 − e − λ ). I W ) is generated by elements { � By definition, the ideal ( � e ω i } i =1 .. 4 m corresponding to the fundamental weights ω i . An element x can be written as 4 m � where n i ∈ Z and δ i ∈ � n i � e ω i + δ i � e ω i , x = I . (4) i =1 29 / 31
Invariants vs. Representation Theory How to show that Inv 3 ( G , 2) dec = Inv 3 ( G , 2) sdec for simple groups of type C 4 n ? We want to show that c 2 ( x ) ∈ 2 Z q for every element I W ) ∩ Z [ T ∗ ]. x ∈ ( � Given a weight χ ∈ Λ we denote by W ( χ ) its W -orbit and we e χ := � define � λ ∈ W ( χ ) (1 − e − λ ). I W ) is generated by elements { � By definition, the ideal ( � e ω i } i =1 .. 4 m corresponding to the fundamental weights ω i . An element x can be written as 4 m � where n i ∈ Z and δ i ∈ � n i � e ω i + δ i � e ω i , x = I . (4) i =1 29 / 31
Invariants vs. Representation Theory How to show that Inv 3 ( G , 2) dec = Inv 3 ( G , 2) sdec for simple groups of type C 4 n ? We want to show that c 2 ( x ) ∈ 2 Z q for every element I W ) ∩ Z [ T ∗ ]. x ∈ ( � Given a weight χ ∈ Λ we denote by W ( χ ) its W -orbit and we e χ := � define � λ ∈ W ( χ ) (1 − e − λ ). I W ) is generated by elements { � By definition, the ideal ( � e ω i } i =1 .. 4 m corresponding to the fundamental weights ω i . An element x can be written as 4 m � where n i ∈ Z and δ i ∈ � n i � e ω i + δ i � e ω i , x = I . (4) i =1 29 / 31
Invariants vs. Representation Theory How to show that Inv 3 ( G , 2) dec = Inv 3 ( G , 2) sdec for simple groups of type C 4 n ? We want to show that c 2 ( x ) ∈ 2 Z q for every element I W ) ∩ Z [ T ∗ ]. x ∈ ( � Given a weight χ ∈ Λ we denote by W ( χ ) its W -orbit and we e χ := � define � λ ∈ W ( χ ) (1 − e − λ ). I W ) is generated by elements { � By definition, the ideal ( � e ω i } i =1 .. 4 m corresponding to the fundamental weights ω i . An element x can be written as 4 m � where n i ∈ Z and δ i ∈ � n i � e ω i + δ i � e ω i , x = I . (4) i =1 29 / 31
Invariants vs. Representation Theory How to show that Inv 3 ( G , 2) dec = Inv 3 ( G , 2) sdec for simple groups of type C 4 n ? We want to show that c 2 ( x ) ∈ 2 Z q for every element I W ) ∩ Z [ T ∗ ]. x ∈ ( � Given a weight χ ∈ Λ we denote by W ( χ ) its W -orbit and we e χ := � define � λ ∈ W ( χ ) (1 − e − λ ). I W ) is generated by elements { � By definition, the ideal ( � e ω i } i =1 .. 4 m corresponding to the fundamental weights ω i . An element x can be written as 4 m � where n i ∈ Z and δ i ∈ � n i � e ω i + δ i � e ω i , x = I . (4) i =1 29 / 31
Invariants vs. Representation Theory Consider a ring homomorphism f : Z [Λ] → Z [Λ / T ∗ ] induced by taking the quotient Λ → Λ / T ∗ = C ∗ . We have Λ / T ∗ ≃ Z / 2 Z and Z [Λ / T ∗ ] = Z [ y ] / ( y 2 − 2 y ), where y = f ( e ω 1 − 1). Observe that C ∗ is W -invariant. By definition, f ( I ) = 0, so f ( x ) = 0. Since ω i ∈ T ∗ for all even i , e ω i ) = y for all odd i and f ( δ i ) ∈ f ( � f ( � I ) = ( y ), we get � � n i d i y + m i d i y 2 = ( 0 = f ( x ) = n i + 2 m i ) d i y , i is odd i is odd where m i ∈ Z and d i = 2 i � 4 m � is the cardinality of W ( ω i ), which implies that ( � i i is odd n i + 2 m i ) d i = 0. Dividing this sum by the g.c.d. of all d i ’s and taking the result � n � n modulo 2 (here one uses the fact g . c . d . ( n , k ) | ), we obtain that k the coefficient n 1 in the presentation (4) has to be even. 30 / 31
Invariants vs. Representation Theory Consider a ring homomorphism f : Z [Λ] → Z [Λ / T ∗ ] induced by taking the quotient Λ → Λ / T ∗ = C ∗ . We have Λ / T ∗ ≃ Z / 2 Z and Z [Λ / T ∗ ] = Z [ y ] / ( y 2 − 2 y ), where y = f ( e ω 1 − 1). Observe that C ∗ is W -invariant. By definition, f ( I ) = 0, so f ( x ) = 0. Since ω i ∈ T ∗ for all even i , e ω i ) = y for all odd i and f ( δ i ) ∈ f ( � f ( � I ) = ( y ), we get � � n i d i y + m i d i y 2 = ( 0 = f ( x ) = n i + 2 m i ) d i y , i is odd i is odd where m i ∈ Z and d i = 2 i � 4 m � is the cardinality of W ( ω i ), which implies that ( � i i is odd n i + 2 m i ) d i = 0. Dividing this sum by the g.c.d. of all d i ’s and taking the result � n � n modulo 2 (here one uses the fact g . c . d . ( n , k ) | ), we obtain that k the coefficient n 1 in the presentation (4) has to be even. 30 / 31
Invariants vs. Representation Theory Consider a ring homomorphism f : Z [Λ] → Z [Λ / T ∗ ] induced by taking the quotient Λ → Λ / T ∗ = C ∗ . We have Λ / T ∗ ≃ Z / 2 Z and Z [Λ / T ∗ ] = Z [ y ] / ( y 2 − 2 y ), where y = f ( e ω 1 − 1). Observe that C ∗ is W -invariant. By definition, f ( I ) = 0, so f ( x ) = 0. Since ω i ∈ T ∗ for all even i , e ω i ) = y for all odd i and f ( δ i ) ∈ f ( � f ( � I ) = ( y ), we get � � n i d i y + m i d i y 2 = ( 0 = f ( x ) = n i + 2 m i ) d i y , i is odd i is odd where m i ∈ Z and d i = 2 i � 4 m � is the cardinality of W ( ω i ), which implies that ( � i i is odd n i + 2 m i ) d i = 0. Dividing this sum by the g.c.d. of all d i ’s and taking the result � n � n modulo 2 (here one uses the fact g . c . d . ( n , k ) | ), we obtain that k the coefficient n 1 in the presentation (4) has to be even. 30 / 31
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