Orbit closures of quiver representations Kavita Sutar Chennai Mathematical Institute September 5, 2012
Outline 1 Orbit closures 2 Geometric technique 3 Calculations 4 Results
Quiver Q = (Q 0, Q 1 ) is a directed graph with set of vertices Q 0 and set of arrows Q 1. (Notation: ta a ha) Dynkin quiver - underlying unoriented graph is a Dynkin diagram.
A n D n E 6 E 7 E 8
Quiver Q = (Q 0, Q 1 ) is a directed graph with set of vertices Q 0 and set of arrows Q 1. (Notation: ta a ha) Dynkin quiver - underlying unoriented graph is a Dynkin diagram. A representation of Q is a pair V = ((V i ) i Q0, (V(a)) a Q1 ), where V i are finite-dimensional vector spaces over algebraically closed field k V(a) and V ta V ha. Dimension vector d = (dimv i ) i Q0. The representation space of dimension type d = (d i ) i Q0 N Q0 Rep(Q, d) = {V = ((V i ) i Q0, (V(a)) a Q1 ) dim(v i ) = d i } = a Q 1 Hom(V ta, V ha ) = A N.
GL(d) = i Q 0 GL(d i ) GL(d) acts on Rep(Q, d) by simultaneous change of basis at each vertex ((g i ) i Q0, (V(a)) a Q1 ) = (g ha V(a) g 1 ta ) a Q1 {GL(d) orbits} {isomorphism classes of representations of Q} O V [V] Orbit closure O V is a subvariety of the affine space Rep(Q, d).
Example 1 a 2 b 3 Q = (Q 0, Q 1 ); Q 0 = {1, 2, 3}, Q 1 = {a, b} V 1 V a V 2 V b V 3 Rep(Q, d) = Hom(V 1, V 2) Hom(V 2, V 3) d = (dim V 1, dim V 2, dim V 3) k 3 k 4 k 3 V a V b Rep(Q, (3, 4, 3)) = Hom(k 3, k 4 ) Hom(k 4, k 3 ) = A 24 k 3 V k 4 k 3 a V b Gl(3, k) Gl(4, k) Gl(3, k) GL(d) = GL(3) GL(4) GL(3) (g 1, g 2, g 3) (V a, V b ) := (g 2 V a g 1 1, g 3 V b g 1 2 ) O V is an affine variety in A 24.
Example V x V(a) dimv x = d Rep(Q, d) = M d d (k) Group action: conjugation Orbits: conjugacy classes of matrices in M(d, k) Geometry: normal, Cohen-Macaulay varieties with rational singularities. For nilpotent V(a), if char k > 0 then O V(a) is a Frobenius split variety. if char k = 0 then O V(a) is Gorenstein; defining ideal generated by minors of various sizes.
Example V (a) V 1 V 2 d = (d 1, d 2 ) Rep(Q, d) = M d2 d 1 (k) Group action: (g 1, g 2 ) V(a) = g 1 V(a)g 1 2 Orbits: O r = matrices of rank r, 0 r m = min(d 1, d 2 ) O r = j r O j (determinantal varieties) = {W Rep(Q, d) rank W(a) r} Geometry: normal, Cohen-Macaulay varieties with rational singularities; Gorenstein if r = 0, r = m or d 1 = d 2 ; regular if r = 0 or r = m; Resolution of defining ideal (Lascoux); defining ideal generated by (r + 1) (r + 1) minors.
Why study orbit closures? types of singularities degenerations desingularization normality, C-M, unibranchness tangent spaces defining equations etc.
Some results (1981) Abeasis, Del Fra, Kraft Equi-oriented quiver of type A n : orbit closures are normal, Cohen-Macaulay with rational singularities. (1998) Laxmibai, Magyar Extended the above result to arbitrary characteristic. (2001,2002) Bobinski, Zwara Proved the above result for A n with arbitrary orientation and for D n. Defining ideals? What about quivers of type E?
Goal To study orbit closures by calculating resolutions Strategy Use geometric technique
p Z X V V q q Y X X: affine space Y: subvariety V: a projective variety
Z X V q q p Y X V (affine space) (projective variety) Let Z = tot(η) and X V = tot(e). Exact sequence of vector bundles over V: 0 η E τ 0 Define ξ = τ K(ξ) : 0 t ξ 2 (p ξ) p ξ O X V resolves O Z as O X V -module.
Z X V q q p Y X V (affine space) (projective variety) Let Z = tot(η) and X V = tot(e). Exact sequence of vector bundles over V: 0 η E τ 0 Define ξ = τ K(ξ) : 0 t ξ 2 (p ξ) p ξ O X V q F
Z X V q q p Y X V (affine space) (projective variety) Main theorem [Weyman] F i = i+j H j (V, ξ) K A( i j) where A = K[X]. j 0 F i = 0 for i < 0 F is a finite free resolution of the normalization of K[Y]. If F i = 0 for i < 0 and F 0 = A Y is normal, Cohen-Macaulay and has rational singularities.
(Reineke desingularization) Z Rep(Q, d) x Q 0 F lag(d s (x),..., K d(x) ) q q p O V Rep(Q, d) x Q 0 F lag(d s (x),..., K d(x) ) F i = j 0 Hj (V, i+j ξ) K A( i j) where A = K[Rep(Q, d)]. If F i = 0 for i < 0 F is a finite free resolution of the normalization of K[O V ]. If F i = 0 for i < 0 and F 0 = A O V is normal and has rational singularities.
Steps in the construction Construct the Auslander-Reiten quiver of Q. Find a directed partition of the AR quiver (with s parts). Define a subset of Z I,V of Rep(Q, d) x Q 0 Flag(β, V x ) Z I,V = {(V, (R s (x) R 2 (x) V x ) x Q0 ) a Q 1, t, V(a)(R t (ta)) R t (ha)} Z I,V q Rep(Q, d) Flag(β, V x ) x Q 0 O V Rep(Q, d) q Theorem [Reineke] q(z I,V) = O V. q is a proper birational isomorphism of Z I,V and O V.
Example - determinantal variety KK 1 2 0K K0 V = a(0k) b(kk) c(k0) Z : V 1 V 2 β i = dim R i R 1 R 2 2 Z Rep(Q, Hom(V 1 d), V 2 ) Gr(β Flag(β 2, V i, 2 V) i ) i=1
Example - determinantal variety V φ 1 V 2 dim V 1 = 2 dim V 2 = 3 r = rank φ = 2 X = Hom(V 1, V 2 ) = V1 V 2 Orbit closure O V = {φ : V 1 V 2 rk φ 2} V = Gr(2, V 2 ) Z = tot(v1 R) = {(φ, R) im(φ) R} A = K[X] = Sym(V 1 V2 ) Lascoux resolution 0 S 21 V 1 3 V2 A( 3) d2 2 V 1 2 V2 A( 2) d1 A 0 d 1 = (x 11 x 22 x 12 x 21, x 11 x 32 x 12 x 31, x 21 x 32 x 22 x 31 ) x 31 x 32 d 2 = x 21 x 22 x 11 x 12
Non-equioriented A 3 0KK K00 1 2 3 0K0 KKK KK0 00K V = m α X α Z : V1 V 2 V α R 3 + β i = dim R i R 1 R 2 R 3 3 Z Rep(Q, d) Flag(β Gr(β i, i V, i V) i ) i=1
Non-equioriented A 3 0KK K00 1 2 3 0K0 KKK KK0 00K V = α R + X α Z : V 1 V 2 V 3 d = (3, 4, 3) β = (2, 1, 2) R 1 R 2 R 3 ξ = R 1 Q 2 R 3 Q 2
A 4 (source-sink ) K000 0KK0 000K 1 2 3 4 KKK0 0KKK 00K0 KKKK 0K00 00KK KK00 V = m α X α Z : V1 V 2 V 3 V α R 4 + R 1 R 2 R 3 R 4 Z Rep(Q, d) S 1 4 Flag(βi 2, βi 1, V i ) i=1 S 2 S 3 S 4
A 4 (source-sink ) K000 0KK0 000K 1 2 3 4 KKK0 0KKK 00K0 KKKK 0K00 00KK KK00 V = m α X α Z : V1 V 2 V 3 V α R 4 + R 1 R 2 R 3 R 4 S 1 ξ = (R 1 2 Q 1 1 + R 2 2 Q 2 1 ) (R 1 2 Q 1 3 + R 2 2 Q 2 3 ) (R 1 4 Q 1 3 + R 2 4 Q 2 3 ) S 2 S 3 S 4
ξ = a Q 1 ( t ξ = s t=1 R ta t a Q 1 λ(a) =t Q ha ) ( t ξ = R ta t a Q 1 S λ(a) R ta t Assign weights to each summand. S λ(a) Q ha t Q ha ) t Apply Bott s theorem to these weights to calculate F i = i+j H j (V, ξ) K A( i j), j 0 j = #exchanges D(λ) := a Q 1 λ(a) j = (i + j) j = i. Rewrite F D(λ) = i+j H j (V, ξ) K A( i j) j 0
Non-equioriented A 3 V 1 V 3 V 2 F D(λ) = c ν λ,µ(s λ V 1 S µ V 2 S ν V3 ) λ + µ =t Theorem (-) D(λ) E Q F 0 = A and F i = 0 for i < 0. Thus O V is normal and has rational singularities.
V 1 φ V 3 ψ V2 Theorem(-) Minimal generators of the defining ideal: Let p =rank(φ), q =rank(ψ), r =rank(φ + ψ), the minimal generators of O V are (p + 1) (p + 1) minors of φ, (q + 1) (q + 1) minors of ψ, (r + 1) (r + 1) minors of φ + ψ, taken by choosing b + k + 1 columns of φ and c + l + 1 columns of ψ such that k + l = d 1. (b, c, d are multiplicities of some indecomposable representations) Characterization of Gorenstein orbits.
0KK K00 1 2 3 0K0 KKK KK0 00K V = α R + X α Z : V 1 V 2 V 3 d = (3, 4, 3) R 1 R 2 R 3 ξ = R 1 Q 2 R 3 Q 2 A = Sym(V 1 V 3 ) Sym(V 2 V 3 )
A ( 3 V 1 3 V 3 A( 3)) ( 3 V 2 3 V 3 A( 3)) ( 2 V 1 2 V 2 4 V 3 A( 4)) (S 211 V 1 4 V 3 A( 4)) (S 211 V 2 4 V 3 A( 4)) ( 3 V 1 2 V 2 S 2111 V 3 A( 5)) ( 2 V 1 3 V 2 S 2111 V 3 A( 5)) 3 V 1 3 V 2 S 222 V 3 A( 6) (S 211 V 1 3 V 2 S 2221 V 3 A( 7)) ( 3 V 1 S 211 V 2 S 2221 V 3 A( 7)) ( 2 V 1 S 222 V 2 S 2222 V 3 A( 8)) (S 222 V 1 2 V 2 S 2222 V 3 A( 8)) 3 V 1 3 V 2 S 3111 V 3 A( 6) (S 211 V 1 S 211 V 2 S 2222 V 3 A( 8)) (S 222 V 1 3 V 2 S 3222 V 3 A( 9)) ( 3 V 1 S 222 V 2 S 3222 V 3 A( 9)) (S 222 V 1 S 222 V 2 S 3333 V 3 A( 12))
Source-sink quivers Theorem(-) Q be a source-sink quiver and V be a representation of Q such that the orbit closure O V admits a 1-step desingularization Z. Then D(λ) E Q Corollary If Q is Dynkin then O V is normal and has rational singularities. Orbit closures of type E 6, E 7 and E 8 admitting a 1-step desingularization are normal! Corollary If Q is extended Dynkin then F is a minimal free resolution of the normalization of O V.
Equioriented A n Theorem(-) Let Q be an equioriented quiver of type A n and β, γ be two dimension vectors. Let α = β + γ and V Rep(Q, α). Then D(λ) E Q F(β, γ) i = 0 for i < 0. F(β, γ) 0 = A. The summands of F(β, γ) 1 are of the form γ i+β j+1 V i γ i+β j+1 V j where 1 i < j n.
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