A combinatorial Fourier transform for quiver representation varieties in type A
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1 A combinatorial Fourier transform for quiver representation varieties in type A Pramod N. Achar, Maitreyee Kulkarni, and Jacob P. Matherne Louisiana State University and University of Massachusetts Amherst April 8, 8 Maurice Auslander Distinguished Lectures and International Conference
2 Goal Consider the quiver ÝÑ ÝÑ ÝÑ. Notation: Epwq - space of representations for dimension vector w pw,..., w n q Gpwq GLpw q ˆ ˆ GLpw n q w pw n,..., w q - the reverse dimension vector / 7
3 Goal Consider the quiver ÝÑ ÝÑ ÝÑ. Notation: Epwq - space of representations for dimension vector w pw,..., w n q Gpwq GLpw q ˆ ˆ GLpw n q w pw n,..., w q - the reverse dimension vector Can we give a combinatorial description of the Fourier Sato transform: D b Gpwq pepwqq T ÝÑ D b Gpw q pepw qq F ÞÝÑ q! q pfqrdim Epwqs for simple perverse sheaves F? / 7
4 / 7 Outline Quiver representation varieties Some combinatorics Fourier Sato transform Combinatorial Fourier transform
5 / 7 Outline Quiver representation varieties Some combinatorics Fourier Sato transform Combinatorial Fourier transform
6 4 / 7 Quiver representations Consider the type A n equioriented quiver Q n ÝÑ ÝÑ ÝÑ. A quiver representation is: A finite-dimensional C-vector space M i for each vertex. A linear map x i for each arrow. x M x M x n M n
7 4 / 7 Quiver representations Consider the type A n equioriented quiver Q n ÝÑ ÝÑ ÝÑ. A quiver representation is: A finite-dimensional C-vector space M i for each vertex. A linear map x i for each arrow. x M x M x n M n ReppQ n q - abelian category of finite-dimensional complex representations of Q n
8 5 / 7 Quiver representation varieties Fix a dimension vector w pw, w,..., w n q. A quiver representation variety Epwq is the space of all quiver representations for a fixed dimension vector w. Note that Epwq is an affine variety: Epwq» A w w `w w ` `w n w n.
9 Quiver representation varieties Fix a dimension vector w pw, w,..., w n q. A quiver representation variety Epwq is the space of all quiver representations for a fixed dimension vector w. Note that Epwq is an affine variety: Epwq» A w w `w w ` `w n w n. Gpwq GLpw q ˆ ˆ GLpw n q acts on Epwq by pg,..., g n q px,..., x n q pg x g,..., g nx n g n q giving it a stratification by orbits. Note that two points x, y P Epwq are in the same Gpwq-orbit if and only if they are isomorphic objects of ReppQ n q. 5 / 7
10 6 / 7 Classifying the orbits Theorem (Gabriel s Theorem) There is a bijection tindec. objects in ReppQ n qu{ ÐÑ tpos. roots for A n root systemu.
11 6 / 7 Classifying the orbits Theorem (Gabriel s Theorem) There is a bijection tindec. objects in ReppQ n qu{ ÐÑ tpos. roots for A n root systemu. To an indecomposable representation R ij Ñ Ñ Ñ C vertex i id ÝÑ ÝÑ id we associate its dimension vector, the positive root C Ñ Ñ Ñ. vertex j γ ij p,...,,,...,,,..., q. position i position j
12 6 / 7 Classifying the orbits Theorem (Gabriel s Theorem) There is a bijection tindec. objects in ReppQ n qu{ ÐÑ tpos. roots for A n root systemu. To an indecomposable representation R ij Ñ Ñ Ñ C vertex i id ÝÑ ÝÑ id we associate its dimension vector, the positive root Corollary There is a bijection C Ñ Ñ Ñ. vertex j γ ij p,...,,,...,,,..., q. position i position j tgpwq-orbits in Epwqu ÐÑ Bpwq : tb ij ÿ b ij γ ij wu.
13 7 / 7 Outline Quiver representation varieties Some combinatorics Fourier Sato transform Combinatorial Fourier transform
14 8 / 7 Triangular arrays ladder chute Define the set Ppwq of triangular arrays of nonnegative integers such the entries in the j th chute sum to w j. Ladders are weakly decreasing.
15 8 / 7 Triangular arrays ladder chute Define the set Ppwq of triangular arrays of nonnegative integers such the entries in the j th chute sum to w j. Ladders are weakly decreasing. For w p,, q, P Ppwq We will write y ij for the entry in the i th chute and j th column.
16 9 / 7 Classifying the orbits combinatorially Lemma (Achar K. Matherne) There is a bijection Bpwq : tb ij ÿ b ij γ ij wu ÐÑ Ppwq. b b b ` b b b ` b b ` b ` b b b b b b b
17 / 7 Running Example (A ) Let w p,, q. C C C rank C C C rank C C C rank rank C C C
18 / 7 Partial orders on Ppwq If Y P Ppwq, we write O Y for the corresponding Gpwq-orbit in Epwq. Let Y, Y P Ppwq.
19 / 7 Partial orders on Ppwq If Y P Ppwq, we write O Y for the corresponding Gpwq-orbit in Epwq. Let Y, Y P Ppwq. Geometric partial order on Ppwq: Y ď g Y if and only if O Y Ă O Y. Combinatorial partial order on Ppwq: Y ď c Y if for all i and j, jÿ jÿ y ik ě k k y ik.
20 / 7 Partial orders on Ppwq If Y P Ppwq, we write O Y for the corresponding Gpwq-orbit in Epwq. Let Y, Y P Ppwq. Geometric partial order on Ppwq: Y ď g Y if and only if O Y Ă O Y. Combinatorial partial order on Ppwq: Y ď c Y if for all i and j, jÿ jÿ y ik ě k k y ik. Theorem (Achar K. Matherne) The geometric and combinatorial partial orders coincide.
21 Running Example (A ) / 7
22 / 7
23 4 / 7 Some observations from the combinatorics Let Y P Ppwq. Denote by MpYq a representation in the orbit O Y.
24 4 / 7 Some observations from the combinatorics Let Y P Ppwq. Denote by MpYq a representation in the orbit O Y. Theorem (Achar K. Matherne) ÿ dim O Y y ij y ik ` ďiďn ďjăkďn i` ÿ ďiďn ďjăkďn i` y i`,j y ik.
25 4 / 7 Some observations from the combinatorics Let Y P Ppwq. Denote by MpYq a representation in the orbit O Y. Theorem (Achar K. Matherne) ÿ dim O Y y ij y ik ` ďiďn ďjăkďn i` ÿ ďiďn ďjăkďn i` y i`,j y ik. MpYq is an injective object in ReppQ n q if and only if Y is constant along ladders. MpYq is a projective object in ReppQ n q if and only if Y has nonzero entries only in the last ladder.
26 5 / 7 Outline Quiver representation varieties Some combinatorics Fourier Sato transform Combinatorial Fourier transform
27 6 / 7 Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology)
28 Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology) ÐÑ tgpwq-orbits in Epwqu tsimple perverse sheaves on Epwqu. O Y ÞÝÑ ICpO Y q 6 / 7
29 Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology) ÐÑ tgpwq-orbits in Epwqu tsimple perverse sheaves on Epwqu. O Y ÞÝÑ ICpO Y q So, get a bijection: ÐÑ Ppwq tsimple perverse sheaves on Epwqu. Y ÞÝÑ ICpO Y q 6 / 7
30 7 / 7 Fourier Sato transform Can we give a combinatorial description of the Fourier Sato transform: D b Gpwq pepwqq for simple perverse sheaves F? T ÝÑ D b Gpw q pepw qq F ÞÝÑ F ^rdim Epwqs
31 8 / 7 Running example Ppwq Ppw q
32 9 / 7 Some properties and applications of the Fourier transform Properties: t-exact for the perverse t-structure and sends simples to simples. equivalence of categories almost an involution
33 9 / 7 Some properties and applications of the Fourier transform Properties: t-exact for the perverse t-structure and sends simples to simples. equivalence of categories almost an involution Applications: character formula for quantum loop algebras uses Fourier transform on graded quiver varieties (Nakajima) monoidal categorification of certain cluster algebras (Nakajima)
34 / 7 Outline Quiver representation varieties Some combinatorics Fourier Sato transform Combinatorial Fourier transform
35 / 7 Combinatorial Fourier transform Theorem (Achar K. Matherne) There is a bijection T Ppwq ÝÑ Ppw q defined inductively by T Y y,n τ y,n n τ y,n y,n n τ y n, y n, y n, TpY q where Tpaq a.
36 / 7 Sliding at j j th chute Define τ j : Ppwq Ñ Ppw ` e `... ` e j q by: Add as far down the j th chute as possible, drawing an impassable vertical line there. Repeat for chutes j,..., not crossing lines.
37 / 7 Example of T T T T τ τ
38 4 / 7 Running example Ppwq Ppw q
39 5 / 7 Main theorem Theorem (Achar K. Matherne) The bijection T : Ppwq Ñ Ppw q determines T : D b Gpwq pepwqq Ñ Db Gpw q pepw qq for simple perverse sheaves; that is, TpICpO Y qq ICpO TpYq q.
40 Outline of the proof Geometric Fourier transform Combinatorial Fourier transform
41 Outline of the proof Geometric Fourier transform Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
42 Outline of the proof Geometric Unique dense Fourier transform open orbit in the commuting variety Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
43 Outline of the proof Unique dense open orbit in the commuting variety Pyasetskiĭ Evens Mirković Geometric Fourier transform Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
44 Outline of the proof Unique dense open orbit in the commuting variety Pyasetskiĭ Evens Mirković A K M Geometric Fourier transform Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
45 Outline of the proof Unique dense open orbit in the commuting variety Pyasetskiĭ Evens Mirković A K M Geometric Fourier transform Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
46 Outline of the proof Unique dense open orbit in the commuting variety Pyasetskiĭ Evens Mirković A K M Geometric Fourier transform Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q
47 Outline of the proof Unique dense open orbit in the commuting variety Pyasetskiĭ Evens Mirković A K M Geometric Fourier transform Knight-Zelevinsky Combinatorial Fourier transform Multisegment duality Bpwq Ñ Bpw q A K M Inverse combinatorial Fourier transform 6 / 7
48 7 / 7
49 Thanks! 7 / 7
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