A combinatorial Fourier transform for quiver representation - PowerPoint PPT Presentation

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 28, 2018 Maurice Auslander Distinguished Lectures and International Conference

Goal Consider the quiver ‚ Ý Ñ ‚ Ý Ñ ¨ ¨ ¨ Ý Ñ ‚ . Notation: E p w q - space of representations for dimension vector 1 w “ p w 1 , . . . , w n q G p w q “ GL p w 1 q ˆ ¨ ¨ ¨ ˆ GL p w n q 2 w ˚ “ p w n , . . . , w 1 q - the reverse dimension vector 3 1 / 27

Goal Consider the quiver ‚ Ý Ñ ‚ Ý Ñ ¨ ¨ ¨ Ý Ñ ‚ . Notation: E p w q - space of representations for dimension vector 1 w “ p w 1 , . . . , w n q G p w q “ GL p w 1 q ˆ ¨ ¨ ¨ ˆ GL p w n q 2 w ˚ “ p w n , . . . , w 1 q - the reverse dimension vector 3 Can we give a combinatorial description of the Fourier–Sato transform: T G p w ˚ q p E p w ˚ qq D b D b G p w q p E p w qq Ý Ñ q 2 ! q ˚ F ÞÝ Ñ 1 p F qr dim E p w qs for simple perverse sheaves F ? 1 / 27

Outline Quiver representation varieties Some combinatorics Fourier–Sato transform Combinatorial Fourier transform 2 / 27

Outline Quiver representation varieties Some combinatorics Fourier–Sato transform Combinatorial Fourier transform 3 / 27

Quiver representations Consider the type A n equioriented quiver Q n “ ‚ Ý Ñ ‚ Ý Ñ ¨ ¨ ¨ Ý Ñ ‚ . A quiver representation is: A finite-dimensional x n ´ 1 x 1 x 2 C -vector space M i for M 1 M 2 ¨ ¨ ¨ M n each vertex. A linear map x i for each arrow. 4 / 27

Quiver representations Consider the type A n equioriented quiver Q n “ ‚ Ý Ñ ‚ Ý Ñ ¨ ¨ ¨ Ý Ñ ‚ . A quiver representation is: A finite-dimensional x n ´ 1 x 1 x 2 C -vector space M i for M 1 M 2 ¨ ¨ ¨ M n each vertex. A linear map x i for each arrow. Rep p Q n q - abelian category of finite-dimensional complex representations of Q n 4 / 27

Quiver representation varieties Fix a dimension vector w “ p w 1 , w 2 , . . . , w n q . A quiver representation variety E p w q is the space of all quiver representations for a fixed dimension vector w . Note that E p w q is an affine variety: E p w q » A w 1 w 2 ` w 2 w 3 `¨¨¨` w n ´ 1 w n . 5 / 27

Quiver representation varieties Fix a dimension vector w “ p w 1 , w 2 , . . . , w n q . A quiver representation variety E p w q is the space of all quiver representations for a fixed dimension vector w . Note that E p w q is an affine variety: E p w q » A w 1 w 2 ` w 2 w 3 `¨¨¨` w n ´ 1 w n . G p w q “ GL p w 1 q ˆ ¨ ¨ ¨ ˆ GL p w n q acts on E p w q by p g 1 , . . . , g n q ¨ p x 1 , . . . , x n ´ 1 q “ p g 2 x 1 g ´ 1 1 , . . . , g n x n ´ 1 g ´ 1 n ´ 1 q giving it a stratification by orbits. Note that two points x , y P E p w q are in the same G p w q -orbit if and only if they are isomorphic objects of Rep p Q n q . 5 / 27

Classifying the orbits Theorem (Gabriel’s Theorem) There is a bijection t indec. objects in Rep p Q n qu{„ 1 ´ 1 Ð Ñ t pos. roots for A n root system u . 6 / 27

Classifying the orbits Theorem (Gabriel’s Theorem) There is a bijection t indec. objects in Rep p Q n qu{„ 1 ´ 1 Ð Ñ t pos. roots for A n root system u . To an indecomposable representation Ñ ¨ ¨ ¨ id id R ij “ 0 Ñ ¨ ¨ ¨ Ñ 0 Ñ C Ý Ý Ñ vertex j Ñ 0 Ñ ¨ ¨ ¨ Ñ 0 . C vertex i we associate its dimension vector, the positive root γ ij “ p 0 , . . . , 0 , position j , 0 , . . . , 0 q . 1 1 position i , . . . , 6 / 27

Classifying the orbits Theorem (Gabriel’s Theorem) There is a bijection t indec. objects in Rep p Q n qu{„ 1 ´ 1 Ð Ñ t pos. roots for A n root system u . To an indecomposable representation Ñ ¨ ¨ ¨ id id R ij “ 0 Ñ ¨ ¨ ¨ Ñ 0 Ñ C Ý Ý Ñ vertex j Ñ 0 Ñ ¨ ¨ ¨ Ñ 0 . C vertex i we associate its dimension vector, the positive root γ ij “ p 0 , . . . , 0 , position j , 0 , . . . , 0 q . 1 1 position i , . . . , Corollary There is a bijection t G p w q -orbits in E p w qu 1 ´ 1 ÿ Ð Ñ B p w q : “ t b ij | b ij γ ij “ w u . 6 / 27

Outline Quiver representation varieties Some combinatorics Fourier–Sato transform Combinatorial Fourier transform 7 / 27

Triangular arrays ladder Define the set P p w q of triangular arrays of nonnegative integers such that: @ j , the entries in the j th chute sum to w j . Ladders are weakly decreasing. chute 8 / 27

Triangular arrays ladder Define the set P p w q of triangular arrays of nonnegative integers such that: @ j , the entries in the j th chute sum to w j . Ladders are weakly decreasing. chute 1 0 For w “ p 1 , 1 , 2 q , P P p w q 0 1 0 2 We will write y ij for the entry in the i th chute and j th column. 8 / 27

Classifying the orbits combinatorially Lemma (Achar–K.–Matherne) There is a bijection b ij γ ij “ w u 1 ´ 1 ÿ B p w q : “ t b ij | Ð Ñ P p w q . b 11 b 11 ‚ ‚ b 12 b 12 ‚ ‚ b 22 b 12 ` b 22 b 13 b 13 ‚ ‚ ‚ ‚ b 23 b 13 ` b 23 ‚ ‚ b 33 b 13 ` b 23 ` b 33 ‚ ‚ 9 / 27

Running Example ( A 3 ) Let w “ p 1 , 1 , 2 q . 1 0 0 0 C 2 C C 1 0 0 2 ˜ ¸ 0 0 0 rank 1 1 C 2 C C 1 0 0 2 1 0 rank 1 0 C C C 2 0 1 0 2 0 rank 1 rank 1 0 C 2 C C 0 1 1 2 10 / 27

Partial orders on P p w q If Y P P p w q , we write O Y for the corresponding G p w q -orbit in E p w q . Let Y , Y 1 P P p w q . 11 / 27

Partial orders on P p w q If Y P P p w q , we write O Y for the corresponding G p w q -orbit in E p w q . Let Y , Y 1 P P p w q . Geometric partial order on P p w q : Y ď g Y 1 O Y Ă O Y 1 . if and only if Combinatorial partial order on P p w q : j j Y ď c Y 1 ÿ ÿ y 1 y ik ě if for all i and j , ik . k “ 1 k “ 1 11 / 27

Partial orders on P p w q If Y P P p w q , we write O Y for the corresponding G p w q -orbit in E p w q . Let Y , Y 1 P P p w q . Geometric partial order on P p w q : Y ď g Y 1 O Y Ă O Y 1 . if and only if Combinatorial partial order on P p w q : j j Y ď c Y 1 ÿ ÿ y 1 y ik ě if for all i and j , ik . k “ 1 k “ 1 Theorem (Achar–K.–Matherne) The geometric and combinatorial partial orders coincide. 11 / 27

Running Example ( A 3 ) 0 0 0 1 1 2 1 0 0 1 0 2 0 1 1 0 0 2 1 0 1 0 0 2 12 / 27

0 0 18 0 3 3 3 0 1 1 0 17 1 2 2 0 3 2 3 3 1 0 16 1 2 2 3 1 15 1 1 2 1 3 0 2 14 2 0 2 1 1 0 3 1 3 3 1 2 13 1 0 2 1 1 1 2 1 3 3 1 2 11 2 1 2 1 0 1 2 0 3 3 2 10 0 2 1 1 3 0 2 3 9 3 1 0 3 0 0 2 1 0 0 3 0 3 3 3 1 3 8 2 0 3 0 0 1 2 0 3 3 2 3 5 1 0 3 0 0 2 1 0 3 3 3 0 0 3 0 0 3 13 / 27

Some observations from the combinatorics Let Y P P p w q . Denote by M p Y q a representation in the orbit O Y . 14 / 27

Some observations from the combinatorics Let Y P P p w q . Denote by M p Y q a representation in the orbit O Y . Theorem (Achar–K.–Matherne) ÿ ÿ dim O Y “ y ij y ik ` y i ` 1 , j y ik . 1 1 ď i ď n ´ 1 1 ď i ď n ´ 1 1 ď j ă k ď n ´ i ` 1 1 ď j ă k ď n ´ i ` 1 14 / 27

Some observations from the combinatorics Let Y P P p w q . Denote by M p Y q a representation in the orbit O Y . Theorem (Achar–K.–Matherne) ÿ ÿ dim O Y “ y ij y ik ` y i ` 1 , j y ik . 1 1 ď i ď n ´ 1 1 ď i ď n ´ 1 1 ď j ă k ď n ´ i ` 1 1 ď j ă k ď n ´ i ` 1 M p Y q is an injective object in Rep p Q n q if and only if Y is 2 constant along ladders. M p Y q is a projective object in Rep p Q n q if and only if Y has 3 nonzero entries only in the last ladder. 14 / 27

Outline Quiver representation varieties Some combinatorics Fourier–Sato transform Combinatorial Fourier transform 15 / 27

Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology) 16 / 27

Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology) 1 ´ 1 t G p w q -orbits in E p w qu Ð Ñ t simple perverse sheaves on E p w qu . O Y ÞÝ Ñ IC p O Y q 16 / 27

Some basics about perverse sheaves Perverse sheaves - complexes of sheaves that encode information about the singularities of a space (intersection cohomology) 1 ´ 1 t G p w q -orbits in E p w qu Ð Ñ t simple perverse sheaves on E p w qu . O Y ÞÝ Ñ IC p O Y q So, get a bijection: 1 ´ 1 P p w q Ð Ñ t simple perverse sheaves on E p w qu . Y ÞÝ Ñ IC p O Y q 16 / 27

Fourier–Sato transform Can we give a combinatorial description of the Fourier–Sato transform: T D b D b G p w ˚ q p E p w ˚ qq G p w q p E p w qq Ý Ñ F ^ r dim E p w qs ÞÝ Ñ F for simple perverse sheaves F ? 17 / 27

Running example P p w ˚ q P p w q 1 2 0 0 1 0 0 1 0 0 2 1 0 1 1 1 1 0 0 1 0 0 2 1 1 2 0 0 0 1 0 0 1 0 2 1 0 1 0 0 0 1 1 0 1 1 2 1 18 / 27

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 19 / 27