Categories of noncrossing partitions

Transcription

1 Categories of noncrossing partitions Kiyoshi Igusa, Brandeis University KIAS, Dec 15, 214

2 Topology of categories The classifying space of a small category C is a union of simplices k : BC = X X 1 X k k / BC has one vertex for every object, one edge for every morphism, one triangle 2 for every commuting triangle X = K(G, 1) means π 1 X = G and universal covering space is contractible: X. 1

3 Two categories of noncrossing partitions 1. N P: Objects: Noncrossing partitions Morphisms: (unions of) Binary trees Constructs K(π, 1) for picture group of A n. 2. HK: Hubery-Krause category of noncrossing partitions. Q: What are they and how are they related? 2

4 A: There is a third category related to both. N P IT HK 3. IT : Cluster morphism category (This category has clusters as morphisms.)

5 Co-authors I-Orr-Todorov-Weyman We introduce the picture group G(Q) for Dynkin quivers Q and we compute its cohomology in type A n. I-Todorov We construct a category G(Q) for any modulated quiver Q so that, when Q is Dynkin, BG(Q) = K(G(Q), 1). I In the special case of Q = A n with straight orientation, I gave a combinatorial construction of the category G(A n ). 3

6 1st category of noncrossing partitions N P Objects: Noncrossing partitions of the ordered set. [n] = {, 1, 2,, n} Example of noncrossing partition: 3 This is a partition of [3] with three parts: {, 2}, {1}, {3} Here {, 2}, {1, 3} is forbidden: This is crossing. 3 4

7 All five objects of N P[2]: S = Ω = 5

8 Morphisms in N P[2] Morphisms are given by vertical refinement : When you pull {, 1}, {2} apart, you have to say which is on top. So, there are two morphisms: {, 1, 2} {, 1}, {2} But, when {1} is pulled out, it must be underneath. So, there is only one morphism: {, 1, 2} {, 2}, {1} 6

9 Irreducible morphisms in N P[2]: S = Ω = 7

10 Two of the morphisms S Ω: S = Ω = 8

11 Green is green Green edge i j is c-vector e i+1 + e i e j [ ] 1 2 C = = C Corresponding cluster are negative rows of C 1 E 1 = [ ] [ ] = So, cluster is P 1 [1], S 2 [ ] 1 1 9

12 All five morphisms S Ω These are the five binary tree structures on the ordered set {, 1, 2}. 1

13 Geometric realization of N P[n] THEOREM The geometric realization of the category N P[n] is BN P[n] = K(G(A n ), 1) where G(A n ) is the picture group of type A n (straight orientation). This group has generators: x ij, i < j n and relations: [x ij, x jk ] = x ik and [x ij, x kl ] = 1 if {i, j}, {k, l} are noncrossing (i < j < k < l, k < i < j < l or...) [x, y] := y 1 xyx 1 11

14 noncrossing partitions = exceptional sequences. DEFINITION Suppose that A = mod-λ for some fin dim hereditary algebra Λ. An object E in A is exceptional if it is indecomposable without self-extensions. A sequence of exceptional objects: E 1, E 2,, E k is called exceptional if Hom(E j, E i ) = = Ext(E j, E i ) when j > i. The sequence is complete if k is maximal. (i.e, k = n, the number of vertices in quiver.) 12

15 Linearization of exceptional sequences Exceptional objects E are uniquely determined by their dimension vectors dim E. A sequence of such vectors β 1,, β n is an exceptional sequence iff βj, β i = for all i < j. Hubery and Krause define a category abstractly and show that their category can be described as follows. 13

16 Hubery-Krause category: HK For hereditary algebra Λ, let K (Λ) = Z n with the Euler pairing:, so that dimm, dimn = dim Hom(M, N) dim Ext(M, N) The objects of HK are pairs (Γ, E) where Γ = K (Λ) and E = (β 1,, β n ) so that β i Γ are the dimension vectors of a complete exceptional sequence in mod-λ. For example: (Z 2, (2e 1, 4e 2 )) 14

17 Morphisms in HK A morphism (Γ 1, E 1 ) (Γ 2, E 2 ) is an isometric embedding Γ 1 Γ 2 so that the image of E 1 in Γ 2 is, up to sign, an exceptional sequence. Two examples: Multiplication by 1 is an automorphism of any (Γ, E), Aut(K (KA 2 )) = Z/6 = {id, φ, φ 2, φ 3, φ 4, φ 5 } 15

18 Linearization functor E(K) is the category whose objects are hereditary abelian categories A over K and whose morphisms are exact embeddings A B whose images are extension closed. Take functor: K : E(K) HK sending a category A to its root space and the exceptional sequence given by the dimension vectors of its simple objects in admissible order. 16

19 Example: A 2 β γ α α = dim S 1, β = dim S 2, γ = dim P 2 K (KA 2 ). (S 2, S 1 ) is an exceptional sequence. So K (KA 2 ) = (Z 2, (β, α)) HK 17

20 Cluster morphism category IT Given an exact extension closed embedding j : A B in E(K) There exists a partial cluster tilting object T in the cluster category C B of B so that A = T. But T is not unique. Adding a choice of T gives a larger category Ẽ(K) with the same objects as E(K) but with morphisms (j, T ) : A B DEFINITION IT K = Ẽ(K)op 18

21 Composition of C (A B C, ib = T ) (i,t ) B (j,t ) A (j, T )(i, T ) = (i j, τ 1 + (τ +(T ), τ + (T )) τ + is the bijection from ordered clusters to signed exceptional sequences given by τ + (v 1,, v k ) = (w 1,, w k ) (1) w j v j is a linear combination of v i for i > j. (2) v i, w j = for i > j. 19

22 Example: A 2 Let A = mod-ka 2 where A 2 :. There is a unique morphism j : A in E(K). But there are five morphisms (j, T ) : A in Ẽ(K). Let F : Ẽ(K) E(K) be the forgetful functor. 2

23 Comma category Given a functor F : C D and an object X in D, the comma category F X is the category with objects pairs (Y, f) where Y C and f : F Y X is a morphism in D. A morphism (Y, f) (Z, g) is a morphism h : Y Z in C so that f = g F h : F Y F Z X. THEOREM The opposite category N P[n] op is isomorphic to a full subcategory of the comma category K F K (KA n ). 21

24 Example: K F K (KA 2 ) (KA 1, α) (KA 1, γ) (KA 1, β) (KA 1, α) (KA 1, γ) (KA 1, β) (KA 2, id) (KA 2, φ) (KA 2, φ 2 ) (KA 2, φ 3 ) (KA 2, φ 4 ) (KA 2, φ 5 ) N P[2] op is the full subcategory in blue. 22

25 Irreducible morphisms in N P[2] op : 23

26 As full subcategory of K K (KA 2 ): (KA 2, id) P 2 [1] P 2 S 2 P 1 [1] P 1 (KA 1, α ) (KA 1, γ ) (KA 1, β ) P 1 [1] P 1 P 2 (, ) S 2 S 2 [1] 24

27 Signed exceptional sequences There are n!c(n + 1) = = 1 signed exceptional sequence: (±P 1, ±P 2 ), (±P 2, S 2 ), (±S 2, ±P 1 ) Each corresponds to a total ordering of the five clusters under the twist map τ +. E.g, for the ordered cluster (P 1 [1], S 2 ): τ + ( α, β) = ( α β, β) = ( γ, β) gives signed exceptional sequence ( P 2, S 2 ). 25

28 Repeating the formula: (1) w j v j is a linear comb. of v i for i > j. (2) v i, w j = for i > j. Summary of KA 2 example on last slide:

29 Kβ KA 2 S 2 [1] P 1 Kα KA 2 S 2 1 P 1 [1] P 1 2 P 2 [1] 1 P S 2 [1] 2 P 2 Kα KA 2 KA 2 S 2 P 1 [1] 1 1 Kβ KA 2 2 P 1 [1] 2 P 2 Kγ KA 2 P 2 [1] S 2 26