Categorification of perfect matchings Alastair King, Bath work in - PowerPoint PPT Presentation

Categorification of perfect matchings Alastair King, Bath work in progress with I. Canakci & M. Pressland [CKP] and with B.T. Jensen & X. Su [JKS3] Ann Arbor, Aug 2020

Toy model: perfect matchings on a circle On a circular graph ( C 0 , C 1 ) with n vertices, a (perfect) matching is a choice of orientation for each edge. A matching is specified by its label J = ( J • , J ◦ ) with J • ⊆ C 1 being the anti-clockwise edges and J ◦ the clockwise edges. A matching has chirality (helicity) k = ( k • , k ◦ ) ∈ N {• , ◦} , where k • = # J • and k ◦ = # J ◦ , so that k • + k ◦ = n . A new matching of the same chirality is obtained by flipping a source to a sink or vice versa. Chirality is the only invariant of flipping.

Cochains on a closed string Fatten the circle to a quiver with faces Q = ( Q 0 , Q 1 , Q 2 ), i.e. a 2-complex s.t. ∂ f is an oriented cycle, for all f ∈ Q 2 . For any such Q , a matching is a function µ ∈ N Q 1 s.t d µ = 1, on all faces f ∈ Q 2 , in ptic, in lattice M = { µ ∈ Z Q 1 : d µ ∈ c ( Z ) } . c d d Z Q 0 Z Q 1 Z Q 2 Z c i deg c d Z Q 0 Z M Z d is coboundary, c is constants, i is inclusion, deg is restriction of d. Now flip is adding/subtracting d ( s i ) for s i basic in Z Q 0 . Denote by M + = M ∩ N Q 1 the cone of multi-matchings . For string, rank M = n + 1 and M + is the cone on a unit n -cube.

Another example: double (or r -fold) dimers • 0 1 • 0 1 • 1 • 1 0 0 • 0 0 • 2 0 • deg µ = 2 Σ Question: why is wt(Σ) = 2 ? Guess: because χ ( P 1 ) = 2 and some appropriate quiver Grassmannian is P 1 .

Chirality revisited Let H 1 = M / d � Z Q 0 � and h : M → H 1 be the quotient. Note: deg = dg ◦ h for dg: H 1 → Z and dg − 1 (0) = H 1 ( Q ). For the closed string: H 1 ∼ = { ( h • , h ◦ ) ∈ Z {• , ◦} : h • + h ◦ ∈ n Z } . Explicitly, write Q 1 = Q • 1 ∪ Q ◦ 1 and, for ∗ ∈ {• , ◦} , define two closed 1-cycles a ∗ = � 1 a . Then h ∗ ( µ ) = µ ( a ∗ ) = � 1 µ ( a ) a ∈ Q ∗ a ∈ Q ∗ and deg( µ ) = 1 n ( h • ( µ ) + h ◦ ( µ )). Fixed chirality k = ( k • , k ◦ ) and define M k = h − 1 � k � , deg c d → Z Q 0 then rank M k = n and Z − − → M k − → Z is exact. Fact: M k ∼ = the sublattices of the weight lattice of GL ( n ) that n n grade the homogeneous coordinate rings C [ˆ k • ] and C [ˆ Gr Gr k ◦ ] and deg gives the usual degree (Pl¨ ucker coords ∆ J have deg 1).

Categorification of matchings Let Z = C [[ t ]] and Q = ( Q 0 , Q 1 , Q 2 ) be a quiver with faces s.t. (i) the associated topological space | Q | is connected (ii) every arrow a ∈ Q 1 is in the boundary of some face f ∈ Q 2 (iii) Q admits virtual matchings, i.e. deg M → Z is surjective. The category of matrix factorizations MF( Q ; t ) consists of representations M , φ of Q s.t. (i) each M i : i ∈ Q 0 is a f.g. free Z -module of rank r := rk M (ii) the maps φ a : M ta → M ha satisfy φ a s ◦ · · · ◦ φ a 1 = t , when a s + · · · + a 1 = ∂ f is the boundary of a face f ∈ Q 2 . There is a natural invariant ν : K(MF( Q ; t )) → M : [ M ] �→ ν M given by ν M ( a ) = dim coker φ a and such that deg ν M = rk M . For each (deg 1) matching µ ∈ M + , there is a rk 1 rep’n M ( µ ) , φ with ν M ( µ ) = µ , given by M ( µ ) i = Z and φ a = t µ ( a ) . A flip corresponds to simple extension/shortening of the rep’n.

The closed string categorified y 5 y 1 x 5 x 1 For x j ∈ Q ◦ 1 , y j ∈ Q • 1 and fixed k = ( k • , k ◦ ), x 4 x 2 y 4 y 2 define Z -algebra C k as path algebra ZQ mod x 3 xy = t = yx , x k • = y k ◦ ( ⇒ x n = t k ◦ , y n = t k • ). y 3 The category CM C k of f.g. C k -modules free over Z is a full exact ∼ = subcategory of MF( Q ; t ) and ν : K(CM C k ) − → M k ⊆ Wt GL ( n ). CM C k contains M ( µ ) for all µ of chirality k ; these are all the rk 1 modules (up to isom). n Theorem [JKS1] There is a cluster character Ψ : CM C k → C [ˆ Gr k ] such that wt Ψ M = ν M . In particular, Ψ M ( J ) = ∆ J . Fact: C k is thin , i.e. each component e i Ce j , for i , j ∈ Q 0 , is a free Z -module of rank 1. Hence, for all i ∈ Q 0 , projectives P i = Ce i and (CM-)injectives I i = ( e i C ) ∨ := Hom Z ( e i C , Z ) are matching modules M ( J ), in fact, for J some cyclic interval.

Plabic graph G and dual quiver with faces Q x 5 5 • 1 y 1 • y 5 x 1 y 4 x 4 • • 4 x 2 y 3 • y 2 2 • x 3 3 Arrow directions follows strands. Boundary arrows and backwards paths in boundary faces are x = x ◦ or y = x • from orientation. Dimer algebra A has CM A = MF( Q ; t ), with K(CM A ) = M and all rk 1 modules are matching modules M ( µ ). All M ∈ CM A satisfy chirality relation x k • = y k ◦ for some fixed k .

Matchings on G are cochains on Q (Poincar´ e duality) c d d Z Q 0 Z Q 1 Z Q 2 Z c i deg c d Z Q 0 Z M Z Since | Q | is a disc, both horizontal sequences are exact and hence rank M = # Q 0 . There is a bdry value map d : M → M k (compatible with deg) that is dual to inclusion of chains: path x ∗ �→ � arrows in x ∗ . Explicitly d µ = J = ( J • , J ◦ ), where J ∗ = { j ∈ C 1 : µ ( x ∗ j ) = 1 } . The restriction ρ AC : CM A → CM C categorifies d .

Projectives and injectives Consistency ⇒ A is thin, so projectives P i = Ae i and injectives I i = ( e i A ) ∨ are matching modules M ( µ ) .. but which? [Mu-Sp] define bases of matchings m s / t : Z Q 0 → M , whose bdry values dm s / t give source (s) and target (t) labellings for G . j Prop [CKP] For all j ∈ Q 0 , we have [ P j ] = m s j and [ I j ] = m t j . ⋆ 5 5 • 1 (s • ) (t • ) 13 45 1 1 P ⋆ • • 15 34 3 • • 4 35 23 4 35 15 ⋆ 5 45 23 • I ⋆ 4 34 2 25 2 • • • 3 3 •

Boundary algebra, necklace and positroid � Bdry algebra B = eAe , where e = e i is bdry idempotent. i ∈C 0 ρ AB ρ BC Restriction ρ AC factorises as CM A − → CM B − → CM C , where ρ AB : X �→ eX and ρ BC is a fully faithful embedding. If i ∈ C 0 , then ρ AB : Ae i �→ Be i and ( e i A ) ∨ �→ ( e i B ) ∨ , so these are the matching modules M ( N i ) and M ( N ′ i ) for necklace N and reverse necklace N ′ . In other words, the necklace is B . Matching module M ( J ) is in CM B iff J is in the positroid.

Projective resolution ∼ Can view m = m s as the map K( P A ) = − → K(CM A ) induced by inclusion of category P A of projective A -modules, thus m − 1 comes from projective resolution. Thm [CKP] Each M = M ( µ ) in CM A has a projective resolution � � � Ae ta → Ae ha → Ae i a ∈ µ a �∈ µ i ∈ Q 0 int [Ma-Sc] define weights for internal arrows wt ( a ) ∈ Z Q 0 = K( P A ). Cor [CKP] For µ ∈ M , wt ( G ) wt ( µ ) � �� � � �� � � � � m − 1 ( µ ) = µ ( a )[ P ha ] + deg( µ ) [ P i ] − µ ( a ) wt ( a ) a ∈ Q 1 i ∈ Q 0 a ∈ Q 1 ext int int Prop [CKP] For all j ∈ Q 0 , we have m − 1 ( m s j ) = [ P j ].

Newton-Okounkov cone The restriction functor ρ AB : CM A → CM B : X �→ eA ⊗ A X has a right adjoint F : CM B → CM A : M �→ Hom B ( eA , M ). Here the counit η X : X → FeX is an embedding, i.e., if eX = M , then X ⊆ FM , so FM is the maximal module which restricts to M . For M ( J ) in CM B , FM ( J ) is a matching module M ( µ ) and µ is the minimal matching with d µ = J in the flip partial order. n k case, z [ FM ] is the leading Claim [JKS3] For M ∈ CM C , i.e. the ˆ Gr monomial (a la [Ri-Wi]) in network coords of the clus. char. Ψ M . See [JKS2] for Ψ M ( J ) = ∆ J , which is given in � z µ Z J = network coords by the dimer partition function: µ : d µ = J Expectation: (a) The set { [ FM ] ∈ M : M in CM C } is precisely the n integral points in the Ri-Wi Newton-Okounkov cone for ˆ Gr k . n (b) a basis of C [ˆ Gr k ] is given by { Ψ M : M general in CM C } . (c) Similar holds for positroid ˆ Gr π , by replacing C by B .

Background: network torus and Muller-Speyer twist M ⊇ deg − 1 (0) ∼ = Z Q 0 / c Z , which is the character lattice of the usual network torus, in monodromy coordinates. Thus M is the character lattice of a torus M ∗ that lifts the network ◦ torus to the positroid cone ˆ Gr π , using the dimer part. fun. � � ˆ ◦ � � M ∗ � z µ C Gr → C : ∆ J �→ Z J := π µ : d µ = J Note: for J ∈ N , i.e. ∆ J frozen, Z J is a monomial so invertible. ◦ ◦ Thm: [Mu-Sp] There is an automorphism τ : ˆ π → ˆ Gr Gr π s.t. ( − m − 1 ) ∗· � M ∗ � � ( C ∗ ) Q 0 � C C network cluster τ · � ˆ ◦ � � ˆ ◦ � C Gr C Gr π π

Application: Marsh-Scott twist Rearrange the m − 1 formula, when µ is a (deg 1) matching to get � [ P ha ] − m − 1 ( µ ) wt ( µ ) − wt ( G ) = ( ∗ ) a ∈ d µ n n Recall: [Ma-Sc] define a twist σ • : ˆ k ��� ˆ Gr Gr k and prove that := z − wt ( G ) � σ · • (∆ J ) = Z MS z wt ( µ ) in cluster coords J µ : d µ = J Define p • : M k → Z Q 0 : J �→ � a ∈ J • [ P ha ]. Then dmp • ([ M ]) = [ P • M ] ∈ M k for a projective cover P • M → M . = z p • ( J ) � z − m − 1 ( µ ) Z MS ( ∗ ) ⇒ J µ : d µ = J Thm [CKP] For M ( J ) in CM C , we have σ · • (∆ J ) = Ψ Ω • M ( J ) , where Ω • M is the syzygy ker P • M → M .