Electroids and Positroids 2014 Circular planar electrical networks We consider planar weighted graphs Γ embedded into the disk, with distinguished boundary vertices 1̄, 2̄, . . . , n̄ on the the boundary of the disk. 4̄ 3̄ 5.3 2̄ 5̄ 2 1̄ Kenyon-Wilson groves A grove F in Γ is a subforest such that every interior vertex is connected to some boundary vertex. 4̄ 3̄ 5̄ 2̄ 1̄ Boundary partitions The boundary partition σ(F ) of a grove F is the non-crossing partition whose parts are boundary vertices belonging to the same component of F . 4̄ 3̄ 5̄ 2̄ 1̄ σ(F ) = {2̄, 3̄, 4̄|1̄, 5̄} Planarity =⇒ non-crossing. Grove measurements Let N C n denote the set of non-crossing partitions on {1̄, . . . , n̄}. For σ ∈ N C n , define the grove measurement X Lσ (Γ) = wt(F ) σ(F )=σ where the weight of a grove F is the product of weights of edges belonging to F . Γ 7−→ L(Γ) = (Lσ (Γ))σ∈N C n ∈ PN C n . Grove measurements Let N C n denote the set of non-crossing partitions on {1̄, . . . , n̄}. For σ ∈ N C n , define the grove measurement X Lσ (Γ) = wt(F ) σ(F )=σ where the weight of a grove F is the product of weights of edges belonging to F . Γ 7−→ L(Γ) = (Lσ (Γ))σ∈N C n ∈ PN C n . Two graphs are electrically equivalent if you cannot distinguish them by electrical experiments made at the boundary. Theorem (Kirchhoff, Kenyon-Wilson) Γ and Γ0 are electrically equivalent if and only if Lσ (Γ) = Lσ (Γ0 ). Y − ∆-transformation 1̄ 1̄ C a b B 2̄ 2̄ c 3̄ A 3̄ Γ L1̄|2̄|3̄| = a + b + c, L1̄3̄|2̄ = ac, L1̄2̄|3̄ = ab, Γ0 L1̄|2̄3̄ = bc, L1̄2̄3̄ = abc and L01̄|2̄|3̄| = 1, L01̄3̄|2̄ = B, L01̄2̄|3̄ = C , L01̄|2̄3̄ = A, L01̄2̄3̄ = AB + BC + AC . Y − ∆-transformation 1̄ 1̄ C a b B 2̄ 2̄ c 3̄ A 3̄ Γ L1̄|2̄|3̄| = a + b + c, L1̄3̄|2̄ = ac, L1̄2̄|3̄ = ab, Γ0 L1̄|2̄3̄ = bc, L1̄2̄3̄ = abc and L01̄|2̄|3̄| = 1, L01̄3̄|2̄ = B, A= L01̄2̄|3̄ = C , L01̄|2̄3̄ = A, L01̄2̄3̄ = AB + BC + AC . bc ac ab , B= , C= , a+b+c a+b+c a+b+c Compactification Let us use nonnegative edge weights. The image of the map Γ 7→ L(Γ) is not compact. We let En ⊂ PN C n denote the closure of the image, called the compactified space of circular planar electrical networks. Roughly speaking, a point L ∈ En is represented by an electrical network where some of the boundary points have been glued together, in a planar way. (This is a good compactification. e.g. it’s quite different from the one-point compactification.) Electroids The electroid E(Γ) of Γ ∈ En is the set E(Γ) = {σ | Lσ (Γ) 6= 0} ⊂ N C n . These are non-crossing partitions for which there exist groves inducing such a partition. (We think of this set as something like a matroid.) Question What are all possible electroids? Electroids The electroid E(Γ) of Γ ∈ En is the set E(Γ) = {σ | Lσ (Γ) 6= 0} ⊂ N C n . These are non-crossing partitions for which there exist groves inducing such a partition. (We think of this set as something like a matroid.) Question What are all possible electroids? We have the electroid stratification G En = EE . E Medial graph t6 t7 4̄ t6 t8 t7 t8 3̄ t9 t5 t9 t5 5̄ t4 t4 t10 2̄ t3 t1 t2 1̄ t10 t3 t1 t2 (Critical) Planar graph −→ (Reduced) Medial graph −→ Matching on [2n] Uncrossing poset for matchings The set Pn of matchings is a graded poset with rank function c(τ ) = number of crossings. (Studied by Alman-Lian-Tran, by Kenyon, by Huang-Wen-Zie, by Kim-Lee...) Noncrossing partitions to noncrossing matchings For σ ∈ N C n , we have a natural τ (σ) ∈ Pn which is a noncrossing matching. 2̄ 4 3̄ 5 3 1̄ 6 2 7 1 8 4̄ 12 6̄ 5̄ 9 11 10 σ = (1̄, 4̄, 6̄|2̄, 3̄|5̄) τ (σ) = {(1, 12), (2, 7), (3, 6), (4, 5), (8, 11), (9, 10)} Matchings classify electroid strata Theorem (L.) There is a bijection τ ↔ E(τ ) between matchings and electroids, given by E(τ ) = {σ|τ (σ) ≤ τ } so that we have En = G Eτ τ ∈Pn where c(τ ) Eτ ' R>0 and Eτ = G Eτ 0 . τ 0 ≤τ This result depends on a large theory developed by Curtis-Ingerman-Morrow, and Colin de Verdière, Gitler, and Vertigan. Planar bipartite graphs Assumption: boundary vertices of N are always 1-valent. 3 2 4 1 5 8 6 7 Dimer configurations in planar bipartite graphs Rule: Π must use all interior vertices; boundary vertices may or may not be used. Boundary subset I (Π) = black boundary vertices used union white boundary vertices not used. Boundary measurements |I (Π)| = k(N) for some k(N) that depends only on the planar bipartite graph N. Define the boundary measurement X ∆I (N) = wt(Π). I (Π)=I ) The map [n] N 7−→ M(N) = (∆I (N))I ∈([n]) ∈ P( k ) k is a version of the dimer partition function. TNN Grassmannian Theorem (Kuo, Postnikov) 1 2 [n] The point M(N) lies in the Grassmannian Gr(k, n) ⊂ P( k ) . The image is the totally nonnegative Grassmannian Gr(k, n)≥0 consisting of points represented by nonnegative real coordinates. Analogies! Planar bipartite graph N Dimer configurations in N [n] Plücker space P( k ) Electrical network Γ Groves in Γ Non-crossing partition space PN C n [n] k Grassmannian Gr(k, n) ⊂ P( ) Alternating strand diagram Bounded affine permutations f Bruhat order Subsets I ∈ [n] k Positroids M ⊂ [n] k Grassmann necklaces GL(n)-action cluster algebra Zariski closure of En ⊂ PN C n Medial graph Medial pairings τ “Uncrossing” partial order Non-crossing partitions σ ∈ N C n Electroids E ⊂ N C n Partition necklaces Electrical Lie group action Laurent phenomenon algebrfa [Oh, Knutson-L.-Speyer, Thurston, Goncharov-Kenyon, Talaska, Postnikov-Speyer-Williams, ...] [Curtis-Ingerman-Morrow, Colin de Verdière-Gitler-Vertigan, Kenyon-Wilson, L.-Pylyavskyy, ...] Kenyon-Propp-Wilson’s generalized Temperley trick Γ 7−→ N(Γ) Γ N(Γ) Everything is compatible Theorem (L.) The map Γ 7→ N(Γ) induces an injection ι : En → Gr(n − 1, 2n)≥0 given by ∆I (N(Γ)) = X aI σ Lσ(Γ) σ compatible with all the analogies. The matrix (aI σ ) is a 0-1 matrix indexed by [n] k × N C n which seems to be very interesting. It captures the algebraic structure of the generalized Temperley trick. Posets The closure partial order for planar bipartite case is given by a subposet Bound(k, n) ⊂ S̃n of the affine symmetric group equipped with Bruhat order. Theorem (L.) We have a map ι : Pn ,→ Bound(n − 1, 2n) expressing Pn as an induced subposet. Electroids and positroids G En = E electroid EE and Gr(k, n)≥0 G Π̊M . M positroid Here a positroid M is the matroid of a point in Gr(k, n)≥0 . Theorem (L.) The map ι : Pn ,→ Bound(n − 1, 2n) induces a map E 7→ M(E) such that ι(EE ) = ι(En ) ∩ Π̊M(E) . The end Thanks!

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