Electrical Networks, Hyperplane Arrangements and Matroids Bob Lutz Mathematical Sciences Research Institute December 2, 2019 Partially supported by NSF grants DMS-1401224, DMS-1701576 and DMS-1440140
Section I: Electrical Networks 1 0
What is an electrical network? ◮ A connected graph G = ( V , E ) (edges = wires) ◮ A set ∂ V ⊆ V of at least 2 boundary nodes ◮ A (real or complex) voltage v j at every boundary node j 2 3 1 1 0 4 0 Wheatstone bridge Star network
The Dirichlet problem ◮ Electrical current flows from higher voltages to lower voltages ◮ Consider the interior V ◦ = V \ ∂ V ◮ What are the voltages at the interior nodes? 1 0 1 0
The Dirichlet solution ◮ Every wire ij ∈ E has a (real or complex) conductance c ij ◮ Voltages and conductances satisfy � c ij ( v i − v j ) = 0 j ∼ i at every interior node i ∈ V ◦ ◮ “The current across every interior node is 0” ◮ Uniquely determines the interior voltages (for generic c )
Energies Definition The energy dissipated by an edge ij ∈ E is e ij = c ij ( v i − v j ) 2 Example Let ∆ = ( c 1 + c 3 + c 4 )( c 2 + c 3 + c 5 ) − c 2 3 . We have 1 c 1 c 2 c 1 ( c 3 c 5 + c 4 ( c 2 + c 3 + c 5 )) 2 e 1 c 2 ( c 3 c 4 + ( c 1 + c 3 + c 4 ) c 5 ) 2 c 3 e 2 = 1 c 3 ( c 2 c 4 − c 1 c 5 ) 2 e 3 ∆ 2 c 4 ( c 2 c 3 + c 1 ( c 2 + c 3 + c 5 )) 2 e 4 c 4 c 5 c 5 ( c 1 c 3 + c 2 ( c 1 + c 3 + c 4 )) 2 e 5 0
Question ◮ The map c �→ e is rational (polynomial / polynomial) ◮ What do the fibers look like? ◮ Equivalently, which interior voltages produce the energies e ? Example Let ∆ = ( c 1 + c 3 + c 4 )( c 2 + c 3 + c 5 ) − c 2 3 . We have 1 c 1 c 2 c 1 ( c 3 c 5 + c 4 ( c 2 + c 3 + c 5 )) 2 e 1 c 2 ( c 3 c 4 + ( c 1 + c 3 + c 4 ) c 5 ) 2 c 3 e 2 = 1 c 3 ( c 2 c 4 − c 1 c 5 ) 2 e 3 ∆ 2 c 4 ( c 2 c 3 + c 1 ( c 2 + c 3 + c 5 )) 2 e 4 c 4 c 5 c 5 ( c 1 c 3 + c 2 ( c 1 + c 3 + c 4 )) 2 e 5 0
e -harmonic functions Definition Fix boundary voltages v and energies e ∈ C E . A function h ∈ C V is e -harmonic on ( G , v ) if 1. There are conductances c for which h is the voltage function for the network 2. The resulting energies are e . Interesting Problem (Abrams–Kenyon 2017) Describe the set of e -harmonic functions for a given e .
Example Let all e ≡ 1. Fix boundary voltages 0 and 1. There are two e -harmonic functions, with conductances labeled: a b 5 a − 5 5 b − 5 5 b − 5 5 a − 5 5 5 1 0 1 0 5 b − 5 5 a − 5 5 a − 5 5 b − 5 a b √ √ where a = 1 5) and b = 1 2 (5 − 2 (5 + 5).
Section II: Dirichlet Arrangements
Dirichlet arrangements Definition The Dirichlet arrangement A G , v consists of two types of hyperplanes, corresponding to the edges of G , with coordinates indexed by interior nodes: ◮ A hyperplane x i = v j for every edge ij with j ∈ ∂ V ◮ A hyperplane x i = x j for every edge ij not meeting ∂ V r x r = x s x s = 1 1 0 x s = 0 s x r = 0 x r = 1
Master functions ◮ Let A be an arrangement of k hyperplanes in C n , defined by affine functions f 1 , . . . , f k : C n → C ◮ Master function of A with weights b ∈ C k is multivalued C n → C given by k � f i ( x ) b i Φ b ( x ) = i =1 ◮ A critical point x ∈ C n of Φ b : k � ∂ ∂ f i b i log Φ b ( x ) = f i ( x ) = 0 ∂ x j ∂ x j i =1 for all j = 1 , . . . , n
Example ◮ Arrangement A in C 2 defined by A : x = 0 , y = 0 , x + y − 1 = 0 ◮ Master function with weights b ∈ C 3 : Φ b ( x , y ) = x b 1 y b 2 ( x + y − 1) b 3 ◮ One critical point: � � b 1 b 2 ( x , y ) = , b 1 + b 2 + b 3 b 1 + b 2 + b 3
Why Dirichlet arrangements? Theorem (L. 2019) The e -harmonic functions on ( G , v ) are the critical points of the master function of A G , v with weights e . Example √ √ Let a = 1 5) and b = 1 10 (5 − 10 (5 + 5) with all energies 1: a or b ( b , a ) 1 0 ( a , b ) b or a
So what? Critical points of master functions are well studied! ◮ Interior point methods: logarithmic barrier functions and analytic centers ◮ Algebraic statistics: maximum likelihood estimation . . . especially for certain Dirichlet arrangements. ◮ Real algebraic geometry: solution of B. and M. Shapiro conjecture on Wronskians ◮ Quantum integrable systems: Bethe ansatz in the Gaudin model
Current flow ◮ Real-valued e -harmonic functions induce current flows ◮ Edges directed from higher voltages to lower ◮ # nonzero current flows = # e -harmonic functions = # bounded chambers Example In the running example: a b 1 0 1 0 a b √ √ where a = 1 5) and b = 1 10 (5 − 10 (5 + 5).
Counting bounded chambers ◮ Chromatic polynomial χ G counts proper vertex colorings of G ◮ The beta invariant is β ( G ) = | χ ′ G (1) | ◮ β ( G ) = 0 iff G is disconnected by removing single vertex Theorem (L. 2019) Construct � G from G by adding edges between all boundary vertices. For generic energies e (including all positive) we have # current flows β ( � G ) = # e -harmonic functions # bounded chambers ( | ∂ V | − 2)!
Visibility arrangements Definition Let P be a convex n -polytope in R n . The visibility arrangement vis( P ) is the set of affine spans of the top-dimensional faces of P
Visibility sets Sets of top-dimensional faces of P Chambers of vis( P ) � visible from different points in R n
Order polytopes Definition Let P be a finite poset. The order polytope O ( P ) is the convex polytope in R P of all order-preserving functions P → [0 , 1] Example If every pair in P = { x 1 , . . . , x n } is incomparable, then O ( P ) = [0 , 1] n is the unit hypercube in R n Example If P = { x 1 , . . . , x n } is totally ordered, then O ( P ) is an n -simplex in R n
Visibility arrangements of order polytopes Proposition (Stanley 2015) Let P be a finite poset. The visibility arrangement vis( O ( P )) of the order polytope of P is a Dirichlet arrangement A G , v . 1 Hasse diagram of P 0 � G ( G , v )
Counting visibility sets Theorem (L. 2019) The number of visibility sets of the order polytope O ( P ) is 1 2 α ( � G ) , where α ( � G ) is the number of acyclic orientations of � G . Of the visibility sets, all but β ( � G ) are visible from far away, where β ( � G ) is the beta invariant.
Section III: Network Duals & Matroid Quotients
Graphic matroids ◮ Graphic matroids M ( G ) are a fundamental class of matroids ◮ Recall: circuits of M ( G ) are (edge sets of) cycles of G ◮ Circuits of dual matroid M ∗ ( G ) are bonds of G a b Cycles = abc , abde , cde c Bonds = ab , ace , bce , de e d
Cellularly embedded graphs ◮ Let Σ be a compact surface ◮ Suppose G ֒ → Σ such that every face is a 2-cell ◮ Can define a geometric dual G ∗
Planar duality Theorem (Whitney 1932) If G is planar, then M ∗ ( G ) ∼ = M ( G ∗ ). In other words, Cycles of G ∗ Bonds of G =
Generalization ◮ Ranks of M ∗ ( G ) and M ( G ∗ ) differ by 2 − χ (Σ), so no isomorphism in general ◮ There is a quotient map M ։ N if every circuit of M is a union of circuits of N ◮ Quotient maps are the bijective morphisms of matroids Theorem (Richter–Shank 1984) → Σ, then there is a quotient map M ∗ ( G ) ։ M ( G ∗ ). If G ֒
Network duals ◮ Let Σ be a compact surface with boundary ◮ Suppose G ֒ → Σ is cellularly embedded with ∂ V ⊂ ∂ Σ and no face having > 1 boundary component ◮ Given D = ( G , ∂ V ), can define dual network D ∗
Dirichlet matroids ◮ Dirichlet arrangement A G , v defines matroid M ( D ) depending only on D = ( G , ∂ V ) ◮ Circuits of M ( D ) keep track of cycles and paths between boundary nodes ◮ M ( D ) arises from almost-balanced biased graphs and complete principal truncations of graphic matroids | ∂ V | = 2 M ( D ) ∼ = M ( � G ) is graphic D = star network M ( D ) ∼ = U 2 , | V | is uniform
Quotient map for networks Theorem (L. 2019) → Σ, then there is a quotient map M ∗ ( D ) ։ M ( D ∗ ). If D ֒ ◮ Ranks differ by | ∂ V | − χ (Σ) − 1 ◮ Recover graphic case when | ∂ V | = 2 ◮ Neither M ∗ ( D ) nor M ( D ∗ ) is graphic/cographic in general!
Thanks Thank you!
Recommend
More recommend
