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Root system chip-firing PhD Thesis Defense Sam Hopkins Massachusetts Institute of Technology April 27th, 2018 Includes joint work with Pavel Galashin, Thomas McConville, Alexander Postnikov, and James Propp Sam Hopkins (2018) Root system chip-firing April 27th, 2018 1 / 47

Classical chip-firing and labeled chip-firing Section 1 Classical chip-firing and labeled chip-firing Sam Hopkins (2018) Root system chip-firing April 27th, 2018 2 / 47

Classical chip-firing and labeled chip-firing Classical chip-firing Classical chip-firing (as introduced by Björner-Lovász-Shor, 1991) is a discrete dynamical system that takes place on a graph. The states are configurations of chips on the vertices. We may fire a vertex that has at least as many chips as neighbors, sending one chip to each neighbor: A key property of this system is that it is confluent: from a given initial configuration, either all sequences of firings go on forever, or they all terminate at the same stable configuration (called the stabilization). Sam Hopkins (2018) Root system chip-firing April 27th, 2018 3 / 47

Classical chip-firing and labeled chip-firing Connections between chip-firing and other areas of math This same system was introduced in statistical mechanics under the name of the Abelian Sandpile Model by Bak-Tang-Wiesenfeld, 1987 and Dhar, 1990. It has subsequently been studied in many different parts of math: Algebraic geometry: discrete version of divisor theory for curves, e.g. Riemann-Roch for graphs (Baker-Norine, 2007); Probability: remarkable scaling limit behavior (Pegden-Smart, 2013); Theoretical computer science: decidability of stabilization; Representation theory: chip-firing for group representation and Hopf algebra modules (Benkart-Klivans-Reiner, 2016; Gaetz, 2016; Grinberg-Huang-Reiner, 2017). Our work is mostly closely related to this last area. The aspect of chip-firing we are interested in is the aforementioned confluence property. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 4 / 47

Classical chip-firing and labeled chip-firing Chip-firing on a line One of the first articles to discuss chip-firing was Disks, Balls, and Walls by Anderson et al., in the American Math Monthly, 1989. They studied chip firing on Z (the infinite path graph): Even in this simple setting we can ask questions such as: which stable configuration does n chips at the origin stabilize to? Sam Hopkins (2018) Root system chip-firing April 27th, 2018 5 / 47

Classical chip-firing and labeled chip-firing A brief aside... Things are a bit different in two dimensions: the stabilization of four million chips at the origin of Z 2 is Sam Hopkins (2018) Root system chip-firing April 27th, 2018 6 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing Jim Propp recently introduced a labeled variant of chip-firing on Z. All the chips are now distinguishable, given labels from N. Whenever two chips occupy the same vertex we can fire them together, moving the lesser-labeled chip leftwards one vertex and the greater-labeled chip rightwards one vertex: 4 3 2 1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 7 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing Jim Propp recently introduced a labeled variant of chip-firing on Z. All the chips are now distinguishable, given labels from N. Whenever two chips occupy the same vertex we can fire them together, moving the lesser-labeled chip leftwards one vertex and the greater-labeled chip rightwards one vertex: 4 1 3 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 7 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing Jim Propp recently introduced a labeled variant of chip-firing on Z. All the chips are now distinguishable, given labels from N. Whenever two chips occupy the same vertex we can fire them together, moving the lesser-labeled chip leftwards one vertex and the greater-labeled chip rightwards one vertex: 3 4 1 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 7 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing Jim Propp recently introduced a labeled variant of chip-firing on Z. All the chips are now distinguishable, given labels from N. Whenever two chips occupy the same vertex we can fire them together, moving the lesser-labeled chip leftwards one vertex and the greater-labeled chip rightwards one vertex: 4 1 3 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 7 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing Jim Propp recently introduced a labeled variant of chip-firing on Z. All the chips are now distinguishable, given labels from N. Whenever two chips occupy the same vertex we can fire them together, moving the lesser-labeled chip leftwards one vertex and the greater-labeled chip rightwards one vertex: 1 2 3 4 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 7 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing is not confluent in general What if we started with three chips at the origin: 3 2 1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 8 / 47

Classical chip-firing and labeled chip-firing Labeled chip-firing is not confluent in general What if we started with three chips at the origin: 2 1 3 Not sorted and not confluent! Sam Hopkins (2018) Root system chip-firing April 27th, 2018 8 / 47

Classical chip-firing and labeled chip-firing Sorting an even number of chips Theorem (H.-McConville-Propp) Suppose n = 2m is even. Then starting from n labeled chips at the origin, the chip-firing process sorts the chips to a unique stable configuration: 1 2 m m+1 m+2 2m -m 0 m Sam Hopkins (2018) Root system chip-firing April 27th, 2018 9 / 47

Classical chip-firing and labeled chip-firing A Type B version of labeled chip-firing Consider a modified version of labeled chip-firing on Z where we allow the following three kinds of moves: (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 10 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 3 2 1 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 1 3 2 0 (I) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 3 1 2 0 (II) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 3 0 (III) 1 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 0 (III) 3 1 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 3 1 2 0 (I) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 1 2 3 0 (I) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 0 (III) 2 1 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 1 2 3 0 (I) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B labeled chip-firing example (I) for i < j, if i and j occupy the same vertex, move i leftwards one vertex and j rightwards one vertex (this is the same as before); (II) for any i, j, if i is at vertex a and j is at vertex a, move both i and j rightwards one vertex; (III) for any i, if i is at the origin, move i rightwards one vertex. 0 (III) 1 2 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 11 / 47

Classical chip-firing and labeled chip-firing Type B sorting Theorem (H.-McConville-Propp) For any n, starting from n labeled chips at the origin, the Type B labeled chip-firing process (with moves (I), (II), and (III)) sorts the chips to the following unique stable configuration: 1 2 n -n 0 n Sam Hopkins (2018) Root system chip-firing April 27th, 2018 12 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: 3 2 1-1 -2-3 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: 3-1 -2 1-3 2 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: 1 3 2-2 -3-1 0 (I) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: 3 1 2-2 -3-1 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: -2 3 1-1 -3 0 (II) 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Classical chip-firing and labeled chip-firing Proof of Type B sorting via symmetry Use positive chips 1, 2,..., n and negative chips n, 2,..., 1. Start with all of them at the origin, and carry out symmetrical labeled chip-firing moves (whenever we fire i and j we also fire i and j ). The way the positive chips evolve corresponds exactly to the Type B moves (I), (II), and (III) above: 1-2 -3-1 0 (III) 3 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 13 / 47

Central-firing Section 2 Central-firing Sam Hopkins (2018) Root system chip-firing April 27th, 2018 14 / 47

Central-firing Root system reformulation of Propp s labeled chip-firing For any configuration of n labeled chips, if we set c := (c 1,..., c n ) Z n where c i := the position of the chip i, then, for 1 i < j n, we are allowed to fire chips i and j in this configuration as long as c is orthogonal to e j e i ; and doing so replaces the vector c by c + (e j e i ). Key observation (first made by Pavel Galashin): the vectors e j e i for 1 i < j n are exactly (one choice of) the positive roots Φ + of the root system Φ of Type A n 1, suggesting a natural way to generalize labeled chip-firing to any root system Φ. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 15 / 47

Central-firing Root system basics Root systems are certain highly symmetrical finite sets of vectors in Euclidean space. The key property of a root system Φ is that it s preserved by the reflection s α across the hyperplane orthogonal to any root α Φ. A 1 A 1 A2 B 2 G 2 Root systems were first introduced in the late 19th century in the context of Lie theory because (irreducible, crystallographic) root systems correspond bijectively to simple Lie algebras over the complex numbers. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 16 / 47

Central-firing Root system classification and Dynkin diagrams A convenient way to represent a root system is via a decorated graph called its Dynkin diagram. The root systems have been classified into Cartan-Killing types whose Dynkin diagrams are: 1 2 n 1 n A n 1 2 n 1 n B n 1 2 n 1 n C n n 1 2 n 2 n 1 1 1 1 2 G 2 1 2 3 4 F 4 2 3 4 5 6 E 6 2 3 4 5 6 7 E 7 2 D n 1 3 4 5 6 7 8 E 8 Same ADE classification arises in: singularity theory (Arnol d); quiver representations (Gabriel); finite subgroups of SU(2) (McKay). Sam Hopkins (2018) Root system chip-firing April 27th, 2018 17 / 47

Central-firing Root system notation Type A is the realm of classical combinatorics (e.g., permutations). Let us go over basic notation for root systems, with Type A as an example: Object Notation Type A n 1 Ambient v.s. V, dim V = r {v (1, 1,..., 1) R n } Roots α Φ ±(e i e j ), i < j Coroots α = 2 α,α α Φ ±(e i e j ), i < j Positive roots α Φ + e i e j, i < j Simple roots α 1,..., α r α i = e i e i+1 Root lattice Q = Z[Φ] {v (1, 1,..., 1) Z n } Weight lattice P = {v : v, α Z, α Φ} Z n /(1, 1,..., 1) Fundamental weights {}}{ ω 1,..., ω r ω i = ( 1,..., 1, 0,..., 0) Weyl group W = s α, α Φ GL(V ) Symmetric group S n Sam Hopkins (2018) Root system chip-firing April 27th, 2018 18 / 47 i

Central-firing Central-firing for root systems The description of labeled chip-firing in terms of positive roots of A n 1 generalizes naturally to any root system Φ: for a weight λ P, we allow the firing moves λ λ + α for a positive root α Φ + whenever λ is orthogonal to α. We call the resulting system the central-firing process for Φ (because we allow firing from a weight λ when λ belongs to the Coxeter hyperplane arrangement, which is a central arrangement). You can check that the previously described Type B labeled chip-firing really is central-firing for Φ = B n. Other classical types have similar description of central-firing in terms of chips. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 19 / 47

Central-firing Confluence of central-firing Question For any root system Φ and weight λ P, when is central-firing confluent from λ? Answer: it s complicated. But it seems somewhat related to the Weyl vector: := 1 α = 2 α Φ + r ω i. i=1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 20 / 47

Central-firing Classification of confluence for origin/fundamental weights Conjecture (G.-H.-M.-P.) Confluence of central-firing from λ for λ = 0 or λ a fundamental weight is determined by the table on the right. To first order, central-firing is confluent from λ iff λ modulo Q. Exceptions to this are in red or green....... A2n 0 1 n 1 n n + 1 n + 2 2n A2n+1...... 0 1 2 n n + 1 n + 2 2n 2n + 1... Bn 0 1 2 n 1 n... C4n 0 1 2 3 4 4n 3 4n 2 4n 1 4n C4n+1... 0 1 2 3 4 4n 2 4n 1 4n 4n + 1 C4n+2... 0 1 2 3 4 4n 1 4n 4n + 1 4n + 2 C4n+3... 0 1 2 3 4 4n 4n + 1 4n + 2 4n + 3... D4n 4n 1 0 1 2 3 4 4n 3 4n 2 4n D4n+1... 4n 0 1 2 3 4 4n 2 4n 1 4n + 1 D4n+2... 4n + 1 0 1 2 3 4 4n 1 4n 4n + 2 This conjecture is proved in some but not all cases (e.g. for λ = 0 and Φ = A r or B r, it follows from H.-M.-P. theorems above). E6 E7 E8 0 D4n+3 0 1 2 2 3 1 3 4 5 6 2 0 1 3 4 5 0 1 3 2 4 5 4 6 7 6 7... F4 8 4n + 2 4n 4n + 1 4n + 3 0 1 2 G2 0 1 3 2 4 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 21 / 47

Central-firing Confluence of central-firing modulo the Weyl group Theorem (Galashin-H.-McConville-Postnikov) For any root system Φ, and from any initial weight λ, central-firing is confluent modulo the action of the Weyl group W. In Type A the Weyl group is the symmetric group, so modding out by the Weyl group is the same as forgetting the labels of chips. Thus this theorem gives a generalization of unlabeled chip-firing on a line to any root system. Note: this is very different from the Cartan matrix chip-firing studied by Benkart-Klivans-Reiner, 2016 (e.g., for Φ = A n 1, ours corresponds to chip-firing of n chips on the infinite path, whereas B.-K.-R. corresponds to chip-firing of any number of chips on the n-cycle). For simply laced root systems can even describe unlabeled central-firing as a certain numbers game on the Dynkin diagram. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 22 / 47

Central-firing Chip-firing on a line versus unlabeled central-firing 2 1 0 1 2 0 0 0 2 1 1 2 2 1 0 1 2 1 0 1 2 1 1 2 2 1 0 1 2 0 2 0 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 0 1 2 2 1 0 1 2 2 1 0 2 1 0 1 2 2 1 0 1 2 2 0 2 2 1 1 2 1 0 1 2 1 2 1 2 1 0 1 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 23 / 47

Interval-firing: confluence Section 3 Interval-firing: confluence Sam Hopkins (2018) Root system chip-firing April 27th, 2018 24 / 47

Interval-firing: confluence Interval-firing Central-firing allows the firing move λ λ + α whenever λ, α = 0 for λ P and α Φ +. We found remarkable deformations of this process. For any k N, define the symmetric interval-firing process by λ λ + α if λ, α + 1 { k, k + 1,..., k} and the truncated interval-firing process by λ λ + α if λ, α + 1 { k + 1, k + 2..., k}. (These are analogous to the (extended) Φ -Catalan and Φ -Shi hyperplane arrangements, respectively.) Note: undirected graph of the symmetric process is W -invariant. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 25 / 47

Interval-firing: confluence Pictures of interval-firing for A 2 k truncated symmetric 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 26 / 47

Interval-firing: confluence Pictures of interval-firing for A 2 k truncated symmetric 1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 27 / 47

Interval-firing: confluence Pictures of interval-firing for A 2 k truncated symmetric 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 28 / 47

Interval-firing: confluence Pictures of interval-firing for B 2 k truncated symmetric 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 29 / 47

Interval-firing: confluence Pictures of interval-firing for B 2 k truncated symmetric 1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 30 / 47

Interval-firing: confluence Pictures of interval-firing for B 2 k truncated symmetric 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 31 / 47

Interval-firing: confluence Pictures of interval-firing for G 2 k truncated symmetric 0 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 32 / 47

Interval-firing: confluence Pictures of interval-firing for G 2 k truncated symmetric 1 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 33 / 47

Interval-firing: confluence Pictures of interval-firing for G 2 k truncated symmetric 2 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 34 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 3 6 1 5 4 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 2 1 4 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 3 1 6 5 4 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 2 1 4 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 3 1 6 4 5 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 2 1 4 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 1 3 6 4 5 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 2 1 4 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 1 3 6 4 5 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 1 2 4 3 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 1 3 6 4 5 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 1 3 2 4 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing in Type A via chips When Φ = A n 1, we can interpret interval-firing in terms of chips. The smallest nontrivial case is symmetric k = 0 interval-firing, which has λ λ + α for λ P, α Φ + when λ, α = 1. This corresponds to allowing (adjacent) transpositions of i and j if they re out of order: 2 1 3 6 4 5 Here confluence is obvious. The next smallest case is truncated k = 1 interval-firing, which has λ λ + α when λ, α { 1, 0}. This corresponds to allowing transpositions as well as the usual labeled chip-firing moves: 1 2 3 4 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 35 / 47

Interval-firing: confluence Interval-firing is confluent Theorem (Galashin-H.-McConville-Postnikov) For any root system Φ, and any k 0, the symmetric and truncated interval-firing processes are confluent (from all initial weights). I ll now go over some (geometric) ideas that go into the proof. The main ingredient is a formula for traverse lengths of permutohedra. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 36 / 47

Interval-firing: confluence Traverse lengths of permutohedra For λ P, we define the (W -)permutohedron Π(λ) := ConvexHullW (λ). We use Π Q (λ) := Π(λ) (Q + λ) to denote the lattice points in Π(λ). An α-string of length k is a collection {µ, µ α,..., µ kα} P. An α-traverse in Π(λ) is a maximal (by containment) α-string inside Π Q (λ). We define l λ (α), the traverse length of Π(λ) in direction α, to be the minimum length of an α-traverse in Π(λ). Examples for Φ = A 2 : l ω1 (α) = 0 α Φ l ω1 +ω 2 (α) = 1 α Φ Π(ω 1 ) Π(ω 1 + ω 2 ) Intuition: the minimal length α-traverse should be an edge of Π(λ). Sam Hopkins (2018) Root system chip-firing April 27th, 2018 37 / 47

Interval-firing: confluence Funny weights and traverse length formula Counterexample to (almost correct) intuition: for Φ = B 2, l ω1 (α 1 ) = 0 0 Π(ω 1 ) Definition If Φ is not simply laced, then there are l and s such that the long simple root α l and short simple root α s are adjacent in the Dynkin diagram. We say the dominant weight λ = r i=1 c iω i is funny if c s = 0, c l 1, and c i c l for all long α i. (No weight is funny for simply laced Φ.) Theorem (Galashin-H.-McConville-Postnikov) For a dominant weight λ = r i=1 c iω i, set m λ (α) := min{c i : α i W (α)}. { m λ (α) 1 if λ is funny and α is long, l λ (α) = m λ (α) otherwise. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 38 / 47

Interval-firing: confluence Relevance of traverse length to interval-firing Lemma (Galashin-H.-McConville-Postnikov) Proof. l λ (α) = min{ µ, α : µ Π Q (λ), µ + α / Π Q (λ)} Let {µ, µ α,..., µ kα} be an α-traverse in Π Q (λ). By the W -invariance of Π(λ) we have s α (µ iα) = µ (k i)α for i = 0,..., k. Thus in particular µ µ, α α = s α (µ) = µ kα, so µ, α = k. By definition l λ (α) is the minimal such k. So the formula for traverse lengths says that interval-firing processes get trapped inside certain permutohedra, leading to a proof of confluence. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 39 / 47

Interval-firing: Ehrhart-like polynomials Section 4 Interval-firing: Ehrhart-like polynomials Sam Hopkins (2018) Root system chip-firing April 27th, 2018 40 / 47

Interval-firing: Ehrhart-like polynomials The map η We define η : P P to be the following piecewise-linear dilation operator; if λ P is dominant (belongs to fundamental chamber C 0 of the Coxeter arrangement) then η(λ) = λ + : C 0 C 0 η = η(0) Sam Hopkins (2018) Root system chip-firing April 27th, 2018 41 / 47

Interval-firing: Ehrhart-like polynomials The stable points of interval-firing Lemma (Galashin-H.-McConville-Postnikov) The stable points of symmetric interval-firing are {η k (λ): λ P, λ, α 1 for all α Φ + }, and the stable points of truncated interval-firing are {η k (λ): λ P}. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 42 / 47

Interval-firing: Ehrhart-like polynomials Ehrhart-like polynomials In the pictures above, we saw that the set of weights with interval-firing stabilization η k (λ) looks the same across all values of k, except that it gets dilated as k grows. Following Ehrhart theory, for k 0 we define the quantities L sym λ (k) := #µ with symmetric interval-firing stabilization η k (λ); L tr λ (k) := #µ with truncated interval-firing stabilization ηk (λ). Theorem (Galashin-H.-McConville-Postnikov) For all Φ and all λ P, L sym λ (k) is a polynomial in k. Theorem (Galashin-H.-McConville-Postnikov) For simply laced Φ and all λ P, L tr λ (k) is a polynomial in k. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 43 / 47

Interval-firing: Ehrhart-like polynomials Positivity conjectures Conjecture (Galashin-H.-McConville-Postnikov) For all Φ and all λ P, both L sym λ (k) and L tr λ (k) are polynomials with nonnegative integer coefficients in k. In subsequent work with Alex Postnikov, we proved the half of this conjecture concerning L sym λ (k). Theorem (H.-Postnikov) Let λ P be such that λ, α i {0, 1} for all 1 i r. Then L sym λ (k) = # X Φ +, µ W (λ): µ, α {0, 1} for all α Φ + Span R (X ) rvol(x ) k#x. lin. ind. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 44 / 47

Interval-firing: Ehrhart-like polynomials Lattice points in a polytope plus dilating zonotope The starting point for the formula for L sym λ (k) is a 1980 theorem of Stanley which says that for lattice vectors v 1,..., v m Z n, the Ehrhart polynomial of the zonotope Z := m i=1 [0, v i] is given by #(kz Z n ) = rvol(x ) k #X, X {v 1,...,v m}, lin. ind. where rvol(x ) is the g.c.d. of the maximal minors of the matrix whose columns are the elements of X. Theorem (H.-Postnikov) Let P be any polytope in R n and Z as above. Then, #((kz +P) Z n ) = #(quot X (P) quot X (Z n )) rvol(x ) k #X, X {v 1,...,v m}, lin. ind. where quot X : R n R n /Span R (X ) is the canonical quotient map. Sam Hopkins (2018) Root system chip-firing April 27th, 2018 45 / 47

Interval-firing: Ehrhart-like polynomials Integrality property of slices of permutohedra Note that in general, quot X (P Z n ) quot X (P) quot X (Z n ): P = ConvexHull{(1, 0), (0, 1), (0, 2)} X = {(1, 1)} #quot X (P Z n ) = 3 #quot X (P) quot X (Z n ) = 4 Lemma (H.-Postnikov) For λ P and X Φ +, quot X (Π Q (λ)) = quot X (Π(λ)) quot X (Q + λ). Our proof is case-by-case: is there a uniform proof? Sam Hopkins (2018) Root system chip-firing April 27th, 2018 46 / 47

Interval-firing: Ehrhart-like polynomials Thank you! References: Hopkins, McConville, Propp. Sorting via chip-firing. Electronic Journal of Combinatorics, 24(3), 2017. Galashin, Hopkins, McConville, Postnikov. Root system chip-firing I: Interval-firing. arxiv:1708.04850. Galashin, Hopkins, McConville, Postnikov. Root system chip-firing II: Central-firing. arxiv:1708.04849. Hopkins, Postnikov. A positive formula for the Ehrhart-like polynomials from root system chip-firing. arxiv:1803.08472 Sam Hopkins (2018) Root system chip-firing April 27th, 2018 47 / 47