Combinatorics of Gelfand-Tsetlin Polytopes Yibo Gao, Ben Krakoff, - PowerPoint PPT Presentation

Combinatorics of Gelfand-Tsetlin Polytopes Yibo Gao, Ben Krakoff, Lisa Yang July 27, 2016 Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 1 / 34

Overview Introduction and Preliminaries 1 GT Polytopes Main Results Ladder Diagrams and Face Posets Combinatorial Diameter 2 Proof Combinatorial Automorphisms 3 Generators Automorphism Groups Facet Chains Proof Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 2 / 34

GT Polytopes Definition (GT Polytope) Given a partition λ = ( λ 1 , λ 2 , . . . , λ n ), the Gelfand-Tsetlin Polytope GT λ x = ( x i , j ) 1 ≤ j ≤ i ≤ n ∈ R n ( n +1) / 2 with x i , i = λ i satisfying is the set of points � the following inequalities: 1 x i − 1 , j ≤ x i , j ≤ x i +1 , j , 2 x i , j − 1 ≤ x i , j ≤ x i , j +1 . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 3 / 34

GT Polytopes λ 1 ≤ x 2 , 1 ≤ λ 2 ≤ ≤ x 3 , 1 ≤ x 3 , 2 ≤ λ 3 ≤ ≤ ≤ x 4 , 1 ≤ x 4 , 2 ≤ x 4 , 3 ≤ λ 4 . . . . . . . . . . . . x n , 1 ≤ x n , 2 ≤ . . . ≤ x n , n − 1 ≤ λ n Figure: Inequality constraints of GT polytopes. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 4 / 34

Main Results Theorem (Diameter) diam(GT λ ) = 2 m − 2 − δ 1 , a 1 − δ 1 , a m . Theorem ( m = 2 Automorphism Group) Suppose λ = (1 a 1 , 2 a 2 ) and a 1 , a 2 ≥ 2 . Then δ a 1 , a 2 � =2 Aut ( GT λ ) = D 4 × Z 2 × Z . 2 Theorem ( m ≥ 3 Automorphism Group) Suppose λ = 1 a 1 . . . m a m and m ≥ 3 . Let t = 1 if λ is reverse symmetric and let t = 0 otherwise. Let j be the number of pairs a k , a k +1 ≥ 2 . Then δ 1 , a 1 × S δ 1 , am Aut ( GT λ ) ∼ a m − 1 × Z j +1 = Z t 2 ⋉ ϕ ( S ) a 2 2 Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 5 / 34

Ladder Diagrams Definition (Ladder Diagrams) For λ = (1 a 1 , . . . , m a m ), the grid Γ λ is an induced subgraph of Q constructed as follows. Let the origin be the vertex (0 , 0). Set s j := � j i =1 a i , and define terminal vertices t j = ( s j , n − s j ) for 0 ≤ j ≤ m . Γ λ consists of all vertices and edges appearing on any North-East path between the origin and a terminal vertex. A ladder diagram is a subgraph of Γ λ such that 1 the origin is connected to every terminal vertex by some North-East path. 2 every edge in the graph is on a North-East path from the origin to some terminal vertex. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 6 / 34

Face Posets Theorem (ACK) Let F (Γ λ ) denote the poset of ladder diagrams induced by λ ordered by inclusion. Then F ( GT λ ) ∼ = F (Γ λ ) . 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 4 2 2 2 4 4 4 4 4 2 3 3 4 4 4 4 4 4 7 2 3 3 4 4 7 7 7 7 7 4 4 4 4 5 7 7 7 7 7 7 7 7 7 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 6 6 6 7 7 7 7 8 8 8 8 Figure: Let λ = (1 2 , 2 1 , 4 2 , 7 3 , 8 1 ). From left to right: Γ λ with origin and terminal vertices in red and a dashed line indicating the main diagonal, ladder diagram for a point in GT λ , ladder diagram for a 0-dimensional face (vertex), and ladder diagram for a 2-dimensional face. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 7 / 34

Diameter Theorem By the previous Theorem, it suffices to consider λ = (1 a 1 , . . . , m a m ). Our proofs will use ladder diagrams to model faces of GT λ . Theorem (Diameter) diam(GT λ ) = 2 m − 2 − δ 1 , a 1 − δ 1 , a m . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 8 / 34

Diameter Upper Bound Lemma Any two vertices v and w of GT λ are separated by at most 2 m − 2 − δ 1 , a 1 − δ 1 , a m edges. As ladder diagrams, a vertex is a set of m − 1 noncrossing paths. Figure: Vertices v and w . For each terminal vertex t i , there is a path v i ∈ v and a path w i ∈ w . We want to change each v i to w i by traveling along edges. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 9 / 34

Diameter Lower Bound: Phase 1 Traveling along an edge corresponds to moving a subpath of the diagram. We call this a move . Formally, two vertices are adjacent iff the union of two vertices is (the ladder diagram of) an edge. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 10 / 34

Diameter Lower Bound: Phase 1 Figure: Phase 1 of the algorithm. v → v ′ , w → w ′ (= w ). Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 11 / 34

Diameter Lower Bound: Phase 2 Figure: Phase 2 of the algorithm. First line: v ′ → u . Second line: w ′ → u . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 12 / 34

Diameter Lower Bound Lemma There exist two vertices separated by ≥ 2 m − 2 − δ 1 , a 1 − δ 1 , a m edges. We construct the vertices z h and z v that have this separation. Definition (Zigzag lattice path) Figure: Vertices z h and z v of GT λ . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 13 / 34

Diameter Lower Bound Lemma There exist two vertices separated by ≥ 2 m − 2 − δ 1 , a 1 − δ 1 , a m edges. Proof outline. One would like to argue that each path of z h requires two moves to be changed into the corresponding path of z v . But a single move can alter multiple paths. To do this, paths must be merged together first. We create sets to account for the merges that occur before altering ≥ 2 paths simultaneously. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 14 / 34

Diameter Lower Bound Proof outline cont. For any sequence of ℓ edges (moves) between z h and z v , we can associate sets X 1 , . . . , X ℓ where X i is the set of indices of paths altered by the i th move. Claim: X 1 , . . . , X ℓ satisfies the following conditions: 1 Any index (except possibly 1 and m − 1) appears in at least two sets. 2 The last set one index appears cannot be the last set another index appears in. 3 If X k = { i , i + 1 , . . . , j } , then at least j − i of i , i + 1 , . . . , j appear in sets before X k . 4 If X k = { i , i + 1 , . . . , j } and is the last set an index appears in, then each of i , i + 1 , . . . , j appears in sets before X k . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 15 / 34

Diameter Lower Bound Proof outline cont. 1 Any index (except possibly 1 and m − 1) appears in at least two sets. 2 The last set one index appears cannot be the last set another index appears in. 3 If X k = { i , i + 1 , . . . , j } , then at least j − i of i , i + 1 , . . . , j appear in sets before X k . 4 If X k = { i , i + 1 , . . . , j } and is the last set an index appears in, then each of i , i + 1 , . . . , j appears in sets before X k . Claim: Any sequence of sets satisfying these conditions has length ≥ 2 m − 2 − δ 1 , a 1 − δ 1 , a m . Starting at the end of the sequence X 1 , . . . , X ℓ , we replace any Idea: tuples by singletons. After each replacement, the sequence still satisfy these conditions. At the end, we are left with ≥ 2 m − 2 − δ 1 , a 1 − δ 1 , a m singletons. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 16 / 34

Proof of Diameter Theorem (Diameter) diam(GT λ ) = 2 m − 2 − δ 1 , a 1 − δ 1 , a m . Proof. Combine the upper and lower bounds in the previous two lemmas. Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 17 / 34

Generators Definition (The Corner Symmetry) For any λ , there is a Z 2 automorphism µ on F (Γ λ ) given by swapping two pairs of edges ((0 , 0) , (1 , 0)) with ((0 , 0) , (0 , 1)) and ((1 , 0) , (1 , 1)) with ((0 , 1) , (1 , 1)) in any positive path leaving (0 , 0) Figure: Action of µ Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 18 / 34

Generators Definition (The k -Corner Symmetry) Denote the k th terminal vertex by ( n − i , i ), and suppose that a k , a k +1 ≥ 2. There is a Z 2 automorphism µ k on F (Γ λ ) given by swapping two pairs of edges, (( n − i , i )( n − i , i − 1)) with (( n − i , i )( i − 1 , i ) and (( n − i , i − 1) , ( n − i − 1 , i − 1)) with (( n − i − 1 , i ) , ( n − i − 1 , i − 1)) in any positive path going to ( n − i , i ). Figure: Action of µ k Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 19 / 34

Generators Definition (Symmetric Group Symmetry) Suppose that a 1 = 1. Then there is a S a 2 automorphism group acting on F (Γ λ ) in the following way. Take the first column of possible horizontal edges, and label the top a 2 edges 1 though a 2 . S a 2 then acts by if σ ( i ) = j , the edges corresponding to i are mapped to edges corresponding to j . Figure: Action of (123) Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 20 / 34

Generators Definition (The Flip Symmetry) Suppose that λ = (1 a 1 , 2 a 2 , . . . , m a m ) = (1 a m , 2 a m − 1 , . . . , m a 1 ) =: λ ′ . There is a Z 2 automorphism ρ on F (Γ λ ) given by reflecting a subgraph over the line y = x . Figure: Action of ρ . Yibo Gao, Ben Krakoff, Lisa Yang Combinatorics of GT Polytopes July 27, 2016 21 / 34