Grid Graphs, Gorenstein Polytopes, and Domino Stackings Matthias Beck (San Francisco State) math.sfsu.edu/beck Joint with Christian Haase (FU Berlin) & Steven Sam (MIT) arXiv:0711.4151
“The hardest thing being with a mathematician is that they always have problems.” Tendai Chitewere
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d , i.e., the vertices of P are in Z n L P ( t ) := # ( t P ∩ Z n ) (discrete volume of P ) Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 2
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d , i.e., the vertices of P are in Z n L P ( t ) := # ( t P ∩ Z n ) (discrete volume of P ) Ehrhart’s Theorem (1962) L P ( t ) is a polynomial in t ∈ N . Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 2
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d , i.e., the vertices of P are in Z n L P ( t ) := # ( t P ∩ Z n ) (discrete volume of P ) Ehrhart’s Theorem (1962) L P ( t ) is a polynomial in t ∈ N . Equivalently, h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 , t ≥ 1 where h ( z ) is a polynomial, the Ehrhart h-vector of P . Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 2
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d , i.e., the vertices of P are in Z n L P ( t ) := # ( t P ∩ Z n ) (discrete volume of P ) Ehrhart’s Theorem (1962) L P ( t ) is a polynomial in t ∈ N . Equivalently, h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 , t ≥ 1 where h ( z ) is a polynomial, the Ehrhart h-vector of P . (Serious) Open Problem Classify Ehrhart polynomials/h-vectors. Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 2
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d , i.e., the vertices of P are in Z n L P ( t ) := # ( t P ∩ Z n ) (discrete volume of P ) Ehrhart’s Theorem (1962) L P ( t ) is a polynomial in t ∈ N . Equivalently, h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 , t ≥ 1 where h ( z ) is a polynomial, the Ehrhart h-vector of P . (Serious) Open Problem Classify Ehrhart polynomials/h-vectors. (Easier) Open Problem Construct and study special classes of lattice poly- topes. Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 2
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 3
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Some sample problems Find P for which the Ehrhart h-vector h ( z ) is palindromic. ◮ Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 3
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Some sample problems Find P for which the Ehrhart h-vector h ( z ) is palindromic. ◮ For which P is the Ehrhart h-vector h ( z ) unimodal, i.e., ◮ h 0 ≤ · · · ≤ h j − 1 ≤ h j ≥ h j +1 ≥ · · · ≥ h d ? Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 3
Ehrhart Polynomials P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 Some sample problems Find P for which the Ehrhart h-vector h ( z ) is palindromic. ◮ For which P is the Ehrhart h-vector h ( z ) unimodal, i.e., ◮ h 0 ≤ · · · ≤ h j − 1 ≤ h j ≥ h j +1 ≥ · · · ≥ h d ? Study Ehrhart h-vectors of special classes, e.g., simplicial polytopes. ◮ Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 3
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) P is Gorenstein if there is a k ∈ N such that L P ◦ ( t ) = L P ( t − k ) for all t ≥ k and L P ◦ ( t ) = 0 for 0 < t < k . We call k the index of P . Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 4
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) P is Gorenstein if there is a k ∈ N such that L P ◦ ( t ) = L P ( t − k ) for all t ≥ k and L P ◦ ( t ) = 0 for 0 < t < k . We call k the index of P . Examples the unit cube ✷ = [0 , 1] d with L ✷ ( t ) = ( t + 1) d and L ✷ ◦ ( t ) = ( t − 1) d ◮ Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 4
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) P is Gorenstein if there is a k ∈ N such that L P ◦ ( t ) = L P ( t − k ) for all t ≥ k and L P ◦ ( t ) = 0 for 0 < t < k . We call k the index of P . Examples the unit cube ✷ = [0 , 1] d with L ✷ ( t ) = ( t + 1) d and L ✷ ◦ ( t ) = ( t − 1) d ◮ � t + d � the standard simplex ∆ = conv { 0 , e 1 , e 2 , . . . , e d } with L ∆ ( t ) = ◮ d � t − 1 � and L ∆ ◦ ( t ) = d Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 4
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) P is Gorenstein if there is a k ∈ N such that L P ◦ ( t ) = L P ( t − k ) for all t ≥ k and L P ◦ ( t ) = 0 for 0 < t < k . We call k the index of P . Examples the unit cube ✷ = [0 , 1] d with L ✷ ( t ) = ( t + 1) d and L ✷ ◦ ( t ) = ( t − 1) d ◮ � t + d � the standard simplex ∆ = conv { 0 , e 1 , e 2 , . . . , e d } with L ∆ ( t ) = ◮ d � t − 1 � and L ∆ ◦ ( t ) = d the Birkhoff polytope ◮ · · · x 11 x 1 n � j x jk = 1 for all 1 ≤ k ≤ n . . ∈ R n 2 . . . . ≥ 0 : � k x jk = 1 for all 1 ≤ j ≤ n x n 1 . . . x nn Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 4
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 P is Gorenstein if and only if the Ehrhart h-vector h ( z ) of P is palindromic (this is a nice exercise if one knows the Ehrhart–Macdonald reciprocity theorem). Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 5
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 P is Gorenstein if and only if the Ehrhart h-vector h ( z ) of P is palindromic (this is a nice exercise if one knows the Ehrhart–Macdonald reciprocity theorem). Remark The Gorenstein property has an extension to Cohen–Macaulay algebras. Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 5
Gorenstein Polytopes P ⊂ R n – lattice polytope of dimension d L P ( t ) := # ( t P ∩ Z n ) h ( z ) L P ( t ) z t = � Ehr P ( z ) := 1 + (1 − z ) d +1 t ≥ 1 P is Gorenstein if and only if the Ehrhart h-vector h ( z ) of P is palindromic (this is a nice exercise if one knows the Ehrhart–Macdonald reciprocity theorem). Remark The Gorenstein property has an extension to Cohen–Macaulay algebras. Goal Construct classes of Gorenstein polytopes. Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 5
Suggested Tools LattE macchiato ( http://www.math.ucdavis.edu/ ∼ latte/ ) ◮ barvinok ( http://freshmeat.net/projects/barvinok/ ) ◮ ehrhart ◮ ( http://icps.u-strasbg.fr/Ehrhart/program/program.html ) Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 6
Suggested Tools LattE macchiato ( http://www.math.ucdavis.edu/ ∼ latte/ ) ◮ barvinok ( http://freshmeat.net/projects/barvinok/ ) ◮ ehrhart ◮ ( http://icps.u-strasbg.fr/Ehrhart/program/program.html ) Normaliz ◮ ( ftp://ftp.mathematik.uni-osnabrueck.de/pub/osm/kommalg/software/ ) 4ti2 ( www.4ti2.de ) ◮ Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 6
Suggested Tools LattE macchiato ( http://www.math.ucdavis.edu/ ∼ latte/ ) ◮ barvinok ( http://freshmeat.net/projects/barvinok/ ) ◮ ehrhart ◮ ( http://icps.u-strasbg.fr/Ehrhart/program/program.html ) Normaliz ◮ ( ftp://ftp.mathematik.uni-osnabrueck.de/pub/osm/kommalg/software/ ) 4ti2 ( www.4ti2.de ) ◮ polymake ( http://www.math.tu-berlin.de/polymake/ ) ◮ Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 6
Perfect Matchings and Magic Labellings of Graphs A perfect matching of a graph G is a subset M ⊆ E ( G ) such that every vertex of G is incident with exactly one edge of M . Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 7
Perfect Matchings and Magic Labellings of Graphs A perfect matching of a graph G is a subset M ⊆ E ( G ) such that every vertex of G is incident with exactly one edge of M . More generally, a magic labelling (with sum t ) is a function E ( G ) → Z ≥ 0 such that for each vertex v , the sum of the labels of the edges incident to v equals t . Grid graphs, Gorenstein polytopes, and domino stackings Matthias Beck 7
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