Properties of the coordinate ring of a convex polyomino Claudia - PowerPoint PPT Presentation

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Properties of the coordinate ring of a convex polyomino Claudia Andrei 1 Graduate Student Meeting on Applied Algebra and Combinatorics March 2018 University of Osnabr¨ uck, Germany 1 University of Bucharest, Romania Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Outline 1 Preliminaries 2 Gorenstein convex polyominoes 3 The regularity of K [ P ] 4 The multiplicity of K [ P ] Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries A. Qureshi, Ideals generated by 2 -minors, collections of cells and stack polyominoes , J. Algebra 357 (2012), 279–303. The coordinate ring of a convex polyomino was introduced by Qureshi. x 14 x 24 x 34 x 44 x 13 x 23 x 33 x 43 x 12 x 22 x 32 x 42 x 11 x 21 x 31 x 41 Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries In order to define polyominoes and polyomino ideals, we give some terminology. On N 2 , we consider the natural partial order defined as follows: ( i , j ) ≤ ( k , l ) if and only if i ≤ k and j ≤ l . Let a = ( i , j ), b = ( k , l ) ∈ N 2 and a ≤ b . The set [ a , b ] = { c ∈ N 2 | a ≤ c ≤ b } represents an interval in N 2 . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries The interval C = [ a , a +(1 , 1)] is called a cell in N 2 with lower left corner a . a +(0 , 1) a +(1 , 1) a a +(1 , 0) Figure: A cell in N 2 Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries Let P be a finite collection of cells in N 2 . Two cells A and B of P are connected by a path in P , if there is a sequence of cells of P given by A = A 1 , A 2 , ··· , A n − 1 , A n = B such that A i ∩ A i +1 is an edge of A i and A i +1 for i ∈ { 1 , ··· , n − 1 } . Definition A collection of cells P is called a polyomino if any two cells of P are connected by a path in P . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries A A 2 B A 3 A 4 A 5 A 9 A 10 A 6 A 7 A 8 Figure: A polyomino Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries A column convex polyomino A row convex polyomino Figure: A convex polyomino Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries Let P be a convex polyomino. After a possible translation, we consider [(1 , 1) , ( m , n )] to be the smallest interval which contains the vertices of P . (4 , 4) (1 , 1) We say that P is a convex polyomino on [ m ] × [ n ], where [ m ] = { 1 ,..., m } and [ n ] = { 1 ,..., n } . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries Let P be a convex polyomino on [ m ] × [ n ]. Fix a field K and a polynomial ring S = K [ x ij | ( i , j ) ∈ V ( P )] , where V ( P ) is the set of the vertices of P . The polyomino ideal I P ⊂ S is generated by all binomials x il x kj − x ij x kl for which [( i , j ) , ( k , l )] is an interval in P . The K -algebra S / I P is denoted K [ P ] and is called the coordinate ring of P . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries x 24 x 34 x 13 x 23 x 33 x 43 x 12 x 22 x 32 x 42 x 21 x 31 Figure: For the ”cross”, I P has 11 generators. Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries The ring R = K [ x i y j | ( i , j ) ∈ V ( P )] ⊂ K [ x 1 , ··· , x m , y 1 , ··· , y n ] can be viewed as an edge ring of a bipartite graph G P with vertex set V ( G P ) = X ∪ Y , where X = { x 1 , ··· , x m } and Y = { y 1 , ··· , y n } and edge set E ( G P ) = {{ x i , y j } | ( i , j ) ∈ V ( P ) } . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Preliminaries A. Qureshi, Ideals generated by 2 -minors, collections of cells and stack polyominoes , J. Algebra 357 (2012), 279–303. ( i , j +1) ( i +1 , j +1) x i +1 x i y j y j +1 ( i , j ) ( i +1 , j ) Figure: The bipartite graph attached to a cell in N 2 K [ P ] can be identified with K [ G P ]. Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes A. Qureshi, Ideals generated by 2 -minors, collections of cells and stack polyominoes , J. Algebra 357 (2012), 279–303. Let P be a convex polyomino on [ m ] × [ n ]. Theorem (A. Qureshi, Theorem 2.2) K [ P ] is a Cohen-Macaulay domain with dim K [ P ] = m + n − 1 . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes H. Ohsugi, T. Hibi, Special simplices and Gorenstein toric rings , J. Combinatorial Theory 113 (2006), 718–725. Let X = { x 1 ,..., x m } and Y = { y 1 ,..., y n } . Theorem (H. Ohsugi, T. Hibi, Theorem 2.1) We consider G to be a bipartite graph on X ∪ Y and suppose that G is 2 -connected. Then K [ G ] is Gorenstein if and only if x 1 ··· x m y 1 ··· y n ∈ K [ G ] and one has | N ( T ) | = | T | +1 for every subset T ⊂ X such that G T ∪ N ( T ) is connected and that G ( X ∪ Y ) \ ( T ∪ N ( T )) is a connected graph with at least one edge. Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes Definition Let G be a graph on V . Then we say that G is 2-connected if G together with G V \{ v } for all v ∈ V are connected. Proposition If P is a convex polyomino on [ m ] × [ n ] , then the bipartite graph G P is 2 -connected. Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes Let G be a graph and T ⊂ V ( G ). The set N ( T ) = { y ∈ V ( G ) | { x , y } ∈ E ( G ) for some x ∈ T } represents the set of the neighbors of the subset T ⊂ V ( G ). Let P be a convex polyomino on [ m ] × [ n ]. We set X = { x 1 , ··· , x m } and Y = { y 1 , ··· , y n } and, if needed, we identify the point ( x i , y j ) in the plane with the vertex ( i , j ) ∈ V ( P ). Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes Definition Let T ⊂ X . The set N Y ( T ) = { y ∈ Y | ( x , y ) ∈ V ( P ) for some x ∈ T } is called a neighbor vertical interval if N Y ( T ) = { y a , y a +1 , ··· , y b } with a < b and for every i ∈ { a , a +1 , ··· , b − 1 } there exists x ∈ T such that [( x , y i ) , ( x , y i +1 )] is an edge in P . x 1 x 4 x 1 x 2 Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes Definition Let U ⊂ Y . The set N X ( U ) = { x ∈ X | ( x , y ) ∈ V ( P ) for some y ∈ U } is called a neighbor horizontal interval if N X ( U ) = { x a , x a +1 , ··· , x b } with a < b and for every i ∈ { a , a +1 , ··· , b − 1 } there exists y ∈ U such that [( x i , y ) , ( x i +1 , y )] is an edge in P . y 5 y 3 y 2 y 1 Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes Proposition Let P be a convex polyomino on [ m ] × [ n ] and G P its associated bipartite graph. Then we have x 1 ··· x m y 1 ··· y n ∈ K [ G P ] if and only if | N Y ( T ) | ≥ | T | for every T ⊂ X and | N X ( U ) | ≥ | U | for every U ⊂ Y . Claudia Andrei Properties of the coordinate ring of a convex polyomino

Preliminaries Gorenstein convex polyominoes The regularity of K [ P ] The multiplicity of K [ P ] Gorenstein convex polyominoes x 3 x 4 x 5 Figure: Perfect matching for G P Claudia Andrei Properties of the coordinate ring of a convex polyomino