Catalan Combinatorics of Borel Ideals and Generalizations Eric S. - PowerPoint PPT Presentation

Catalan Combinatorics of Borel Ideals and Generalizations Eric S. Egge Carleton College September 21, 2014 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals and Generalizations September 21, 2014 1 / 19

Borel Ideals GL n := set of invertible n × n matrices over C B ( n ) := set of upper triangular matrices in GL n Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 2 / 19

Borel Ideals GL n := set of invertible n × n matrices over C B ( n ) := set of upper triangular matrices in GL n Fact GL n has a natural action on C [ x 1 , . . . , x n ] , so B ( n ) does, too. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 2 / 19

Borel Ideals GL n := set of invertible n × n matrices over C B ( n ) := set of upper triangular matrices in GL n Fact GL n has a natural action on C [ x 1 , . . . , x n ] , so B ( n ) does, too. � 1 � 2 1 + 5 x 2 ) = ( x 1 + 3 x 2 ) 2 + 5(2 x 1 + 4 x 2 ) · ( x 2 3 4 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 2 / 19

Borel Ideals GL n := set of invertible n × n matrices over C B ( n ) := set of upper triangular matrices in GL n Fact GL n has a natural action on C [ x 1 , . . . , x n ] , so B ( n ) does, too. � 1 � 2 1 + 5 x 2 ) = ( x 1 + 3 x 2 ) 2 + 5(2 x 1 + 4 x 2 ) · ( x 2 3 4 x 1 �→ x 1 + 3 x 2 x 2 �→ 2 x 1 + 4 x 2 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 2 / 19

Borel Ideals GL n := set of invertible n × n matrices over C B ( n ) := set of upper triangular matrices in GL n Fact GL n has a natural action on C [ x 1 , . . . , x n ] , so B ( n ) does, too. Definition A Borel ideal is an ideal in C [ x 1 , . . . , x n ] which is closed under the action of B ( n ). Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 2 / 19

Catalan Combinatorics of Borel Ideals Theorem (Francisco, Mermin, and Schweig) The Borel ideal generated by x 1 x 2 · · · x n has a minimal generating set (as an ordinary ideal) of C n monomials. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 3 / 19

Catalan Combinatorics of Borel Ideals Theorem (Francisco, Mermin, and Schweig) The Borel ideal generated by x 1 x 2 · · · x n has a minimal generating set (as an ordinary ideal) of C n monomials. Idea: x i �→ x j j < i transforms every generating monomial to another generating monomial. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 3 / 19

Catalan Combinatorics of Borel Ideals x 1 x 2 x 3 ✟ ❍❍❍❍❍ ✟ ✟ ✟ ✟ ✟ ❍ x 2 x 1 x 2 1 x 3 2 ❍❍❍❍❍ � ❅ � ❅ ❍ � ❅ x 2 � 1 x 2 ❅ � ❅ � ❅ � � ❅ x 3 1 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 4 / 19

Catalan Combinatorics of Borel Ideals x 1 x 2 x 3 ✟ ❍❍❍❍❍ x 2 → x 1 ✟ x 3 → x 2 ✟ ✟ ✟ ✟ ❍ x 2 x 1 x 2 1 x 3 x 3 → x 1 2 ❍❍❍❍❍ � x 3 → x 2 ❅ � ❅ ❍ � ❅ x 2 � 1 x 2 ❅ � x 3 → x 1 x 2 → x 1 ❅ � x 2 → x 1 ❅ � � ❅ x 3 1 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 4 / 19

Bijection with Catalan Paths Observation: The minimal generators are the monomials of degree n whose total degree in x 1 , . . . , x j is at least j for all j . Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 5 / 19

Bijection with Catalan Paths Observation: The minimal generators are the monomials of degree n whose total degree in x 1 , . . . , x j is at least j for all j . x 4 x 3 x 3 degree x 2 x 2 x 2 x 1 x 1 x 1 x 1 variables Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 5 / 19

Bijection with Catalan Path Example x 2 1 x 2 x 2 3 x 4 �→ Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 6 / 19

Bijection with Catalan Path Example x 4 x 3 x 3 x 2 1 x 2 x 2 3 x 4 �→ x 2 x 1 x 1 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 6 / 19

Betti Numbers of Borel Ideals C n , k := number of minimal generators of � x 1 x 2 · · · x n � B with largest variable x k Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 7 / 19

Betti Numbers of Borel Ideals C n , k := number of minimal generators of � x 1 x 2 · · · x n � B with largest variable x k Observation C n , k is the number of Catalan paths from (0 , 0) to ( k − 1 , n − 1). Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 7 / 19

Betti Numbers of Borel Ideals C n , k := number of minimal generators of � x 1 x 2 · · · x n � B with largest variable x k Observation C n , k is the number of Catalan paths from (0 , 0) to ( k − 1 , n − 1). C n , k = n − k + 1 � n + k − 2 � n k − 1 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 7 / 19

Betti Numbers of Borel Ideals Theorem (Francisco, Mermin, and Schweig) The jth Betti number b n , j of � x 1 x 2 · · · x n � B is the number of ordered pairs ( m , α ) such that m is a minimal generator and α is a square free monomial of degree j whose largest variable is less than the largest variable of m. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 8 / 19

Betti Numbers of Borel Ideals Theorem (Francisco, Mermin, and Schweig) The jth Betti number b n , j of � x 1 x 2 · · · x n � B is the number of ordered pairs ( m , α ) such that m is a minimal generator and α is a square free monomial of degree j whose largest variable is less than the largest variable of m. Corollary n � k − 1 � � b n , j = C n , k j k =1 Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 8 / 19

Betti Numbers of Borel Ideals Theorem (Francisco, Mermin, and Schweig) The jth Betti number b n , j of � x 1 x 2 · · · x n � B is the number of ordered pairs ( m , α ) such that m is a minimal generator and α is a square free monomial of degree j whose largest variable is less than the largest variable of m. Corollary b n , j = 1 � 2 n �� n + j − 1 � n n − j − 1 j Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 8 / 19

Combinatorics of b n , j : Leaf-Marked Trees Theorem (Francisco, Mermin, and Schweig) b n , j is the number of binary trees with j marked leaves and n unmarked vertices, in which the rightmost leaf is not marked. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 9 / 19

Combinatorics of b n , j : Leaf-Marked Trees Theorem (Francisco, Mermin, and Schweig) b n , j is the number of binary trees with j marked leaves and n unmarked vertices, in which the rightmost leaf is not marked. s s s s s s � � � �❅ ❅ ❅ ❅ � � ❅ ❅ �❅ � ❅ s s s s s s s s s s � � � � ❅ ❅ ❅ ❅ ❅ ❅ � � s s s s s s s s Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 9 / 19

Combinatorics of b n , j : Branch-Marked Trees Theorem (Francisco, Mermin, and Schweig) b n , j is the number of binary trees with j marked vertices with two children and n unmarked vertices. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 10 / 19

Combinatorics of b n , j : Branch-Marked Trees Theorem (Francisco, Mermin, and Schweig) b n , j is the number of binary trees with j marked vertices with two children and n unmarked vertices. s s s s s s � � � �❅ ❅ ❅ ❅ � � ❅ ❅ �❅ � ❅ s s s s s s s s s s � � ❅ � ❅ � ❅ � ❅ ❅ ❅ � s s s s s s s s Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 10 / 19

Combinatorics of b n , j : North-Marked Catalan Paths Theorem (Egge, Rubin) b n , j is the number of Catalan paths with j marked North steps, none touching y = x, and n − j unmarked North steps. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 11 / 19

Combinatorics of b n , j : North-Marked Catalan Paths Theorem (Egge, Rubin) b n , j is the number of Catalan paths with j marked North steps, none touching y = x, and n − j unmarked North steps. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 11 / 19

Combinatorics of b n , j : 132-Avoiding Permutations Theorem (Egge) b n , j is the number of 132-avoiding permutations with n unbarred entries, j barred entries, 1 is not barred, every barred entry is a local minimum. Eric S. Egge (Carleton College) Catalan Combinatorics of Borel Ideals September 21, 2014 12 / 19