R q = 1 : rook placements S n = { permutations w of { 1 , 2 , . . . , - PowerPoint PPT Presentation

Counting matrices over finite fields with zeroes on Rothe diagrams Alejandro Morales (LaCIM, Universit´ e du Qu´ ebec ` a Montr´ eal) CanaDAM 2013, Minisymposia enumerative combinatorics June 10, 2013 joint work with Aaron Klein (Brookline high school ! MIT) Joel B. Lewis (Minnesota) R

q = 1 : rook placements S n = { permutations w of { 1 , 2 , . . . , n }} Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n }

q = 1 : rook placements S n = { permutations w of { 1 , 2 , . . . , n }} Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n } p ( n, S ) = # { permutations w = w 1 · · · w n | w i 6 = j if ( i, j ) 2 S } = # { placements of n non-attacking rooks on S } = # { n ⇥ n permutation matrices P | P i,j = 0 if ( i, j ) 2 S }

q = 1 : rook placements S n = { permutations w of { 1 , 2 , . . . , n }} Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n } p ( n, S ) = # { permutations w = w 1 · · · w n | w i 6 = j if ( i, j ) 2 S } = # { placements of n non-attacking rooks on S } = # { n ⇥ n permutation matrices P | P i,j = 0 if ( i, j ) 2 S } 0 0 0 1 R 0 0 1 0 R w = 4312 1 0 0 0 R 0 1 0 0 R

q = 1 : rook placements S n = { permutations w of { 1 , 2 , . . . , n }} Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n } p ( n, S ) = # { permutations w = w 1 · · · w n | w i 6 = j if ( i, j ) 2 S } = # { placements of n non-attacking rooks on S } = # { n ⇥ n permutation matrices P | P i,j = 0 if ( i, j ) 2 S } Examples Ferrers board: Diagram of w: S = D w S = F λ 0 0 0 0 0 0 0 λ = (1 , 2 , 4 , 4) λ = (1 , 2 , 4 , 4) w = 2143

q : matrices over F q with restricted support GL ( n, q ) = { n ⇥ n invertible matrices over F q } F q finite field q = p s elements Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n } m q ( n, S ) = # { matrices A 2 GL ( n, q ) | A ij = 0 if ( i, j ) 2 S } q = 3 0 0 2 0 0 1 1 0 1 2 1 1 0 2 0 1 2 0

q : matrices over F q with restricted support GL ( n, q ) = { n ⇥ n invertible matrices over F q } F q finite field q = p s elements Let S ✓ { 1 , 2 , . . . , n } ⇥ { 1 , 2 , . . . , n } m q ( n, S ) = # { matrices A 2 GL ( n, q ) | A ij = 0 if ( i, j ) 2 S } q = 3 0 0 2 0 0 1 1 0 1 2 1 1 0 2 0 1 2 0 Examples Ferrers board: Diagram of w: S = D w S = F λ 0 0 0 0 0 0 0 λ = (1 , 2 , 4 , 4) w = 2143

q : matrices over F q with restricted support

Examples p ( n, S ) and m q ( n, S ) p ( n, S ) = # { placements of n non-attacking rooks on S } S = F λ n Y 1 choices p ( n, F λ ) = ( λ i � i + 1) 0 0 0 2 � 1 ” i =1 0 0 4 � 2 ” 4 � 3 ” λ = (1 , 2 , 4 , 4) total 2 choices

Examples p ( n, S ) and m q ( n, S ) p ( n, S ) = # { placements of n non-attacking rooks on S } S = F λ n Y 1 choices p ( n, F λ ) = ( λ i � i + 1) 0 0 0 2 � 1 ” i =1 0 0 4 � 2 ” 4 � 3 ” λ = (1 , 2 , 4 , 4) total 2 choices m q ( n, S ) = # { matrices A 2 GL ( n, q ) | A ij = 0 if ( i, j ) 2 S } q 1 � 1 choices 0 0 0 q 2 � q 1 ” 0 0 q 4 � q 2 ” q 4 � q 3 ” λ = (1 , 2 , 4 , 4)

Examples p ( n, S ) and m q ( n, S ) p ( n, S ) = # { placements of n non-attacking rooks on S } S = F λ n Y 1 choices p ( n, F λ ) = ( λ i � i + 1) 0 0 0 2 � 1 ” i =1 0 0 4 � 2 ” 4 � 3 ” λ = (1 , 2 , 4 , 4) total 2 choices m q ( n, S ) = # { matrices A 2 GL ( n, q ) | A ij = 0 if ( i, j ) 2 S } n q 1 � 1 choices 0 0 0 q λ i � q i � 1 Y m q ( n, F λ ) = q 2 � q 1 ” 0 0 i =1 q 4 � q 2 ” q 4 � q 3 ” λ = (1 , 2 , 4 , 4)

Examples p ( n, S ) and m q ( n, S ) p ( n, S ) = # { placements of n non-attacking rooks on S } S = F λ n Y 1 choices p ( n, F λ ) = ( λ i � i + 1) 0 0 0 2 � 1 ” i =1 0 0 4 � 2 ” 4 � 3 ” λ = (1 , 2 , 4 , 4) total 2 choices m q ( n, S ) = # { matrices A 2 GL ( n, q ) | A ij = 0 if ( i, j ) 2 S } n q 1 � 1 choices 0 0 0 q λ i � q i � 1 Y m q ( n, F λ ) = q 2 � q 1 ” 0 0 i =1 q 4 � q 2 ” n = ( q � 1) n q ( n 2 ) Y q 4 � q 3 [ λ i � i + 1] q ” i =1 λ = (1 , 2 , 4 , 4) where [ k ] q = 1 + q + · · · + q k � 1

Remarks and outline Remarks • p ( n, S ) , m q ( n, S ) invariant under permuting rows and columns of S

Remarks and outline Remarks • p ( n, S ) , m q ( n, S ) invariant under permuting rows and columns of S i) m q ( n, F λ ) / ( q � 1) n is in N [ q ] ,

Remarks and outline Remarks • p ( n, S ) , m q ( n, S ) invariant under permuting rows and columns of S i) m q ( n, F λ ) / ( q � 1) n is in N [ q ] , ( ) ii) p ( n, F λ ) = # , an interval in the Bruhat order

Remarks and outline Remarks • p ( n, S ) , m q ( n, S ) invariant under permuting rows and columns of S i) m q ( n, F λ ) / ( q � 1) n is in N [ q ] , ( ) ii) p ( n, F λ ) = # , an interval in the Bruhat order Outline I Is m q ( n, S ) / ( q � 1) n a polynomial in q ? Is it in N [ q ] ? II When are rook placements on S related to Bruhat intervals?

Is m q ( n, S ) / ( q � 1) n a polynomial in N [ q ] ? Example If S = { (1 , 1) , (2 , 2) , (3 , 3) } , 0 m q (3 , S ) = ( q � 1) 3 ( q 3 + 2 q 2 � q ) 0 0

Is m q ( n, S ) / ( q � 1) n a polynomial? Example (Stembridge 1998) 1 2 3 4 5 7 6 Fano plane PG (2 , 2) S P G (2 , 2)

Is m q ( n, S ) / ( q � 1) n a polynomial? Example (Stembridge 1998) 1 2 3 4 5 7 6 Fano plane PG (2 , 2) S P G (2 , 2) ( q − 1) 7 ( q 14 + · · · − 97 q 9 + · · · + q 3 ) ⇢ if q even , m q (7 , S P G (2 , 2) ) = ( q − 1) 7 ( q 14 + · · · − 98 q 9 + · · · − 6 q 5 ) if q odd .

however, m q ( n, S ) is a q -analogue of p ( n, S ) Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ { 1 , . . . , n } ⇥ { 1 , . . . , n } , m q ( n, S ) ⌘ p ( n, S ) · ( q � 1) n (mod ( q � 1) n +1 ) . – If S = { (1 , 1) , (2 , 2) , (3 , 3) } , m q (3 , S ) / ( q � 1) 3 = q 3 + 2 q 2 � q m q (3 , S ) / ( q � 1) 3 � q =1 = 2 �

however, m q ( n, S ) is a q -analogue of p ( n, S ) Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ { 1 , . . . , n } ⇥ { 1 , . . . , n } , m q ( n, S ) ⌘ p ( n, S ) · ( q � 1) n (mod ( q � 1) n +1 ) . Examples – If S = { (1 , 1) , (2 , 2) , (3 , 3) } , m q (3 , S ) / ( q � 1) 3 = q 3 + 2 q 2 � q m q (3 , S ) / ( q � 1) 3 � q =1 = 2 �

however, m q ( n, S ) is a q -analogue of p ( n, S ) Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ { 1 , . . . , n } ⇥ { 1 , . . . , n } , m q ( n, S ) ⌘ p ( n, S ) · ( q � 1) n (mod ( q � 1) n +1 ) . Examples – If S = { (1 , 1) , (2 , 2) , (3 , 3) } , m q (3 , S ) / ( q � 1) 3 = q 3 + 2 q 2 � q m q (3 , S ) / ( q � 1) 3 � q =1 = 2 � ⇢ 24 if q even , – m q (7 , S P G (2 , 2) ) / ( q � 1) 7 � q =1 = � 24 if q odd .

Outline of talk I Is m q ( n, S ) / ( q � 1) n a polynomial in q ?, Is it in N [ q ] ? ⇥ For which S is it in N [ q ] ? – If S is a Ferrer’s board: F λ ? – If S is a diagram of a permutation: D w ? II When are rook placements on S related to Bruhat intervals?

m q ( n, F λ ) / ( q � 1) n is in N [ q ] Theorem (Haglund 1998) X m q ( n, F λ ) = ( q � 1) n q boxNE( w ) . w 2 p ( n,F λ ) boxes in F λ strictly North and East of rooks

m q ( n, F λ ) / ( q � 1) n is in N [ q ] Theorem (Haglund 1998) X m q ( n, F λ ) = ( q � 1) n q boxNE( w ) . w 2 p ( n,F λ ) boxes in F λ strictly North and East of rooks Example λ = (1 , 2 , 4 , 4)

m q ( n, F λ ) / ( q � 1) n is in N [ q ] Theorem (Haglund 1998) X m q ( n, F λ ) = ( q � 1) n q boxNE( w ) . w 2 p ( n,F λ ) boxes in F λ strictly North and East of rooks Example λ = (1 , 2 , 4 , 4) + q 6 q 7 m q (4 , F 1244 ) = ( q � 1) 4 � �

m q ( n, F λ ) / ( q � 1) n is in N [ q ] Theorem (Haglund 1998) X m q ( n, F λ ) = ( q � 1) n q boxNE( w ) . w 2 p ( n,F λ ) boxes in F λ strictly North and East of rooks Example λ = (1 , 2 , 4 , 4) + q 6 q 7 m q (4 , F 1244 ) = ( q � 1) 4 � � Proposition (Klein-Lewis-M 2012) Haglund’s result extends to skew Ferrers shapes S = F λ /µ

m q ( n, F λ ) / ( q � 1) n is in N [ q ] Theorem (Haglund 1998) X m q ( n, F λ ) = ( q � 1) n q boxNE( w ) . w 2 p ( n,F λ ) boxes in F λ strictly North and East of rooks Example λ = (1 , 2 , 4 , 4) µ = (1 , 2) + q 3 q 4 m q (4 , F 1244 / 12 ) = ( q � 1) 4 � � Proposition (Klein-Lewis-M 2012) Haglund’s result extends to skew Ferrers shapes S = F λ /µ

Outline of talk I Is m q ( n, S ) / ( q � 1) n a polynomial in q ?, Is it in N [ q ] ? ⇥ For which S is it in N [ q ] ? X – If S is a Ferrer’s board: F λ ? – If S is a diagram of a permutation: D w ? II When are rook placements on S related to Bruhat intervals?

m q ( n, S ) when S is the diagram of a permutation: D w Examples 0 0 0 0 0 0 w = 2431 w = 2143 D 2431 = { (1 , 1) , (2 , 1) , (3 , 1) , (2 , 3) } D 2143 = { (1 , 1) , (3 , 3) }

m q ( n, S ) when S is the diagram of a permutation: D w Examples 0 0 0 0 0 0 w = 2431 w = 2143 D 2431 = { (1 , 1) , (2 , 1) , (3 , 1) , (2 , 3) } D 2143 = { (1 , 1) , (3 , 3) } Conjecture (Klein-Lewis-M 2012, true n  8 ) m q ( n, D w ) / ( q � 1) n is in N [ q ] . For all w 2 S n ,