Vexillary signed permutations and essential sets Dave Anderson IMJ - PowerPoint PPT Presentation

Nov 12, 2023 •338 likes •430 views

Vexillary signed permutations and essential sets Dave Anderson IMJ > IMPA ICERM March 5, 2013 Diagrams and essential sets 5 3 2 1 ( v ) = 12 ( v ) = (5 , 3 , 2 , 1 , 1) Diagrams and essential sets v = 1 3 2 0 2 3 1

Vexillary signed permutations and essential sets Dave Anderson IMJ − > IMPA ICERM March 5, 2013
Diagrams and essential sets 5 3 2 1 ℓ ( v ) = 12 λ ( v ) = (5 , 3 , 2 , 1 , 1)
Diagrams and essential sets v = ¯ 1 ¯ 3 2 0 ¯ 2 3 1 ¯ 3 ¯ 2 ¯ 1 0 1 2 3 ¯ 3 ¯ 2 1 3 ¯ 1 0 1 1 2 2 3
Diagrams and essential sets w = ¯ 2 3 1 3 ¯ ¯ 2 ¯ 1 ¯ 3 ¯ 2 × 1 ¯ 1 ℓ ( w ) = 3 0 1 × 1 E ss ( w ) = { (1 , 3 , ¯ 1) , (1 , 1 , 2) } 2 3
Diagrams and essential sets ¯ 5 ¯ 4 ¯ 3 ¯ 2 ¯ 1 ¯ 5 ¯ 4 × 4 × ¯ 3 ¯ w = 5 ¯ 2 ¯ × 2 3 4 1 3 ¯ 1 ℓ ( w ) = 11 0 1 × × 2 E ss ( w ) = { (1 , 3 , 4) , (3 , 2 , ¯ 1) , (4 , 2 , ¯ 3) } 2 3 × 1 4 5
vexillary : (fr. Latin vexillum , pl. vexilla , flag).
Vexillary signed permutations τ = ( 2 4 5 , 4 3 1 , 4 1 1 ) ¯ ¯ 6 6 ¯ ¯ 5 5 × × × ¯ ¯ λ ( τ ) = (8 , 7 , 4 , 3 , 1) × × × × 4 4 ¯ ¯ 3 3 × w ( τ ) = ¯ 3 6 ¯ 2 ¯ 5 ¯ 4 ¯ ¯ ¯ 1 × × 2 2 ¯ ¯ 1 1 × × × × × ℓ ( w ) = 23 0 0 4 4 5 5 1 1 2 2 E ss ( w ) = { (2 , 4 , 5) , (4 , 3 , 1) , (5 , 1 , 1) } 3 3 2 2 4 4 5 5 6 6
Formulas for orthogonal flag varieties / degeneracy loci Theorem (A.-Fulton) For a vexillary signed permutation w = w ( τ ) and strict partition λ = λ ( τ ) we have [Ω w ] = 1 2 k s Pfaff ( M ) , where M is the skew-symmetric matrix having ( k , ℓ ) -entry λ ℓ � ( − 1) j c ( k ) λ k + j c ( ℓ ) λ ℓ − j , c ( k ) λ k c ( ℓ ) λ ℓ + 2 j =1 with c ( k i ) = c ( V − E p i − F q i ) and c ( k ) = c ( k i ) for k i − 1 < k ≤ k i .

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