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

Vexillary signed permutations and essential sets

Dave Anderson IMJ − > IMPA ICERM March 5, 2013

Diagrams and essential sets

1 2 3 5

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 1 2 3 ¯ 3 ¯ 2 ¯ 1 0 1 2 3

1 1 2 3

Diagrams and essential sets

w = ¯ 2 3 1 ¯ 3 ¯ 2 ¯ 1 1 2 3 ¯ 3 ¯ 2 ¯ 1

× ×

ℓ(w) = 3

1 1

Ess(w) = { (1, 3, ¯ 1) , (1, 1, 2) }

Diagrams and essential sets

¯ 5 ¯ 4 ¯ 3 ¯ 2 ¯ 1 1 2 3 4 5 ¯ 5 ¯ 4 ¯ 3 ¯ 2 ¯ 1

× × × × × ×

3 4 2 1

w = 5 ¯ 2 ¯ 4 1 3 ℓ(w) = 11 Ess(w) = { (1, 3, 4) , (3, 2, ¯ 1) , (4, 2, ¯ 3) }

vexillary: (fr. Latin vexillum, pl. vexilla, flag).

Vexillary signed permutations

τ = ( 2 4 5, 4 3 1, 4 1 1 ) ¯ 6 ¯ 5 ¯ 4 ¯ 3 ¯ 2 ¯ 1 1 2 3 4 5 6

2 4 5

¯ 6 ¯ 5 ¯ 4 ¯ 3 ¯ 2 ¯ 1 1 2 3 4 5 6

2 4 5

× × × × × × × × × × × × × × ×

λ(τ) = (8, 7, 4, 3, 1) w(τ) = ¯ 3 6 ¯ 2 ¯ 5 ¯ 4 ¯ 1 ℓ(w) = 23 Ess(w) = { (2, 4, 5) , (4, 3, 1) , (5, 1, 1) }

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 [Ωw] = 1 2ks Pfaff(M), where M is the skew-symmetric matrix having (k, ℓ)-entry c(k)λkc(ℓ)λℓ + 2

λℓ

• j=1

(−1)jc(k)λk+jc(ℓ)λℓ−j, with c(ki) = c(V − Epi − Fqi ) and c(k) = c(ki) for ki−1 < k ≤ ki.