Amenable signed permutations Harry Tamvakis University of Maryland November 2, 2019 Harry Tamvakis Amenable signed permutations
Schubert Calculus Giambelli Problem: Find polynomials that represent the cohomology classes of the Schubert varieties X w in G/P . Relative Version (Fulton-Pragacz, 1996): Find Chern class polynomials which represent the cohomology classes of degeneracy loci X w of vector bundles, when G is a classical Lie group. Harry Tamvakis Amenable signed permutations
Schubert Calculus Giambelli Problem: Find polynomials that represent the cohomology classes of the Schubert varieties X w in G/P . Relative Version (Fulton-Pragacz, 1996): Find Chern class polynomials which represent the cohomology classes of degeneracy loci X w of vector bundles, when G is a classical Lie group. Harry Tamvakis Amenable signed permutations
Intrinsic formulas General w (T., 2009): ∃ polynomial formulas for [ X w ] which are native to G/P , for all w ∈ W P . New, intrinsic point of view in Schubert calculus. In special cases, ∃ alternative intrinsic formulas: In Lie type A For ̟ ∈ S n Grassmannian : Thom-Porteous (1970); Kempf-Laksov (1974). More generally, for ̟ ∈ S n vexillary : Lascoux-Sch¨ utzenberger (1982), et. al. Harry Tamvakis Amenable signed permutations
Intrinsic formulas General w (T., 2009): ∃ polynomial formulas for [ X w ] which are native to G/P , for all w ∈ W P . New, intrinsic point of view in Schubert calculus. In special cases, ∃ alternative intrinsic formulas: In Lie type A For ̟ ∈ S n Grassmannian : Thom-Porteous (1970); Kempf-Laksov (1974). More generally, for ̟ ∈ S n vexillary : Lascoux-Sch¨ utzenberger (1982), et. al. Harry Tamvakis Amenable signed permutations
Intrinsic formulas General w (T., 2009): ∃ polynomial formulas for [ X w ] which are native to G/P , for all w ∈ W P . New, intrinsic point of view in Schubert calculus. In special cases, ∃ alternative intrinsic formulas: In Lie type A For ̟ ∈ S n Grassmannian : Thom-Porteous (1970); Kempf-Laksov (1974). More generally, for ̟ ∈ S n vexillary : Lascoux-Sch¨ utzenberger (1982), et. al. Harry Tamvakis Amenable signed permutations
Permutations ̟ = ( ̟ 1 , . . . , ̟ n ) ∈ S n , where ̟ i = ̟ ( i ) . Code γ = γ ( ̟ ) with γ i := # { j > i | ̟ j < ̟ i } . Shape λ = λ ( ̟ ) obtained by reordering the γ i . Example w = (2 , 1 , 5 , 4 , 3) , γ = (1 , 0 , 2 , 1 , 0) , λ = (2 , 1 , 1) . S n = � s 1 , . . . , s n − 1 � . ̟ has a right/left descent at i if ℓ ( ̟s i ) < ℓ ( ̟ ) (resp. ℓ ( s i ̟ ) < ℓ ( ̟ ) ). ̟ is Grassmannian if ℓ ( ̟s i ) > ℓ ( ̟ ) , ∀ i � = k . ̟ is vexillary if λ ( ̟ − 1 ) = λ ( ̟ ) ′ . Harry Tamvakis Amenable signed permutations
Permutations ̟ = ( ̟ 1 , . . . , ̟ n ) ∈ S n , where ̟ i = ̟ ( i ) . Code γ = γ ( ̟ ) with γ i := # { j > i | ̟ j < ̟ i } . Shape λ = λ ( ̟ ) obtained by reordering the γ i . Example w = (2 , 1 , 5 , 4 , 3) , γ = (1 , 0 , 2 , 1 , 0) , λ = (2 , 1 , 1) . S n = � s 1 , . . . , s n − 1 � . ̟ has a right/left descent at i if ℓ ( ̟s i ) < ℓ ( ̟ ) (resp. ℓ ( s i ̟ ) < ℓ ( ̟ ) ). ̟ is Grassmannian if ℓ ( ̟s i ) > ℓ ( ̟ ) , ∀ i � = k . ̟ is vexillary if λ ( ̟ − 1 ) = λ ( ̟ ) ′ . Harry Tamvakis Amenable signed permutations
Type A degeneracy loci E → X is a vector bundle of rank n and ̟ ∈ S n . 0 � E 1 � · · · � E n = E 0 � F 1 � · · · � F n = E . X ̟ := { x ∈ X | dim( E r ( x ) ∩ F s ( x )) ≥ d ̟ ( r, s ) , ∀ r, s } where d ̟ ( r, s ) := # { i ≤ r | ̟ i > n − s } . Assume: X ̟ has pure codimension ℓ ( ̟ ) in X . Harry Tamvakis Amenable signed permutations
Type A degeneracy loci E → X is a vector bundle of rank n and ̟ ∈ S n . 0 � E 1 � · · · � E n = E 0 � F 1 � · · · � F n = E . X ̟ := { x ∈ X | dim( E r ( x ) ∩ F s ( x )) ≥ d ̟ ( r, s ) , ∀ r, s } where d ̟ ( r, s ) := # { i ≤ r | ̟ i > n − s } . Assume: X ̟ has pure codimension ℓ ( ̟ ) in X . Harry Tamvakis Amenable signed permutations
Type A degeneracy loci E → X is a vector bundle of rank n and ̟ ∈ S n . 0 � E 1 � · · · � E n = E 0 � F 1 � · · · � F n = E . X ̟ := { x ∈ X | dim( E r ( x ) ∩ F s ( x )) ≥ d ̟ ( r, s ) , ∀ r, s } where d ̟ ( r, s ) := # { i ≤ r | ̟ i > n − s } . Assume: X ̟ has pure codimension ℓ ( ̟ ) in X . Harry Tamvakis Amenable signed permutations
Type A vexillary degeneracy loci Theorem (L.-S., Wachs, Macdonald, Fulton (1992)) Suppose that ̟ is a vexillary permutation of shape λ = ( λ 1 , . . . , λ ℓ ) . Then there exist sequences f = ( f 1 ≤ · · · ≤ f ℓ ) and g = ( g 1 ≥ · · · ≥ g ℓ ) consisting of right and left descents of ̟ , respectively, such that [ X ̟ ] = det( c λ i + j − i ( E − E f i − F n − g i )) 1 ≤ i,j ≤ ℓ holds in H ∗ ( X ) . Here c p ( E − E ′ − E ′′ ) is defined by c ( E − E ′ − E ′′ ) := c ( E ) c ( E ′ ) − 1 c ( E ′′ ) − 1 . Harry Tamvakis Amenable signed permutations
Raising Operator Form � [ X ̟ ] = (1 − R ij ) c λ ( E − E f − F n − g ) (1) i<j For i < j and α = ( α 1 , α 2 , . . . ) an integer sequence, R ij α := ( α 1 , . . . , α i + 1 , . . . , α j − 1 , . . . ) If c α := c α 1 c α 2 · · · , then R ij c α := c R ij α . (1) is the image of � i<j (1 − R ij ) c λ under the Z -linear map sending c α to � i c α i ( E − E f i − F n − g i ) , for every integer sequence α . Harry Tamvakis Amenable signed permutations
Raising Operator Form � [ X ̟ ] = (1 − R ij ) c λ ( E − E f − F n − g ) (1) i<j For i < j and α = ( α 1 , α 2 , . . . ) an integer sequence, R ij α := ( α 1 , . . . , α i + 1 , . . . , α j − 1 , . . . ) If c α := c α 1 c α 2 · · · , then R ij c α := c R ij α . (1) is the image of � i<j (1 − R ij ) c λ under the Z -linear map sending c α to � i c α i ( E − E f i − F n − g i ) , for every integer sequence α . Harry Tamvakis Amenable signed permutations
Amenable signed permutations Question: What is the analogue of ‘vexillary’ in the other classical Lie types B, C, and D? Prior attempts at a definition of vexillary signed permutations: Billey-Lam (1998) & Anderson-Fulton (2012). Both of these definitions are wrong . Reason: According to either of them, the Grassmannian signed permutations are not all vexillary. Harry Tamvakis Amenable signed permutations
Amenable signed permutations Question: What is the analogue of ‘vexillary’ in the other classical Lie types B, C, and D? Prior attempts at a definition of vexillary signed permutations: Billey-Lam (1998) & Anderson-Fulton (2012). Both of these definitions are wrong . Reason: According to either of them, the Grassmannian signed permutations are not all vexillary. Harry Tamvakis Amenable signed permutations
Amenable signed permutations Question: What is the analogue of ‘vexillary’ in the other classical Lie types B, C, and D? Prior attempts at a definition of vexillary signed permutations: Billey-Lam (1998) & Anderson-Fulton (2012). Both of these definitions are wrong . Reason: According to either of them, the Grassmannian signed permutations are not all vexillary. Harry Tamvakis Amenable signed permutations
Amenable signed permutations Question: What is the analogue of ‘vexillary’ in the other classical Lie types B, C, and D? Prior attempts at a definition of vexillary signed permutations: Billey-Lam (1998) & Anderson-Fulton (2012). Both of these definitions are wrong . Reason: According to either of them, the Grassmannian signed permutations are not all vexillary. Harry Tamvakis Amenable signed permutations
Amenable signed permutations: Type C w = ( w 1 , . . . , w n ) ∈ W n = � s 0 , s 1 , . . . , s n − 1 � . A-code γ = γ ( w ) with γ i := # { j > i | w j < w i } . Definition (T., 2019) w is leading if the A-code � γ of the extended sequence (0 , w 1 , . . . , w n ) is unimodal. { Leading elements } ⊃ { Grassmannian elements } Definition (T., 2019) w is amenable if w is a modification of a leading element. Harry Tamvakis Amenable signed permutations
Amenable signed permutations: Type C w = ( w 1 , . . . , w n ) ∈ W n = � s 0 , s 1 , . . . , s n − 1 � . A-code γ = γ ( w ) with γ i := # { j > i | w j < w i } . Definition (T., 2019) w is leading if the A-code � γ of the extended sequence (0 , w 1 , . . . , w n ) is unimodal. { Leading elements } ⊃ { Grassmannian elements } Definition (T., 2019) w is amenable if w is a modification of a leading element. Harry Tamvakis Amenable signed permutations
Amenable signed permutations: Type C w = ( w 1 , . . . , w n ) ∈ W n = � s 0 , s 1 , . . . , s n − 1 � . A-code γ = γ ( w ) with γ i := # { j > i | w j < w i } . Definition (T., 2019) w is leading if the A-code � γ of the extended sequence (0 , w 1 , . . . , w n ) is unimodal. { Leading elements } ⊃ { Grassmannian elements } Definition (T., 2019) w is amenable if w is a modification of a leading element. Harry Tamvakis Amenable signed permutations
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