B n 1 -orbits on the flag variety II Mark Colarusso, University of - PDF document

B n − 1 -orbits on the flag variety II Mark Colarusso, University of South Alabama and Sam Evens, University of Notre Dame November 2, 2019

General Notation In this talk G i = GL ( i ) for i = 1 , . . . , n . We have chain of inclusions G 1 ⊂ G 2 ⊂ . . . ⊂ G i ⊂ G i +1 ⊂ G. Let G n − 1 = K and G n = G B i ⊂ G i =standard upper triangular Borel sub- group. Q K = a K -orbit on G/B . Q = a B n − 1 -orbit on G/B . 1

Overview of Talk: 1) Discuss combinatorial model involving par- titions for B n − 1 \ G/B. Get e.g.f and explicit formula for | B n − 1 \ G/B | . 2) Use (1) to develop explicit set of represen- tatives for B n − 1 -orbits in terms of flags. Can use these representatives to study the weak order. 3) In progress: Develop second combinatorial model involving Dyck paths for B n − 1 -orbits us- ing (2) and refined geometric data from first talk. 2

First combinatorial model of B n − 1 \ G/B . B n − 1 -orbits on G/B are modeled by PILS. PILS = partitions into lists. A list of the set { 1 , . . . , n } is any ordered non- empty subset. Notation: σ = ( a 1 a 2 . . . a k ). A PIL of the set { 1 , . . . , n } is any partition of the set { 1 , . . . , n } into lists. Notation: Σ = { σ 1 , . . . , σ ℓ } . 3

Examples: For n = 2, there are 3 PILS: { (12) } , { (21) } , { (1) , (2) } . For n = 3, there are 13 PILS. 6 of form { ( i 1 i 2 i 3 ) } , 6 of form { ( i 1 i 2 ) , ( i 3 ) } , { (1) , (2) , (3) } . For n = 4, there are 73 PILS, and for n = 5, there are 501 PILS. 4

Combinatorial Theorem: There is a one-to- one correspondence: PILS ⇔ B n − 1 \ G/B. Remarks: There is a similar correspondence in the or- thogonal case involving partitions into signed lists satisfying certain parity conditions depend- ing on whether G = SO ( n ) is of type B or type D . 5

Exponential Generating Function for | B n − 1 \ G/B | . Corollary : Let a n = | B n − 1 \ G/B | . Then (1) The e.g.f for the sequence { a n } ∞ n =1 is x e 1 − x . (2) � n − 1 � n − 1 � i a n = n ! ( i + 1)! . i =0 The correspondence between PILS and B n − 1 - orbits on G/B is proven using the fibre bundle structure of these orbits discussed in the last talk and structure of K -orbits on G/B . 6

Notation for Flags and Partial Flags: Flag: F := V 1 ⊂ V 2 ⊂ . . . ⊂ V i ⊂ . . . ⊂ V n = C n . with dim V i = i . Notation: Suppose V i = span { v 1 , . . . , v i } , then write F := v 1 ⊂ v 2 ⊂ . . . ⊂ v i ⊂ . . . ⊂ v n . 7

Partial Flag: P = V 1 ⊂ V 2 ⊂ . . . ⊂ V j ⊂ . . . ⊂ V k = C n where dim V j = i j . Notation: Suppose V j = span { v 1 , . . . , v i j } for j = 1 , . . . , k . P = { v 1 , . . . , v i 1 } ⊂ { v i 1 +1 , . . . , v i 2 } ⊂ . . . ⊂ ⊂ { v i 1 + ... + i k − 1 +1 , . . . , v i k } 8

Recall: Q a B n − 1 -orbit, Q ⊂ Q K = K · ˜ b : ∃ θ -stable parabolic subgroup, ˜ B ⊂ P ⊂ G such that π : G/B → G/P endows Q with structure of fibre bundle: “ Q = Q P × Q ℓ ”. BASE: Q P = a B n − 1 -orbit on partial flag va- riety K/ ( K ∩ P ) = π Q K of K , FIBRE: Q ℓ = a B ℓ − 1 -orbit on G ℓ /B ℓ , ℓ ≤ n − 1. 9

Description of K -oribts Notation: { e 1 , . . . , e n } = standard basis for C n . C n − 1 = span { e 1 , . . . , e n − 1 } . For i = 1 , . . . , n − 1 , ˆ e i = e i + e n . n -closed K -orbits: Q i,c , i = 1 , . . . , n . In this case, Q i,c ∼ = K/B n − 1 . Non-closed orbits: Q i,j = K · F i,j , 1 ≤ i < j ≤ n F i,j := ˆ e 1 ⊂ . . . ⊂ e i ⊂ . . . ⊂ e j − 1 ⊂ e n ⊂ e j ⊂ ���� ���� j i . . . e n − 1 . 10

Suppose: Q ⊂ K · F i,j Let P i,j ⊂ G stabilize partial flag: P i,j = e 1 ⊂ . . . ⊂ { e i , . . . , e j − 1 , e n } ⊂ e j ⊂ . . . ⊂ e n − 1 . Note: F i,j ⊂ P i,j . K/ ( K ∩ P i,j ) = K · ( P i,j ∩ C n − 1 ) Q P i,j = B n − 1 -orbit on K/ ( K ∩ P i,j ). Q ℓ ↔ B ℓ − 1 -orbit on G ℓ /B ℓ , where ℓ = j − i. 11

It follows that: Q P i,j is determined by an s ∈ S n − 1 with s ( i ) < s ( i + 1) < . . . < s ( j ). Q P ↔ ( s (1) . . . s ( i − 1) n s ( j ) . . . s ( n − 1)) . ���� i By induction Q ℓ ↔ Σ ℓ where Σ ℓ is a unique PIL of the set { s ( i ) , . . . , s ( j − 1) } . Conclusion: Q ↔ { ( s (1) . . . s ( i − 1) n s ( j ) . . . s ( n − 1)) , Σ ℓ } . ���� i 12

Example: V = C 4 and G = GL (4). Consider B 3 -orbit: Q = B 3 · (ˆ e 3 ⊂ ˆ e 1 ⊂ e 4 ⊂ e 2 ). Q ⊂ Q 1 , 3 = K · (ˆ e 1 ⊂ e 2 ⊂ e 4 ⊂ e 3 . ); ℓ = 2 G/P = G · ( { e 1 , e 2 , e 4 } ⊂ e 3 ) = Gr(3 , C 4 ) . K/ ( K ∩ P 1 , 3 ) = K · ( { e 1 , e 2 } ⊂ e 3 ) = Gr(2 , C 3 ) . Q P 1 , 3 = B 3 · ( { e 1 , e 3 } ⊂ e 2 ) ↔ s = s ǫ 2 − ǫ 3 . Q P ↔ (42) . Q 2 ↔ is open B 1 = C × -orbit on flag variety of C 2 = span { e 1 , e 3 } . Q 2 ↔ (1)(3). Q = Q P × Q 2 ↔ { (42) , (1)(3) } . 13

Minimal Elements in the weak order. Ultimate Goal: Understand strong order (i.e. closure relations) B n − 1 \ G/B As a step in this direction, we prove: Theorem: Any B n − 1 -orbit Q which is minimal in the weak order is closed. Remark: This is not true for orbits of a gen- eral spherical H on G/B . To prove this, we use the theory of PILS to develop a canonical set of representatives for B n − 1 \ G/B. We can then use these representative to under- stand the Richardson-Springer monoid action. 14

Standard Form for a flag in G/B Definition: A flag in C n F := v 1 ⊂ . . . ⊂ v j ⊂ . . . ⊂ v n . with v j = ˆ e i j or v j = e i j is in standard form if (1) If v k = e n , then v j = e i j for j > k . (2) If k < j and v k = ˆ e i k and v j = ˆ e i j , then i k > i j . 15

Example: For V = C 5 , the flag e 1 ⊂ ˆ e 4 ⊂ ˆ e 3 ⊂ e 5 ⊂ e 2 . is in standard form. But the flag e 1 ⊂ ˆ e 3 ⊂ ˆ e 4 ⊂ e 5 ⊂ ˆ e 2 is not. 16

Combinatorial Theorem 2: There is a 1-1 correspondence { PILS } ← → { Flags in standard form } . (This is proven purely combinatorially; no ge- ometry involved.) Using above theorem and an inductive argu- ment using B n − 1 -orbits on Gr( ℓ, C n ), we can show: Prop: Every B n − 1 -orbit Q contains a unique flag in standard form F . 17

To prove theorem about the weak order: Q ⊂ Q K , Q = Q P × Q ℓ . RS Monoid action is compatible Geometry: with fibre bundle structure ⇒ Q c ≤ w Q , Q c = B n − 1 -orbit closed in Q K . An easy computation with Combinatorics: standard forms and PILS and induction shows that Q ′ c ≤ w Q c , Q ′ c closed in G/B . 18

A second more refined combinatorial model: Labelled Dyck Paths Problem: Given two flags F and F ′ in stan- dard form it’s hard to tell if the corresponding B n − 1 -orbits are related in weak (strong) order. Connect the combinatorics of the Solution: standard form to the geometry of the fibre bundle structure of B n − 1 -orbits to develop a more sophisticated combinatorial model in terms of labelled Dyck paths. 19

First step: Iterate fibre bundle characteriza- tion of a B n − 1 -orbit Q to assign to Q the fol- lowing data: Q → [( d 0 , Q i 0 ,j 0 , s 0 ) , ( d 1 , Q i 1 ,j 1 , s 1 ) , . . . , ( d k , Q i k ,j k , s k )] . with d 0 = n > d 1 > d 2 > . . . > d k . Q i ℓ ,j ℓ = a G d ℓ − 1 -orbit on G d ℓ /B d ℓ s ℓ = shortest coset rep of s ℓ S d ℓ +1 in S d ℓ − 1 /S d ℓ +1 . Data corresponds to a labelled Key Idea: Dyck path of length 2n. Labels determined by “Weyl group data” ( s 0 , . . . , s k ). Path determined by “ K -orbit data” ( Q i 0 ,j 0 , . . . , Q i k ,j k ) . 20

How does this work? Take Q a B n − 1 -orbit. Q ⊂ Q K If Q K = Q c is closed, then Q is a B n − 1 -orbit on K/B n − 1 and therefore given by s 0 ∈ S n − 1 and the iteration stops. If Q K = Q i,j then let i 0 := i , j 0 := j and d 1 = j 0 − i 0 . Then Q = Q P i 0 ,j 0 × Q d 1 , Q P i 0 ,j 0 = a B n − 1 -orbit on K/ ( K ∩ P i 0 ,j 0 ) and so determined by s 0 ∈ S n − 1 /S d 1 . Q d 1 = a B d 1 − 1 -orbit on G d 1 /B d 1 . 21

THUS, Q d 1 ⊂ Q i 1 ,j 1 , a G d 1 − 1 -orbit on G d 1 /B d 1 . Let d 2 = j 1 − i 1 . So Q d 1 = Q P i 1 ,j 1 × Q d 2 . CONTINUE until reach a G d k − 1 -orbit Q i k ,j k which is closed in G d k /B d k .

Current State of Affairs: We have an easy algorithm to read off “ K - orbit data” from unique flag in standard from in B n − 1 -orbit Q and produce unlabelled Dyck path. In Progress : Develop algorithm to read off “Weyl group data” from standard form. Conjectures: 1) Weyl group data+ K -orbit data determines the orbit Q completely. 2) Weak (strong) order on B n − 1 \ G/B can be understood in terms on a natural ordering on labelled Dyck paths. 22