Spanning line configurations Brendan Pawlowski (joint with Brendon - PowerPoint PPT Presentation

Spanning line configurations Brendan Pawlowski (joint with Brendon Rhoades) University of Southern California July 3, 2019

Coinvariant algebras Type A n − 1 coinvariant algebra: R n := Z [ x ] / ( e 1 , . . . , e n ) , e d = sum of degree d squarefree monomials in x := { x 1 , . . . , x n } . ◮ R n is a free Z -module of rank n ! ◮ As S n -modules, R n ⊗ Q ≃ Q [ S n ]. ◮ R n ≃ H ∗ (Fl( n )), the integral cohomology ring of the complete flag variety Fl( n ). ◮ Basis of Schubert polynomials coming from H ∗ (Fl( n ) , Z ) picture.

Generalized coinvariant algebras Haglund, Rhoades, and Shimozono: for 1 ≤ k ≤ n , define R n , k := Z [ x ] / ( x k 1 , . . . , x k n , e n − k +1 , . . . , e n ) . Example ◮ R n , n = Z [ x ] / ( e 1 , . . . , e n , x n 1 , . . . , x n n ) = R n free of rank n !. ◮ R n , 1 = Z [ x ] / ( e n , x 1 1 , . . . , x 1 n ) = Z free of rank 1. Let OSP n , k := { partitions of { 1 , . . . , n } into k ordered blocks } . ◮ R n , k is free of rank # OSP n , k . ◮ As S n -modules, R n , k ⊗ Q ≃ Q [ OSP n , k ].

Schubert basis? ◮ Haglund, Rhoades, and Shimozono give several bases of R n , k , but not a Schubert-like basis. ◮ Is R n , k ≃ H ∗ ( X ) for some nice X ? ◮ R n , k isn’t usually rank-symmetric, so such an X can’t be a compact smooth manifold!

Why H ∗ (Fl( n )) ≃ R n ? Recall Fl( n ) is the space of chains of linear subspaces F 1 � F 2 � · · · � F n = C n . Define X n = { ( ℓ 1 , . . . , ℓ n ) ∈ ( P n − 1 ) n : ℓ 1 , . . . , ℓ n span C n } , e.g. � 1 � � 1 � 1 1 ∈ X 2 but not columns of . 0 1 0 0 ◮ Have a map X n → Fl( n ): ( ℓ 1 , . . . , ℓ n ) �→ ℓ 1 ⊆ ℓ 1 ⊕ ℓ 2 ⊆ · · · ◮ This is a homotopy equivalence! = ⇒ H ∗ ( Fl ( n )) ≃ H ∗ ( X n ). ◮ Con: X n isn’t a compact manifold / projective variety. ◮ Pro: S n acts on X n by permuting lines, which induces the S n -action on R n ; no such S n -action on Fl( n ) is evident.

Why H ∗ ( X n ) ≃ R n ? ◮ A rank m vector bundle E over a space X assigns an m -dimensional (complex) vector space to each point of X . ◮ Example: a trivial vector bundle assigns the same vector space to each point. ◮ Example: the rank 1 tautological bundle L i assigns to the point ( ℓ 1 , . . . , ℓ n ) ∈ X n the vector space ℓ i .

Why H ∗ ( X n ) ≃ R n ? Given E a vector bundle over X : ◮ Have Chern classes c d ( E ) ∈ H 2 d ( X ) and the total Chern class d ≥ 0 c d ( E ) = 1 + c 1 ( E ) + · · · + c rank( E ) ( E ) ∈ H ∗ ( X ). c ( E ) = � ◮ If E is trivial then c ( E ) = 1. ◮ Whitney sum formula: c ( E ⊕ F ) = c ( E ) c ( F ) and c ( E / F ) = c ( E ) / c ( F ). Recall the (rank 1) tautological bundles L 1 , . . . , L n over X n . ◮ Set x i = c 1 ( L i ), so c ( L i ) = 1 + x i . ◮ L 1 ⊕ · · · ⊕ L n = C n is trivial on X n , so � � � 1 = c ( L 1 ⊕ · · · ⊕ L n ) = c ( L i ) = (1 + x i ) = e d ( x ) . i i d ≥ 0 ◮ Gives an S n -equivariant map R n = Z [ x ] / ( e 1 , . . . , e n ) → H ∗ ( X n ); in fact an isomorphism.

The moduli space of spanning line configurations For 1 ≤ k ≤ n , define X n , k := { ( ℓ 1 , . . . , ℓ n ) ∈ ( P k − 1 ) n : � ℓ i = C k } . i Example:  1 1 2 0   1 1 2 0   ∈ X 4 , 3 ; not columns of columns of 0 1 0 0 0 0 0 0    − 1 − 1 0 0 1 0 0 1 Theorem (Pawlowski and Rhoades) H ∗ ( X n , k ) ≃ R n , k as rings with S n -action. Example ◮ X n , n = X n ◮ X n , 1 = ( P 0 ) n = { pt }

Why H ∗ ( X n , k ) ≃ R n , k ? Chern class arguments give an S n -equivariant map R n , k = Z [ x ] / ( e n − k +1 , . . . , e n , x k 1 , . . . , x k n ) → H ∗ ( X n , k ); turns out to be an isomorphism. ◮ Set S = L 1 ⊕ · · · ⊕ L n , so c ( S ) = � i c ( L i ) = � d e d ( x ). ◮ Have a short exact sequence 0 → M → S → L 1 + · · · + L n = C k → 0. ◮ Whitney formula: 1 = c ( C k ) = c ( S / M ) = c ( S ) / c ( M ). ◮ So c ( S ) = � d e d ( x ) = c ( M ), which vanishes above degree rank( M ) = n − k . ◮ Get an S n -equivariant map R n , k = Z [ x ] / ( e n − k +1 , . . . , e n , x k 1 , . . . , x k n ) → H ∗ ( X n , k ); turns out to be an isomorphism.

The Schubert decomposition of X n (or Fl( n )) The ( i , j ) entry of the rank table of a matrix A is the rank of the upper-left i × j corner of A :  0 1 0  0 1 1  has rank table 1 0 0 1 2 2  0 0 1 1 2 3 rank table of ℓ • ∈ ( P k − 1 ) n := rank table of matrix with columns ℓ • . ◮ For a permutation matrix w ∈ S n , the set of ℓ • ∈ X n with the same rank table as w is a Schubert cell C w . ◮ Example: ℓ • ∈ C 213 iff the rank table of ℓ • is the one shown above. ◮ Fact: X n = � w ∈ S n C w .

The Schubert decomposition of X n (or Fl( n )) Affine paving of a variety Z : sequence of closed subvarieties Z = Z 0 ⊇ · · · ⊇ Z m = ∅ with Z i \ Z i +1 isomorphic to a disjoint union of affine spaces, the cells of the paving. ◮ The Schubert cells are the cells of an affine paving of X n . ◮ The closed Schubert variety C w determines a cohomology class [ C w ] ∈ H ∗ ( X n ). ◮ Affine paving = ⇒ H ∗ ( X n ) is free on the n ! classes [ C w ]. ◮ Under the iso. R n ≃ H ∗ ( X n ), the Schubert polynomial S w of Lascoux and Sch¨ utzenberger maps to [ C w ].

An affine paving of X n , k Each ℓ • ∈ X n , k is in a “Schubert cell” C w labeled by a length n word w on [ k ] := { 1 , . . . , k } determined as follows: ◮ Say the lex minimal linearly independent subtuple of ℓ • occurs in positions J ; this subtuple is in some Schubert cell C v ⊆ X k : � 0 � � 0 � 0 1 1 ∈ C 21 ⊆ X 2 , J = { 1 , 3 } k = 2 , n = 3 : ℓ • = � 1 1 2 1 2 ◮ Fill the positions of w in J with the letters of v : v = 21 � w = 2?1. ◮ If j / ∈ J , then ℓ j ⊆ ℓ 1 + · · · + ℓ i for some i < j ◮ Set w j = w i for the minimal such i : w = 221. Let C w be the set of ℓ • ⊆ X n , k with word w , e.g. ℓ • ∈ C 221 above.

An affine paving of X n , k For which w is C w nonempty? ◮ A word w = w 1 · · · w n is a Fubini word (packed word) if { w 1 , . . . , w n } = [ k ] for some k . ◮ Example: 31323 is Fubini but not 3133. ◮ Length n Fubini words on [ k ] ← → ordered set partitions of [ n ] into k blocks, e.g. 31323 ← → 2 | 4 | 135. Theorem (Pawlowski and Rhoades) The sets C w as w runs over the length n Fubini words on [ k ] are the cells of an affine paving of X n , k . ◮ Corollary: H ∗ ( X n , k ) is free of rank # OSP ( n , k ). ◮ The classes [ C w ] are represented by certain permuted Schubert polynomials S v ( x σ (1) , . . . , x σ ( n ) ).

Further directions ◮ Rhoades: fix a composition ( d 1 , . . . , d n ), consider the space of tuples ( V 1 , . . . , V n ) with dim V i = d i and V 1 + · · · + V n = C k . ◮ Rhoades and Wilson: require linear independence of some of the lines in ℓ • ∈ X n , k � r -Stirling numbers ◮ Pawlowski, Ramos, and Rhoades (in progress): representation stability for the S n -modules H ∗ ( X n , k ) ≃ R n , k .

Questions Define the Bruhat order on Fubini words by v ≤ w ⇐ ⇒ C w ⊇ C v . 212 221 121 211 112 122 ◮ When n = k this is the usual strong Bruhat order on S n ◮ How to describe covering relations in general? ◮ Our affine paving of X n , k is not a CW decomposition: C w ∩ C v � = ∅ need not imply C w ⊆ C v .

Questions ◮ Λ := ring of symmetric functions over Z ◮ The Delta conjecture of Haglund, Remmel, and Wilson predicts a combinatorial formula for ∆ ′ e k − 1 e n ∈ Λ ⊗ Q ( q , t ). ◮ Haglund, Rhoades, Shimozono: The graded Frobenius � characteristic grFrob ( R n , k ⊗ Q ) is ∆ ′ e k − 1 e n t =0 (up to a twist). � � Does the H ∗ ( X n , k ) picture help here? ◮ HRS give an explicit expansion grFrob ( R n , k ⊗ Q ) = � g λ ( q ) Q ′ λ in terms of dual Hall-Littlewood functions Q ′ λ . ◮ Q ′ λ is also (essentially) the graded Frobenius characteristic of H ∗ (Springer fiber indexed by λ ). ◮ Can X n , k be decomposed using Springer fibers in a way that explains the expansion grFrob( R n , k ⊗ Q ) = � g λ ( q ) Q ′ λ ? ◮ Is there an “ X n , k version” of Springer fibers?