Chow Rings of Matroids University of Minnesota-Twin Cities REU - PowerPoint PPT Presentation

Chow Rings of Matroids University of Minnesota-Twin Cities REU Thomas Hameister, Sujit Rao, Connor Simpson August 2, 2017

Outline 1 Preliminaries 2 Methods for calculating Hilbert series 3 Uniform matroids and M r ( F n q ) 4 Future work and other lattices

Motivation The Chow ring of a ranked atomic lattice L is a graded ring denoted A ( L ). The proof of the Heron-Rota-Welsh conjecture by Adiprasito-Huh-Katz uses properties of A ( L ) when L is the lattice of flats of a matroid M . We are interested in combinatorial information about the lattice L or the matroid M which can be determined from A ( L ). Example L ( U n , r ) = { A ⊆ [ n ] with # A ≤ r − 1 } L ( M r ( F n q )) = { A ≤ F n q with dim A ≤ r − 1 } L ( M ( K n )) = { partitions of [ n ] }

Definitions Definition (Feichtner-Yuzvinsky 2004) Let L be a ranked lattice with atoms a 1 , . . . , a k . The Chow ring of L is A ( L ) = Z [ { x p : p ∈ L , p � = ⊥} ] / ( I + J ) where I = ( x p x q : p and q are incomparable)    �  . J = x q : 1 ≤ i ≤ k q ≥ a i Theorem (Adiprasito-Huh-Katz 2015) The Heron-Rota-Welsh conjecture is true.

Incidence algebra Theorem (Feichtner-Yuzvinsky 2004) m t − t rk x i − rk x i − 1 − 1 � � H ( A ( L ) , t ) = 1 + 1 − t i =1 ⊥ = x 0 < x 1 < ··· < x m Proposition If η, γ ∈ ( Q ( t ))[ L ] are given by rk y − rk x − 1 � t i η ( x , y ) = i =1 and γ = (1 − η ) − 1 ζ , then H ( A ([ x , y ]) , t ) = γ ( x , y ). Proposition γ L × K = (1 − t (1 − γ L ) ⊗ (1 − γ K )) − 1 ( γ L ⊗ γ K ) .

Differential operators and derivations Motivation: What is H ( A ( L × B 1 ) , t )?

Differential operators and derivations Motivation: What is H ( A ( L × B 1 ) , t )? Observation: the FY formula for L × B 1 is Leibniz rule-like.

Differential operators and derivations Motivation: What is H ( A ( L × B 1 ) , t )? Observation: the FY formula for L × B 1 is Leibniz rule-like. Define new multiplicands; use them to get H ( A ( L ) , t , s )

Differential operators and derivations Motivation: What is H ( A ( L × B 1 ) , t )? Observation: the FY formula for L × B 1 is Leibniz rule-like. Define new multiplicands; use them to get H ( A ( L ) , t , s ) H ( A ( L ) , t , 0) = H ( A ( L ) , t )

Differential operators and derivations Motivation: What is H ( A ( L × B 1 ) , t )? Observation: the FY formula for L × B 1 is Leibniz rule-like. Define new multiplicands; use them to get H ( A ( L ) , t , s ) H ( A ( L ) , t , 0) = H ( A ( L ) , t ) Proposition H ( A ( L × B 1 ) , t , s ) = (1 + ∂ s ) H ( A ( L ) , t , s )

Applications of AHK results Motivation: Many families of lattices such that if L is in the family, then [ z , ⊤ ] is in the family too for all z ∈ L .

Applications of AHK results Motivation: Many families of lattices such that if L is in the family, then [ z , ⊤ ] is in the family too for all z ∈ L . AHK gives isomorphisms relating Chow rings of these intervals to the Chow ring of the whole

Applications of AHK results Motivation: Many families of lattices such that if L is in the family, then [ z , ⊤ ] is in the family too for all z ∈ L . AHK gives isomorphisms relating Chow rings of these intervals to the Chow ring of the whole Theorem Let L be a “nicely ranked” atomic lattice with rk L = r + 1 and ⇒ [ z , ⊤ ] ∼ rk( z ) = rk( z ′ ) = = [ z ′ , ⊤ ]. Let z 2 , . . . , z r − 1 ∈ L with rk( z i ) = i . Then r i − 1 dim Z A q ( L ) = 1 + � � dim Z A q − p ([ z i , ⊤ ]) # L i i =2 p =1

Applications of AHK results Motivation: Many families of lattices such that if L is in the family, then [ z , ⊤ ] is in the family too for all z ∈ L . AHK gives isomorphisms relating Chow rings of these intervals to the Chow ring of the whole Theorem (A better one!) Let L be a “nicely ranked” atomic lattice with rk L = r + 1 and ⇒ [ z , ⊤ ] ∼ rk( z ) = rk( z ′ ) = = [ z ′ , ⊤ ]. Let z 2 , . . . , z r − 1 ∈ L with rk( z i ) = i . Then r � H ( A ( L ) , t ) = [ r + 1] t + t # L i [ i − 1] t H ([ z i , ⊤ ] , t ) i =2

Applications of AHK results: examples Uniform: r � � n � H ( U n , r +1 , t ) = [ r + 1] t + t [ i − 1] t H ( U n − i , r +1 − i , t ) . i i =2

Applications of AHK results: examples Uniform: r � � n � H ( U n , r +1 , t ) = [ r + 1] t + t [ i − 1] t H ( U n − i , r +1 − i , t ) . i i =2 Subspaces: r � � n � � � � � M r +1 ( F n � � M r +1 − i ( F n q ) � , t = [ r +1] t + t [ i − 1] t q ) � , t H A H A i i =2 q

Uniform matroids Recall U n , r has lattice of flats the truncation of the boolean lattice at rank r .

Uniform matroids Recall U n , r has lattice of flats the truncation of the boolean lattice at rank r . Some invariants of interest for A ( U n , r ) have combinatorial meaning.

Uniform matroids Recall U n , r has lattice of flats the truncation of the boolean lattice at rank r . Some invariants of interest for A ( U n , r ) have combinatorial meaning. Theorem The Hilbert series of U n , n is the Eulerian polynomial � = � t exc ( σ ) . � A ( U n , n ) , t H σ ∈ S n

Uniform matroids For r < n , there are surjective maps π n , r : A ( U n , r +1 ) → A ( U n , r ).

Uniform matroids For r < n , there are surjective maps π n , r : A ( U n , r +1 ) → A ( U n , r ). Theorem For E n , k := { σ ∈ S n : # fix ( σ ) ≥ k } , the Hilbert series of K n , r = ker( π n , r ) is t r − exc ( σ ) � H ( K n , r , t ) = σ ∈ E n , n − r

Uniform matroids For r < n , there are surjective maps π n , r : A ( U n , r +1 ) → A ( U n , r ). Theorem For E n , k := { σ ∈ S n : # fix ( σ ) ≥ k } , the Hilbert series of K n , r = ker( π n , r ) is t r − exc ( σ ) � H ( K n , r , t ) = σ ∈ E n , n − r Can be used to characterize Hilbert series for H ( A ( U n , r ) , t ) for all r .

Uniform matroids The Charney-Davis quantity of a graded ring R supported in finitely many degrees is H ( R , − 1).

Uniform matroids The Charney-Davis quantity of a graded ring R supported in finitely many degrees is H ( R , − 1). Theorem For odd r , the Charney-Davis quantity for the uniform matroid, U n , r , of rank r and dimension n is r − 1 � � 2 n � E 2 k 2 k k =0 where E 2 ℓ is the ℓ -th secant number, i.e. t 2 ℓ � sech( t ) = E 2 ℓ (2 ℓ )! ℓ ≥ 0

q -analogs of uniform matroids: M r ( F n q ) The lattice of flats of M r ( F n q ) is the lattice of dimension ≤ r subspaces in F n q .

q -analogs of uniform matroids: M r ( F n q ) The lattice of flats of M r ( F n q ) is the lattice of dimension ≤ r subspaces in F n q . Have q -analogues of each piece of data for uniform matroid.

q -analogs of uniform matroids: M r ( F n q ) The lattice of flats of M r ( F n q ) is the lattice of dimension ≤ r subspaces in F n q . Have q -analogues of each piece of data for uniform matroid. Theorem The Hilbert series of M ( F n q ) is � = � A ( M ( F n � q maj ( σ ) − exc ( σ ) t exc ( σ ) . H q )) , t σ ∈ S n

q -analogs of uniform matroids: M r ( F n q ) There are again surjective maps π n , r : A ( M r +1 ( F n q )) → A ( M r ( F n q )). Theorem The Hilbert series of K n , r = ker( π n , r ) is � = � A ( M r ( F n � q maj ( σ ) − exc ( σ ) t r − exc ( σ ) . H q )) , t σ ∈ E n , n − r

q -analogs of uniform matroids: M r ( F n q ) t 2 n � Let cosh q ( t ) = [2 n ] q ! and sech q ( t ) = 1 / cosh q ( t ). n ≥ 0 Theorem For odd r , the Charney Davis quantity of A ( M r ( F n q )) is r − 1 � � 2 n � E 2 k , q 2 k k =0 where E 2 ℓ, q satisfies t 2 ℓ � sech q ( t ) = E 2 ℓ, q [2 ℓ ] q ! ℓ ≥ 0

Beyond matroids Question: Why look at only matroids? Is the Chow ring still nice for more general lattices?

Beyond matroids Question: Why look at only matroids? Is the Chow ring still nice for more general lattices? If L is an atomic lattice with atoms E , let     � d ( x , y ) := min  # S : S ⊆ E , x ∨ s = y  s ∈ S

Beyond matroids Question: Why look at only matroids? Is the Chow ring still nice for more general lattices? If L is an atomic lattice with atoms E , let     � d ( x , y ) := min  # S : S ⊆ E , x ∨ s = y  s ∈ S If d ( x , y ) = rk( y ) − rk( x ), then we get Poincar´ e duality.

Beyond matroids Question: Why look at only matroids? Is the Chow ring still nice for more general lattices? If L is an atomic lattice with atoms E , let     � d ( x , y ) := min  # S : S ⊆ E , x ∨ s = y  s ∈ S If d ( x , y ) = rk( y ) − rk( x ), then we get Poincar´ e duality. Can also generalize some early lemmas needed for hard Leftschetz, etc.

Experimental results Experimentally, the following have symmetric Hilbert series: Polytope face lattices Simplicial complexes Convex closure lattices Various manual examples

Experimental results Experimentally, the following have symmetric Hilbert series: Polytope face lattices Simplicial complexes Convex closure lattices Various manual examples Conjecture All Chow rings of ranked atomic lattices exhibit Poincar´ e duality.