K -classes for matroids and equivariant localization Alex Fink 1 David Speyer 2 1 North Carolina State University 2 University of Michigan arXiv:1004.2403 FPSAC 2011 Fink & Speyer K -classes for matroids and equivariant localization 1 / 16
Overview If you remember one thing. . . You can get the Tutte polynomial of an arbitrary matroid via algebraic geometry. Outline: ◮ Setup: matroids and torus orbits on the Grassmannian; valuations and K -theory ◮ A K -theoretic matroid invariant ◮ Invariants that factor through it, incl. Tutte ◮ Equivariant localization ◮ Some ingredients of proofs Fink & Speyer K -classes for matroids and equivariant localization 2 / 16
Matroids as polytopes Definition (Edmonds; Gelfand-Goresky-MacPherson-Serganova) A matroid M on the ground set [ n ] is a polytope in R n such that ◮ every vertex (basis) of M lies in { 0 , 1 } n ; ◮ every edge of M is parallel to e j − e i for some i , j ∈ [ n ] . 1100 The edges are the exchanges between the bases. 0110 1010 M lies in { � n 1001 i = 1 x i = r } for some r , the rank. 0101 1010 0110 1001 0101 1010 0110 0101 1001 0011 Fink & Speyer K -classes for matroids and equivariant localization 3 / 16
Matroid toric varieties on the Grassmannian The Grassmannian is G ( r , n ) = { configs of n vectors spanning C r } / GL r . T := ( C ∗ ) n � G ( r , n ) by scaling the vectors. If x M ∈ G ( r , n ) represents M , the orbit closure Tx M ⊆ G ( r , n ) is the toric variety of M . Toric degenerations D of Tx M ← → certain matroid subdivisions Σ . Components of D are toric varieties of facets of Σ . Ditto intersections. Schematic example degenerates to . Fink & Speyer K -classes for matroids and equivariant localization 4 / 16
Matroid toric varieties on the Grassmannian The Grassmannian is G ( r , n ) = { configs of n vectors spanning C r } / GL r . T := ( C ∗ ) n � G ( r , n ) by scaling the vectors. If x M ∈ G ( r , n ) represents M , the orbit closure Tx M ⊆ G ( r , n ) is the toric variety of M . Toric degenerations D of Tx M ← → certain matroid subdivisions Σ . Components of D are toric varieties of facets of Σ . Ditto intersections. Schematic example degenerates to . Fink & Speyer K -classes for matroids and equivariant localization 4 / 16
Matroid valuations A matroid valuation f is a function that is additive with inclusion-exclusion in matroid subdivisions. E.g. f ( ) = f ( ) + f ( ) − f ( ) . Examples ◮ Lattice point count. ◮ the Tutte polynomial, M �→ T M ∈ Z [ x , y ] . (Not obvious!) The Tutte polynomial is � ( x − 1 ) corank ( S ) ( y − 1 ) nullity ( S ) . T M = S ⊆ [ n ] Fink & Speyer K -classes for matroids and equivariant localization 5 / 16
Matroid valuations A matroid valuation f is a function that is additive with inclusion-exclusion in matroid subdivisions. E.g. f ( ) = f ( ) + f ( ) − f ( ) . Examples ◮ Lattice point count. ◮ the Tutte polynomial, M �→ T M ∈ Z [ x , y ] . (Not obvious!) The Tutte polynomial is � ( x − 1 ) corank ( S ) ( y − 1 ) nullity ( S ) . T M = S ⊆ [ n ] Fink & Speyer K -classes for matroids and equivariant localization 5 / 16
K -theory: a valuation from algebraic geometry We use the K -theory ring K 0 ( X ) and the T -equivariant K -theory ring K T 0 ( X ) . The class of Y ⊆ X is denoted [ Y ] ∈ K 0 ( X ) , resp. [ Y ] T ∈ K T 0 ( X ) . [ Y ] T determines [ Y ] . Facts ◮ [ · ] T is additive with inclusion-exclusion over components. ◮ [ · ] T is unchanged by toric degenerations. Theorem 1 (Speyer) There is a valuation Y : { matroids } → K T 0 ( G ( r , n )) such that Y ( M ) = [ Tx M ] T for M representable. Fink & Speyer K -classes for matroids and equivariant localization 6 / 16
K -theory: a valuation from algebraic geometry We use the K -theory ring K 0 ( X ) and the T -equivariant K -theory ring K T 0 ( X ) . The class of Y ⊆ X is denoted [ Y ] ∈ K 0 ( X ) , resp. [ Y ] T ∈ K T 0 ( X ) . [ Y ] T determines [ Y ] . Facts ◮ [ · ] T is additive with inclusion-exclusion over components. ◮ [ · ] T is unchanged by toric degenerations. Theorem 1 (Speyer) There is a valuation Y : { matroids } → K T 0 ( G ( r , n )) such that Y ( M ) = [ Tx M ] T for M representable. Fink & Speyer K -classes for matroids and equivariant localization 6 / 16
K -theory: a valuation from algebraic geometry We use the K -theory ring K 0 ( X ) and the T -equivariant K -theory ring K T 0 ( X ) . The class of Y ⊆ X is denoted [ Y ] ∈ K 0 ( X ) , resp. [ Y ] T ∈ K T 0 ( X ) . [ Y ] T determines [ Y ] . Facts ◮ [ · ] T is additive with inclusion-exclusion over components. ◮ [ · ] T is unchanged by toric degenerations. Theorem 1 (Speyer) There is a valuation Y : { matroids } → K T 0 ( G ( r , n )) such that Y ( M ) = [ Tx M ] T for M representable. Fink & Speyer K -classes for matroids and equivariant localization 6 / 16
Invariants that factor through K -theory Theorem 2 (FS) The Tutte polynomial factors through Y. So do the Ehrhart polynomial, and Speyer’s invariant h. Speyer: How many faces can a matroid subdivision have? Construct h : { matroids } → Z [ t ] : ◮ valuative (proved) ◮ positive . . . (open in general) Then h ( uniform matroid ) is an upper bound for the f -vector. h ( � of k series-parallels ) = ( − t ) k . Example ) = − 2 t + t 2 . are products of series-parallels, so h ( Fink & Speyer K -classes for matroids and equivariant localization 7 / 16
Invariants that factor through K -theory Theorem 2 (FS) The Tutte polynomial factors through Y. So do the Ehrhart polynomial, and Speyer’s invariant h. Speyer: How many faces can a matroid subdivision have? Construct h : { matroids } → Z [ t ] : ◮ valuative (proved) ◮ positive . . . (open in general) Then h ( uniform matroid ) is an upper bound for the f -vector. h ( � of k series-parallels ) = ( − t ) k . Example ) = − 2 t + t 2 . are products of series-parallels, so h ( Fink & Speyer K -classes for matroids and equivariant localization 7 / 16
Invariants that factor through K -theory Theorem 2 (FS) The Tutte polynomial factors through Y. So do the Ehrhart polynomial, and Speyer’s invariant h. Speyer: How many faces can a matroid subdivision have? Construct h : { matroids } → Z [ t ] : ◮ valuative (proved) ◮ positive . . . (open in general) Then h ( uniform matroid ) is an upper bound for the f -vector. h ( � of k series-parallels ) = ( − t ) k . Example ) = − 2 t + t 2 . are products of series-parallels, so h ( Fink & Speyer K -classes for matroids and equivariant localization 7 / 16
Invariants that factor through K -theory Theorem 2 (FS) The Tutte polynomial factors through Y. So do the Ehrhart polynomial, and Speyer’s invariant h. Speyer: How many faces can a matroid subdivision have? Construct h : { matroids } → Z [ t ] : ◮ valuative (proved) ◮ positive . . . (open in general) Then h ( uniform matroid ) is an upper bound for the f -vector. h ( � of k series-parallels ) = ( − t ) k . Example ) = − 2 t + t 2 . are products of series-parallels, so h ( Fink & Speyer K -classes for matroids and equivariant localization 7 / 16
Equivariant localization Technique: Equivariant localization ( Goresky-Kottwitz-MacPherson) If X is nice & has an action of a big enough torus T , e.g. G ( r , n ) , K T 0 ( X ) can be constructed from its moment graph Γ . ◮ V (Γ) = { T -fixed points of X } . ◮ E (Γ) = { 1-dimensional T -orbits of X } . = P 1 = T ∪ { 0 , ∞} Their closures ∼ . � �� � endpoints The T -action on an edge factors through a character χ . Keep χ as an edge label. Fink & Speyer K -classes for matroids and equivariant localization 8 / 16
Equivariant localization Technique: Equivariant localization ( Goresky-Kottwitz-MacPherson) If X is nice & has an action of a big enough torus T , e.g. G ( r , n ) , K T 0 ( X ) can be constructed from its moment graph Γ . ◮ V (Γ) = { T -fixed points of X } . ◮ E (Γ) = { 1-dimensional T -orbits of X } . = P 1 = T ∪ { 0 , ∞} Their closures ∼ . � �� � endpoints The T -action on an edge factors through a character χ . Keep χ as an edge label. Fink & Speyer K -classes for matroids and equivariant localization 8 / 16
Equivariant localization for G ( r , n ) { 1 , 2 } For G ( r , n ) , Γ is the union of all t 3 /t 2 1-skeleta of matroids. { 1 , 3 } { 2 , 3 } ◮ V (Γ) ← → r -subsets of n . t 2 /t 1 { 1 , 4 } { 2 , 4 } ◮ E (Γ) ← → exchanges ( S , S \ { i } ∪ { j } ) , . . . with labels t j / t i . t 3 /t 1 { 3 , 4 } Fink & Speyer K -classes for matroids and equivariant localization 9 / 16
Equivariant localization: the K -theory ring 0 ( point ) = Z [ Char T ] = Z [ t ± 1 1 , . . . , t ± 1 K T n ] . Theorem (GKM, . . . ) K T 0 ( X ) equals { functions V (Γ) → K T 0 ( pt ) : χ f ( v ) ∼ = f ( w ) (mod 1 − χ ) for v − − − w an edge of Γ } . Example There’s a class [ O ( 1 )] on G ( r , n ) . t 1 t 2 { 1 , 2 } � [ O ( 1 )] T ( x S ) = t S := t i . t 1 t 3 t 2 t 3 i ∈ S { 1 , 3 } { 2 , 3 } { 1 , 4 } { 2 , 4 } t 1 t 4 t 2 t 4 { 3 , 4 } t 3 t 4 Fink & Speyer K -classes for matroids and equivariant localization 10 / 16
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