Combinatorics and topology of toric arrangements III. Epilogue - PowerPoint PPT Presentation

Combinatorics and topology of toric arrangements III. Epilogue Emanuele Delucchi (SNSF / Universit´ e de Fribourg) Toblach/Dobbiaco February 24, 2017

Finite matroids Rank functions / intersection posets ... of central hyperplane arrangements Representable m. Orientable m. ...of pseudosphere arrangements |R ( A ) | = χ rk ( − 1) matroids / geometric lattices (tropical linear spaces, matroids over the hyperfield K )

Toric arrangements A = [ a 1 , . . . , a n ] ∈ M d × n ( Z ) A in ( Z q ) d ⊆ ( S 1 ) d ⊆ ( C ∗ ) d Discrete tori Poset of layers Topology (in ( C ∗ ) d ) (enumeration) M ( A ) := ( C ∗ ) d \ [ A C ( A ): { χ i } i | ρ ( q ) := | ( Z q ) d \ [ A q | • Poin( M ( A ) , t ) = t d T A ( 2 t +1 , 0) t [Looijenga ‘95, is a quasipolynomial in q , with De Concini–Procesi 2005] χ 1 ( t ) = ( − 1) d T (1 − t, 0), • M ( A ) minimal; presentation Matroid over Z of the ring H ∗ ( M ( A ) , Z ) χ j ( t ) =?, χ ρ ( t ) = ( − 1) d T A (1 − t, 0) [D.–d’Antonio ‘13, Callegaro–D. ‘15] M ( I ) := Z d / h a i i I • Wonderful models [Kamiya–Takemura–Terao ‘08, [Moci ‘12, Gaiffi-De Concini ‘16] Lawrence ‘11, ...] Dissections of ( S 1 ) d Arithmetic matroid Ehrhart theory m ( I ) := | Tor( M ( I )) | The complement ( S 1 ) d \ [ A has (of zonotopes) T A (1 , 0) connected regions. The zonotope Z A := P a i [Lawrence ‘09 and ‘11; Ar. Tutte Poly. has Ehrhart polynomial Ehrenborg–Readdy–Slone ‘09] E Z A ( t ) = ( − 1) d T A ( t +1 t , 1) T A ( x, y ) The “Coxeter case” (= | Z d ∩ tZ A | for t ∈ N ) [Moci ‘08, Aguiar-Petersen ‘14, [d’Adderio–Moci ‘13] D.-Girard ‘16+]

Toric arrangements Combinatorial framework Ansatz: “periodic arrangements” L ( A ↾ ) F ( A ↾ ) Poset of Poset (category) intersections of polyhedral faces A ↾ / Z d / Z d (as acyclic / Z d (as posets) categories) C ( A ) F ( A ) A Characterize axiomatically the involved posets and the group actions.

Semimatroids Recall by way of example K := { I such that \ i ∈ I H i 6 = ; {} , { 1 } , { 2 } , { 3 } , { 4 } 1 { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 4 } , { 2 , 4 } 2 { 1 , 2 , 3 }} These are the central sets. 3 4 A : rk : K ! N , I 7! codim( \ i ∈ I H i ) ( S, K , rk) is a semimatroid . [today: loopless ] L is a geometric semilattice H 1 H 2 H 3 H 4 V Cryptomorphism L ( A ): ):

Group actions on semimatroids Let G be a group A G -semimatroid A G -geometric semilattice S : G � ( S, K , rk) S : G � L is an action of G is an action of G on a semimatroid ( S, K , rk) on a geometric semilattice L by rank- and centrality- pre- by poset-automorphisms. serving bijections of S . Cryptomorphism! X

Group actions on semimatroids c 2 b 0 b 1 b 2 b − 1 a 5 Z 2 � c 1 a 4 d 0 d 1 d 2 a 1 a 2 a 3 b 3 c 0 S := { a i , b j , c k , d l } i,j,k,l ∈ Z , L := poset of intersections, K := {; , a 1 , b 0 , a 1 b 0 , b 1 , a 1 b 1 , . . . } 63 a 1 b 0 c 0 for X 2 K , rk( X ) := codim \ X

Group actions on semimatroids Quotient posets Let G be a group G -semimatroid G -geometric semilattice S : G � ( S, K , rk) S : G � L C S := L /G , the set { Gx | x 2 L} ordered by Gx  Gy i ff x  L gy for some g (This is a poset)

Group actions on semimatroids Example ( G = Z 2 ) d 0 a 1 a 2 b 0 b 1 a 0 c 0 [ a 1 b 0 ] [ a 2 b 1 ] [ a 1 b 1 ] [ b 1 c 0 ] [ a 1 c 0 ] [ a 0 b 0 c 0 d 0 ] a c b d C S := L /G ;

Group actions on semimatroids [ a 0 b 0 c 0 d 0 ] Example ( G = Z 2 ) K S := K /G [ a 0 b 0 c 0 ] [ a 0 b 0 d 0 ] [ a 0 c 0 d 0 ] [ b 0 c 0 d 0 ] d 0 [ a 1 b 1 ] [ a 1 c 0 ] [ b 1 c 0 ] [ a 1 b 0 ] [ a 2 b 1 ] [ a 0 b 0 ] [ a 0 c 0 ] [ a 0 d 0 ] [ b 0 c 0 ] [ b 0 d 0 ] [ c 0 d 0 ] a 1 a 2 a c b d b 0 b 1 G { x 1 ...x k } a 0 Φ ; c 0 { a, b, c, d } (1) { Gx 1 ...Gx k } E S := S/G = { a, b, c, d } { a, b, c } (1) { a, b, d } (1) { a, c, d } (1) { b, c, d } (1) K := {{ Gx 1 ...Gx k }|{ x 1 ...x k }∈K} = { a, b } (4) { a, c } (2) { b, c } (2) { a, d } (1) { b, c } (1) { b, d } (1) a (1) b (1) c (1) d (1) rk( A ) := max { rk( X ) | Φ( GX ) ⊆ A } ; (1) m S ( A ) := | Φ − 1 ( A ) | . X m S ( A )( x − 1) rk( S ) − rk( A ) ( y − 1) | A |− rk( A ) T S ( x, y ) := A ⊆ E S

Group actions on semimatroids Translative actions S is called translative if, for all x 2 S and g 2 G , { x, g ( x ) } 2 K implies x = g ( x ) . The function rk : 2 E S ! N always defines a semimatroid. It Theorem defines a matroid if, and only if, S is translative. In the ‘realizable’ case, this corresponds to the arrangement A 0 , (remember?) Theorem If S is translative, the triple ( E S , rk , m S ) satisfies axiom (P) “pseudo-arithmetic”

Group actions on semimatroids Translative actions S is called translative if, for all x 2 S and g 2 G , { x, g ( x ) } 2 K implies x = g ( x ) . Theorem. If S is translative, the characteristic polynomial of the poset C S = L /G is χ C S ( t ) = ( − 1) r T S (1 − t, 0) . Corollary. If S arises from a translative Z r -action on a rank r oriented semimatroid (“periodic wiggly arrangement”), then the number of regions of the associated toric pseudoarrangement is |R ( A ) | = ( − 1) r T S (1 , 0)

Deletion / Contraction c 2 c 2 e 1 b 0 b 1 b 2 b 0 b 1 b 2 b − 1 b − 1 a 5 a 5 S \ e : S : c 1 c 1 a 4 a 4 d 0 d 1 d 2 e 0 d 0 d 1 d 2 a 1 a 2 a 3 a 1 a 2 a 3 b 3 b 3 c 0 c 0 e := Ge 0 ; stab( e ) := stab( e i ) b 0 b 1 b 2 b 3 S /e : stab( e ) � a 1 a 2 a 3 a 4 d 0 d 1 d 2

Group actions on semimatroids Translative actions Theorem If S is translative, for all e 2 E S we have the recursion T S ( x, y ) = ( x − 1) T S \ e ( x, y ) + ( y − 1) T S /e ( x, y ) , according to whether e is a coloop or a loop of ( E S , K , rk), where S \ e := G � ( S, K , rk) \ e, S /e := stab( e ) � ( S, K , rk) /e . Think: “removing an orbit of hyperplanes”, resp. considering the stab( H e )- periodic arrangement induced in H e (NRDC)

Group actions on semimatroids Towards arithmetic matroids A translative S is called normal if, for all X 2 K , stab( X ) is normal in G . This allows, given X 2 K , to consider the group Γ X := Y G/ stab( x ) x ∈ X If S is translative and normal, ( E S , rk , m S ) satisfies (P), Theorem. (A.1.2) and (A.2). For the “initiated”: moreover, T S ( x, y ) satisfies an “activity decomposition theorem” ` a la Crapo.

Group actions on semimatroids Towards arithmetic matroids A translative S is called normal if, for all X 2 K , stab( X ) is normal in G . This allows, given X 2 K , to consider Γ X := Q x ∈ X G/ stab( x ) , and W ( X ) := { ( g x ) x ∈ X 2 Γ X | { g x x } x ∈ X 2 K} S is called arithmetic if, for all X 2 K , W ( X ) is a subgroup of Γ X . Theorem: If S is arithmetic, then ( E S , rk , m S ) is an arithmetic matroid. Remark 1. There are translative and not normal, and normal but not arithmetic S ’s. In general, it seems very restrictive to require arithmeticity.

Group actions on semimatroids Towards arithmetic matroids A translative S is called normal if, for all X 2 K , stab( X ) is normal in G . This allows, given X 2 K , to consider Γ X := Q x ∈ X G/ stab( x ) , and W ( X ) := { ( g x ) x ∈ X 2 Γ X | { g x x } x ∈ X 2 K} S is called arithmetic if, for all X 2 K , W ( X ) is a subgroup of Γ X . Theorem: If S is arithmetic, then ( E S , rk , m S ) is an arithmetic matroid. Remark 2. W ( X ) parametrizes all elements of Φ − 1 (Φ( GX )). In the case of periodic arrangements, this induces a group structure on the set of connected components of the intersection of the “subtori” in Φ( GX ) ✓ E S .

Representable cases Call S representable if it arises as an action by translations on an affine rank d arrangement A of hyperplanes. In this case, ( E S , rk , m S ) is an arithmetic matroid and C S ' C ( A ). G = { id } ! (Central) arrangements of hyperplanes, G = Z d ! (Centered) toric arrangements* G = Z 2 d ! Elliptic arrangements (*) in this case, the arithmetic matroid ( E S , rk , m S ) is dual to that associ- ated to the list of defining characters by d’Adderio–Br¨ and´ en–Moci

Coarse overview G -semimatroids / G -geometric semilattices ... of periodic hyperplane arrangements Representable Orientable ...of pseudoarrangements S

“Finer” overview Representable G -semimatroids $ G -geometric semilattices G =id: (finite) geometric semilattices Arithmetic matroids ? G =id & centered (finite) matroids & c. S Arithmetic S Almost-arithmetic S Translative T S ( x, y ) sat. (NRDC), χ C S ( t ) = T S (1 − t, 0)

Your turn!

Your turn! 1. Does the theory of AM’s fully embed in G -semimatroids? Construct, for every arithmetic matroid ( E, rk , m ) a G -semimatroid S such that ( E S , rk , m S ) is isomorphic to ( E, rk , m ) – or find obstructions ( !).

Your turn! 1. Does the theory of AM’s fully embed in G -semimatroids? 2. Structure of the posets C S – are these posets shellable ? At least Cohen-Macauley? ( C ( A ) shellable for toric Weyl type A n , B n , C n [D.-Girard ‘17+] ) – characterize intrinsecally the class of these posets (cf. “developability” in Bridson-H¨ afliger)

Your turn! 1. Does the theory of AM’s fully embed in G -semimatroids? 2. Structure of the posets C S 3. Duality theory Construct, for a given arithmetic S , a S ∗ such that ( S ∗ ) ∗ ' S and, for instance, T S ( x, y ) = T S ∗ ( y, x ). Can one do it for general translative S ? One motivation for developing duality is the following item.