The poset of proper divisibility

6th Combinatorics Day Almada - July 14, 2016 The poset of proper divisibility Antonio Macchia joint work with Davide Bolognini, Emanuele Ventura, Volkmar Welker

Classical divisibility and direct product We consider the divisibility of monomials in n variables x 1,...,x n.

Classical divisibility and direct product We consider the divisibility of monomials in n variables x 1,...,x n. Every monomial x a1 1 xan n (a 1,...,a n ) N n. is determined by its exponent vector

Classical divisibility and direct product We consider the divisibility of monomials in n variables x 1,...,x n. Every monomial x a1 1 xan n (a 1,...,a n ) N n. is determined by its exponent vector We study the divisibility of monomials as an order relation inn n. The poset of all divisors of (a 1,...,a n ) can be seen as the direct product of n chains with a 1 + 1,...,a n + 1 elements. (2, 3) (2, 2) (1, 3) 2 1 0 3 2 = 1 0 (2, 1) (1, 2) (0, 3) (2, 0) (1, 1) (0, 2) (1, 0) (0, 1) (0, 0)

Proper divisibility What happens if we consider proper divisibility instead of classical divisibility?

Proper divisibility What happens if we consider proper divisibility instead of classical divisibility? For every (a 1,...,a n ),(b 1,...,b n ) N n, we say that (a 1,...,a n ) properly divides(b 1,...,b n ) if for every 1 i n, either a i = b i = 0 or a i < b i. For example, (3, 2, 4) properly divides(4, 4, 6), (1, 2, 0) properly divides(2, 4, 0), (2, 0, 3) does not divide(3, 0, 3) properly. We study proper divisibility as an order relation, setting (a 1,..., a n ) = (b 1,..., b n ) or (a 1,..., a n ) (b 1,..., b n ) (a 1,..., a n ) properly divides(b 1,..., b n )

For an arbitrary(a 1,...,a n ) N n, we set P(a 1,...,a n ) = {(b 1,...,b n ) N n : (b 1,...,b n ) (a 1,...,a n )} and consider P(a 1,...,a n ) as a poset ordered by proper divisibility. It has a unique minimal element 0 = (0,...,0) and a unique maximal element 1 = (a 1,...,a n ). Hence it is bounded. (3, 3) (3, 2) (2, 3) (3, 1) (2, 2) (1, 3) (3, 0) (2, 1) (1, 2) (0, 3) (2, 0) (1, 1) (0, 2) (1, 0) (0, 1) (0, 0)

For an arbitrary(a 1,...,a n ) N n, we set P(a 1,...,a n ) = {(b 1,...,b n ) N n : (b 1,...,b n ) (a 1,...,a n )} and consider P(a 1,...,a n ) as a poset ordered by proper divisibility. It has a unique minimal element 0 = (0,...,0) and a unique maximal element 1 = (a 1,...,a n ). Hence it is bounded. (3, 3) (4, 4) (3, 2) (2, 3) (3, 1) (2, 2) (1, 3) (3, 0) (3, 1) (3, 2) (3, 3) (2, 3) (1, 3) (0, 3) (3, 0) (2, 1) (1, 2) (0, 3) (2, 0) (2, 1) (2, 2) (1, 2) (0, 2) (2, 0) (1, 1) (0, 2) (1, 0) (0, 1) (1, 0) (1, 1) (0, 1) (0, 0) (0, 0)

Order complex We study P(a 1,...,a n ) from a geometric perspective. We associate to P(a 1,...,a n ) its order complex (P(a 1,...,a n )) which is the simplicial complex consisting of all chains in P(a 1,...,a n )\{ 0, 1}. (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Order complex We study P(a 1,...,a n ) from a geometric perspective. We associate to P(a 1,...,a n ) its order complex (P(a 1,...,a n )) which is the simplicial complex consisting of all chains in P(a 1,...,a n )\{ 0, 1}. (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1)

Motivation Proper divisibility of monomials in Gröbner basis theory.

Motivation Proper divisibility of monomials in Gröbner basis theory. Miller, Sturmfels (1999): proper divisibility and free resolutions.

Motivation Proper divisibility of monomials in Gröbner basis theory. Miller, Sturmfels (1999): proper divisibility and free resolutions. Olteanu, Welker (2015): Buchberger resolution.

Motivation Proper divisibility of monomials in Gröbner basis theory. Miller, Sturmfels (1999): proper divisibility and free resolutions. Olteanu, Welker (2015): Buchberger resolution. Proper divisibility as an order relation.

Some definitions Let P be a poset with order relation.

Some definitions Let P be a poset with order relation. We say that p P covers q P, denoted q p, if q < p and there is no q P with q < q < p. e f c d a b 0

Some definitions Let P be a poset with order relation. We say that p P covers q P, denoted q p, if q < p and there is no q P with q < q < p. A chain of length k in P is a totally ordered subset x 0 < x 1 < < x k of k + 1 elements. e f c d a b 0

Given a poset P, its dual poset is P, on the same elements and with inverse order. P (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Given a poset P, its dual poset is P, on the same elements and with inverse order. P (3, 4) P (0, 0) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (0, 1) (1, 0) (1, 1) (1, 2) (0, 2) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (0, 0) (3, 4)

Basic facts The poset P(a 1,...,a n ) has a 1 a n + 1 elements. (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Basic facts The poset P(a 1,...,a n ) has a 1 a n + 1 elements. The length of P(a 1,...,a n ) is max 1 i n {a i }. (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Basic facts The poset P(a 1,...,a n ) has a 1 a n + 1 elements. The length of P(a 1,...,a n ) is max 1 i n {a i }. In general, P(a 1,...,a n ) is not a lattice. (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Basic facts The poset P(a 1,...,a n ) has a 1 a n + 1 elements. The length of P(a 1,...,a n ) is max 1 i n {a i }. In general, P(a 1,...,a n ) is not a lattice. In general, P(a 1,...,a n ) is not pure. (3, 4) (2, 0) (2, 1) (2, 2) (2, 3) (1, 3) (0, 3) (1, 0) (1, 1) (1, 2) (0, 2) (0, 1) (0, 0)

Nonpure shellability Definition A simplicial complex is shellable if there exists a linear order F 1,...,F t of its facets such that the subcomplex ( k 1 F i ) F k i=1 is pure of dimension dim F k 1 for all k = 2,...,t, where F denotes the union of all subsets of F, including the empty set.

1. CL-shellability A bounded poset P is CL-shellable if there exists a labelingλ(q p) of the covering relations of P by integers, called CL-labeling, such that in every rooted interval there is a unique strictly increasing chain and this is lexicographically strictly first. Theorem (Björner, Wachs, 1997) If a bounded poset P is CLshellable, then (P) is shellable. Theorem (BMVW) For every (a 1,...,a n ) N n, the order complex (P(a 1,...,a n )) is shellable. In particular, its homology groups H ( (P(a 1,...,a n ));Z) are torsion-free.

A question by Wachs Question (Wachs) P CL-shellable P CL-shellable?

A question by Wachs Question (Wachs) P CL-shellable P CL-shellable? The previous theorem does not hold if we replace P(a 1,...,a n ) with P(a 1,...,a n ).

A question by Wachs Question (Wachs) P CL-shellable P CL-shellable? The previous theorem does not hold if we replace P(a 1,...,a n ) with P(a 1,...,a n ). Proposition (BMVW) The poset P(4, 4) is not CL-shellable but its dual P(4, 4) is. In particular, (P(4, 4)) is shellable. (4, 4) (3, 0) (3, 1) (3, 2) (3, 3) (2, 3) (1, 3) (0, 3) (2, 0) (2, 1) (2, 2) (1, 2) (0, 2) (1, 0) (1, 1) (0, 1) (0, 0)

2. Homology of (P(a, b)) From now on we assume n = 2 and we study the homology groups of the order complex (P(a, b)) over Z.

2. Homology of (P(a, b)) From now on we assume n = 2 and we study the homology groups of the order complex (P(a, b)) over Z. Definition Let λ be a CL-labeling of a poset P. A maximal chain 0 = p0 p 1 p i+1 p i+2 = 1 in P is called falling if λ(p 0 p 1 ) > > λ(p r 1 p r ). Theorem (Björner, Wachs, 1996) If a bounded poset P is CLshellable, then rank H i ( (P);Z) = #falling chains of length i + 2 in P.

2.1 Persistence of the homology of (P(a, b)) A remarkable property of these posets is the persistence of the homology, which is rarely observed in naturally defined posets.

2.1 Persistence of the homology of (P(a, b)) A remarkable property of these posets is the persistence of the homology, which is rarely observed in naturally defined posets. Proposition (BMVW) For all 2 a b, there exists an integer t (a,b) 0, which depends on a and b, such that H i ( ( P(a, b);z )) 0 if and only if 0 i t(a,b). For every 1 i t (a,b), we show the existence of a falling chain of length i + 2.

2.2 Contractibility Theorem (Björner, Wachs, 1997) Let P 1, P 2 be bounded posets such that (P 1 ) and (P 2 ) are non-empty and shellable. Then

2.2 Contractibility Theorem (Björner, Wachs, 1997) Let P 1, P 2 be bounded posets such that (P 1 ) and (P 2 ) are non-empty and shellable. Then (P 1 P 2 ) is shellable, H i ( (P 1 P 2 );Z) = j+k=i 2 H j ( (P 1 );Z) H k ( (P 2 );Z). In particular, (P 1 P 2 ) has torsion-free homology. The order complex of the poset of all divisors of(a 1,...,a n ) (divisibility poset) is shellable. If a i 2 for some i, then it has trivial homology. Hence it is contractible and, indeed, collapsible.

Proposition Let 2 a b. Then (P(a, b)) is contractible if and only if a = b = 3. Moreover, (P(3, 3)) is collapsible.

Proposition Let 2 a b. Then (P(a, b)) is contractible if and only if a = b = 3. Moreover, (P(3, 3)) is collapsible. Proof. It follows from the previous Proposition. Indeed, the order complex ( P(3, 3) ) is a connected acyclic graph, hence collapsible. (2, 0) (2, 1) (2, 2) (1, 2) (0, 2) (1, 0) (1, 1) (0, 1)

2.3 Ranks of the homology groups of (P(a, b)) Theorem (BMVW) Let 2 a b and 0 i a 2. Then ( )[( )( ) ( )( )] rank H ( ( ) ) i a 3 i i b 2 i i b 3 i i P(a, b) ;Z = 2 +, t 1 t i t t 1 i t t=0 where we assume ( 1) = 1 by convention. Idea of the proof. Every falling chain in P(a, b) contains exactly one of the elements (1, 1),(1, 0) and(0, 1). Hence rank H i ( ( P(a, b) ) ;Z ) = F (1,1) + F (1,0) + F (0,1), where F (c,d) denotes the number of falling chains in P(a, b) of length i + 2 containing the element(c, d).

3. Euler characteristic of (P(a, b)) Theorem (BMVW) For every 2 a b, the reduced Euler characteristic of ( P(a, b) ) is χ ( ( P(a, b) )) = ( 1) a 2 ( )( ) a 2 b a ( 1) h. h a 2 2h a 2 1 h=0

3. Euler characteristic of (P(a, b)) Theorem (BMVW) For every 2 a b, the reduced Euler characteristic of ( P(a, b) ) is χ ( ( P(a, b) )) = ( 1) a 2 Idea of the proof. ( )( ) a 2 b a ( 1) h. h a 2 2h a 2 1 h=0 We use generating function techniques. The generating function of the left-hand side is f(u, v)= χ ( ( P(a, b) )) ( u a v b = 2 a=2 b=a u 2 v 2 2uv u v+1 u3 v 2 u 2 v u+1 ).

Vanishing of the Euler characteristic Example a = 5 b χ 5 0 6 6 7 12 8 16 9 16 10 10 11-4 a = 6 b χ 6 12 7 12 8 4 9-12 10-34 11-58 12-78 a = 7 b χ 7 0 8-20 9-40 10-50 11-40 12-102 13-68

Vanishing of the Euler characteristic Example a = 5 b χ 5 0 6 6 7 12 8 16 9 16 10 10 11-4 a = 6 b χ 6 12 7 12 8 4 9-12 10-34 11-58 12-78 a = 7 b χ 7 0 8-20 9-40 10-50 11-40 12-102 13-68 Corollary For every 2 a b, χ ( ( P(a, b) )) = 0 if a = b is odd.

Conclusions: Proper division product The posets P(a 1,...,a n ) can be seen as examples of a general construction.

Conclusions: Proper division product The posets P(a 1,...,a n ) can be seen as examples of a general construction. For every 1 i n, let P i be a bounded poset with bottom element 0 i and top element 1 i. For every(a 1,...,a n ),(b 1,...,b n ) P 1 P n we set (a 1,..., a n ) = (b 1,..., b n ) or (a 1,..., a n ) p (b 1,..., b n ) for all i, either a i = b i = 0 i or a i < b i in P i

Then it is easily seen that P(a 1,...,a n ) = C a1+1 p p C an+1, where C l is anl-elements chain. For this reason we call p the proper division product.

Then it is easily seen that P(a 1,...,a n ) = C a1+1 p p C an+1, where C l is anl-elements chain. For this reason we call p the proper division product. Questions Let P and Q be two (pure) bounded shellable posets. 1 Is (P p Q) shellable? 2 Assume (P p Q) is nonempty. Is there an integer t P;Q 0 such that H i ( (P p Q);Z) 0 if and only if 0 i t P;Q?

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