Commutative algebraof generalized permutohedra

Anton Dochtermann, UT Austin

joint with Alex Fink and Raman Sanyal

Memphis AMS sectionalOctober 18, 2015

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

The Koszul resolution = topology of the simplex

Let ∆n = conv(e1, e2, . . . , en) ⊂ Rn denote the(n− 1)-dimensional simplex. Let S = k[x1, x2, . . . , xn].

Identifying ei with the variable xi, we see that the (cellular)homology chain complex of ∆n supports a minimal resolutionof the graded maximal ideal m ⊂ k[x1, x2, . . . , xn].

An ‘edge’ syzygy −x2(x1) + x1(x2) + 0(x3) = 0.

x1 x3

x2

For n = 3 we have I = 〈x1, x2, x3〉:

0← I ← S3 ← S3 ← S ← 0

where S3 ← S3 : d2 =

−x2 0 x3

x1 −x3 00 x2 −x1

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Cellular resolutions

Note that the generators of I naturally label the 0-cells of ∆n.We label the higher dimensional faces with the associatedsyzygy element (= the LCM of the vertices that theycontain).

x1 x3

x2

x1x3

x2x3x1x2

x1x2x3

0← I ← S[−1]3 ← S[−2]3 ← S[−3]← 0.

In general a cellular resolution of an ideal I ⊂ S is apolyhedral (CW-) complex X with 0-cells labeled by thegenerators of I, such that the complex computing cellularhomology of X provides a resolution of I.

Let the topology of the complex X ‘do the work for you’.

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Adding simplices

If P and Q are polytopes in Rd, the Minkowski sum is given by

P + Q = {p + q : p ∈ P,q ∈ Q}.

We’ll be adding simplices. For K ⊆ [n], we let∆K = conv(ek : k ∈ K}.Running example P = ∆1,2,3 + ∆1,2 + ∆2,3

x1 x3

x2

++x1

x2

x3

x2

=

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

An example of a ‘generalized permutohedron’ introduced byPostnikov, et al.

Note that the lattice points of P define a collection ofmonomials that generate an ideal IP .

x1 x3

x2

++x1

x2

x3

x2

=x1x2x3

x1x32x12x3

x2x32

x22x3

x23

x12x2

x1x22

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Mixed subdivisions

If P is a Minkowski sum of simplices, a mixed subdivision Σ isa subdivision of P whose cells consist of Minkowski sums ofsubsets of the summands.

x1 x3

x2

++x1

x2

x3

x2

=

x1x32x12x3

x2x32x12x2

x1x22

x1x2x3

x22x3

x23x1x32x12x3

x2x32x1x2x3

x1 x3

+x1

x2

x3

+

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Theorem (D, Fink, Sanyal)

Suppose P is a sum of simplices. Then any regular mixedsubdivision Σ of P supports a minimal cellular resolution of IP .

In our example:IP = 〈x2

1x2, x21x3, x1x

22, x1x2x3, x1x

23, x

32, x

22x3, x2x

23〉

0← IP ← S[−3]8 ← S[−4]11 ← S[−5]4 ← 0.

x1 x3

x2

++x1

x2

x3

x2

x1x32x12x3

x2x32x12x2

x1x22

x1x2x3

x22x3

x23

Corollary: All regular fine mixed subdivisions of P have thesame f -vector (cf. Postnikov)

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Tropical hyperplane arrangements

Work in the ‘tropical semiring’ T = (R∪∞,⊕,�), where ⊕ ismin and � is addition.

For any a ∈ Tn the tropical hyperplane H(−a) is given by

H(−a) = {x ∈ Tn : min(ak + xk) is achieved at least twice}

Example: a = (0, 0, 0), b = (3, 0,∞), c = (∞, 2, 0)

A =

0 3 ∞0 0 20 ∞ 0

.

a

b

c

(0,0,0)

(0,3,0)

(0,-5,-2) (0,-2,-2)x2 = -2

x2 = x1+3

PUNCHLINE: Each hyperplane H(−a) is a translate of thenormal fan of ∆K , where K is the subset of finite coordinatesof the vector a.

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

So what? Tropical combinatorics

Combinatorics of finite arrangements are governed by regulartriangulations of ∆n ×∆d (Develin/Sturmfels, Fink/Rincon)

The ‘Cayley trick’ then connects the arrangement of tropicalhyperplanes to the regular mixed subdivisions.

PUNCHLINE: Combinatorics/geometry of the regular mixedsubdivision is encoded in the arrangement.

Recover the ideal IP from (tropical) oriented matroid data.

Good notion of (tropical) convexity allows us to establish theresult.

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

A(nother) ‘tropical oriented matroid’ ideal

Each tropical hyperplane in Td has d sides

Label each 0-cell of the arrangement complex according whichregions of which hyperplanes it does not sit in.

a

b

c

1 3

2

x21x32

x12x32

x13x21x12x13

We’ll call the resulting monomial ideal the ‘oriented matroidcotype ideal of A’.

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Detour - Determinantal ideals

Consider an n× d matrix M of indeterminates

X =

x11 x21 x31

x12 x22 x32

x13 x23 x33

Let J2 be the ideal generated by all 2-minors of M .

J2 = 〈x11x22 − x12x21, x11x23 − x13x21, . . . 〉.

The 2 minors form a Grobner basis for J2, S/J2 is a domain[Narasimhan].

in<(J2) is the Stanley-Reisner ring of a shellable simplicialcomplex (⇒ S/J2 is Cohen-Macaulay) [Herzog]

R/J2 is a normal domain [Conca, Hibi]

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Relevance to our work

Given an arrangement of tropical hyperplanes,

Let A be its matrix, and let Σ be the submatrix of Xcorresponding to the finite coordinates.

Let JΣ = ideal generated by all 2-minors of Σ.

In our example A =

0 3 ∞0 0 20 ∞ 0

, so that Σ =

x11 x21

x12 x22 x32

x13 x33

JΣ = 〈x11x22 − x12x21, x12x33 − x13x32〉

The arrangement determines a weight vector (hence a term order):

in<(JΣ) = 〈x12x21, x13x32〉= 〈x21, x32〉 ∩ 〈x12, x32〉 ∩ 〈x21, x13〉 ∩ 〈x12, x13〉.

So that (in<(JΣ))∗ = 〈x21x32, x12x32, x21x13, x12x13〉.Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

This always works for ladders

Theorem (D, Fink, Sanyal)

If JΣ is a ladder determinant ideal then in<(JΣ) is the alexanderdual of the oriented matroid cotype ideal of the arrangement A. Inparticular in<(JΣ) (and hence JΣ) is Cohen-Macaulay.

Idea for the proof goes back to Block/Yu and Sturmfels. Thelatter (!) statement recovers a result of Corso and Nagel.

A =

0 3 ∞0 0 20 ∞ 0

a

b

c

1 3

2

x21x32

x12x32

x13x21x12x13

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra

Further questions

What about Minkowski differences? (e.g. matroid polytopes)What is the right notion of mixed subdivision?

What about the ideal Ivert(P ) generated by just the vertices ofPΣ?

Proposition (DFS)

The ideal Ivert(P ) has a (non minimal) resolution supported onMinkowski sum PΣ (thought of as a polytope).

It is know that a matroidal ideal has a resolution supported onits associated matroid polytope, and one can compute Bettinumbers via Mobius function of the lattice of flats (Ardila).Something similar in general?

Anton Dochtermann, UT Austin Commutative algebra of generalized permutohedra