commutative algebra of generalized...
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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