Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Some recent interactions between
commutative algebra andcombinatorics
FABRIZIO ZANELLO(Michigan Tech)RIMS – Kyoto, Japan
August 4, 2016

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Graded algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

L
Let A = R/I = i≥0 Ai be a standard gradedk -algebra, i.e., R = k [x1 , . . . , xr ], A0 = k is a field, I ⊂ R
is a homogeneous ring ideal, and deg (xi ) = 1.
Often char(k) has suitable restrictions (e.g., = 0).
The Hilbert function (HF) of A is HA (i) = dimk Ai .
P
i
The Hilbert series of A is ∞i=0 HA (i)t , i.e., it’s thegenerating function of the HF of A.
It’s a standard fact from commutative algebra that theHilbert series of A is a rational function with numeratorPeii=0 hi t , where the hi are integers and he 6= 0.
The sequence hA = (h0 = 1, h1 , . . . , he ) is called theh-vector of A, and the index e is its socle degree.

Artinian algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A is artinian if it’s Cohen-Macaulay of Krull-dimension
zero (i.e., every√ homogeneous prime ideal of A is
maximal), or I = (x1 , . . . , xr ), or HA (i) = 0 for i  0(all equivalent; we’ll mostly use the latter).
Thus, in the artinian case, the h-vector and HF of A
coincide. In particular, the hi are nonnegative.
The h-vector of an artinian algebra is also called an
O-sequence (M-sequence, sometimes).
A classical theorem of Macaulay (Proc. London Math.
Soc. 1927) characterized all possible O-sequences.

Artinian algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A is artinian if it’s Cohen-Macaulay of Krull-dimension
zero (i.e., every√ homogeneous prime ideal of A is
maximal), or I = (x1 , . . . , xr ), or HA (i) = 0 for i  0(all equivalent; we’ll mostly use the latter).
Thus, in the artinian case, the h-vector and HF of A
coincide. In particular, the hi are nonnegative.
The h-vector of an artinian algebra is also called an
O-sequence (M-sequence, sometimes).
A classical theorem of Macaulay (Proc. London Math.
Soc. 1927) characterized all possible O-sequences.

Artinian algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A is artinian if it’s Cohen-Macaulay of Krull-dimension
zero (i.e., every√ homogeneous prime ideal of A is
maximal), or I = (x1 , . . . , xr ), or HA (i) = 0 for i  0(all equivalent; we’ll mostly use the latter).
Thus, in the artinian case, the h-vector and HF of A
coincide. In particular, the hi are nonnegative.
The h-vector of an artinian algebra is also called an
O-sequence (M-sequence, sometimes).
A classical theorem of Macaulay (Proc. London Math.
Soc. 1927) characterized all possible O-sequences.

Artinian algebras

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A is artinian if it’s Cohen-Macaulay of Krull-dimension
zero (i.e., every√ homogeneous prime ideal of A is
maximal), or I = (x1 , . . . , xr ), or HA (i) = 0 for i  0(all equivalent; we’ll mostly use the latter).
Thus, in the artinian case, the h-vector and HF of A
coincide. In particular, the hi are nonnegative.
The h-vector of an artinian algebra is also called an
O-sequence (M-sequence, sometimes).
A classical theorem of Macaulay (Proc. London Math.
Soc. 1927) characterized all possible O-sequences.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Level and Gorenstein algebras
The socle of A artinian is soc(A) = 0 : (x1 , . . . , xr ) ⊂ A.
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The socle-vector of A is the HF of the socle; that is,s(A) = (0, s1 , . . . , se ), where si = dimk soc(A)i .
A is level (of type t) if s(A) = (0, . . . , 0, t).
A is Gorenstein if it’s level of type 1; i.e., whens(A) = (0, . . . , 0, t = 1).
Equivalent homological definitions: A is level when thelast module of the graded minimal free resolution of A
contains only one shift; it’s Gorenstein when thatunique shift appears with multiplicity one.
We won’t go into this today, but see e.g. the monographof Geramita et al. (AMS Memoir 2007) for one of thefirst works developing an homological approach to thestudy of level HFs.

Example
Algebraicpreliminaries

Example 1.

Arbitrarylevel
sequences
Pure
O-sequences

A = k [x, y ]/(x 3 , xy 2 , y 4 )Pure
f -vectors

has HF h = (1, 2, 3, 2) and socle-vectors = (0, 0, 0, 2). Hence A is (monomial) level of type 2.

Matroid
h-vectors

Example 2.
A = C[x, y , z]/(z 3 , 2x 2 z + 3yz 2 , 2x 2 y + 3y 2 z, y 3 , x 4 )has HF h = (1, 3, 6, 6, 3, 1) and socle-vectors = (0, 0, 0, 0, 0, 1). Hence A is Gorenstein.

Example
Algebraicpreliminaries

Example 1.

Arbitrarylevel
sequences
Pure
O-sequences

A = k [x, y ]/(x 3 , xy 2 , y 4 )Pure
f -vectors

has HF h = (1, 2, 3, 2) and socle-vectors = (0, 0, 0, 2). Hence A is (monomial) level of type 2.

Matroid
h-vectors

Example 2.
A = C[x, y , z]/(z 3 , 2x 2 z + 3yz 2 , 2x 2 y + 3y 2 z, y 3 , x 4 )has HF h = (1, 3, 6, 6, 3, 1) and socle-vectors = (0, 0, 0, 0, 0, 1). Hence A is Gorenstein.

The general theme of the talk...

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The main (very) general question of the talk is:
Which O-sequences are the Hilbert functions of level (orGorenstein) algebras? How about in the monomial case?
The study of level algebras and their HFs is important
not only in its own right, but because of many
applications to other fields, including algebraic andenumerative combinatorics, plane partitions, matroid
theory, design theory and finite geometries, algebraic
geometry, etc..
Some of these applications have just recently been
discovered or studied systematically. We’ll try to
discuss a few of the more exciting combinatorial
developments in this talk.

The general theme of the talk...

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The main (very) general question of the talk is:
Which O-sequences are the Hilbert functions of level (orGorenstein) algebras? How about in the monomial case?
The study of level algebras and their HFs is important
not only in its own right, but because of many
applications to other fields, including algebraic andenumerative combinatorics, plane partitions, matroid
theory, design theory and finite geometries, algebraic
geometry, etc..
Some of these applications have just recently been
discovered or studied systematically. We’ll try to
discuss a few of the more exciting combinatorial
developments in this talk.

The general theme of the talk...

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The main (very) general question of the talk is:
Which O-sequences are the Hilbert functions of level (orGorenstein) algebras? How about in the monomial case?
The study of level algebras and their HFs is important
not only in its own right, but because of many
applications to other fields, including algebraic andenumerative combinatorics, plane partitions, matroid
theory, design theory and finite geometries, algebraic
geometry, etc..
Some of these applications have just recently been
discovered or studied systematically. We’ll try to
discuss a few of the more exciting combinatorial
developments in this talk.

Unimodality

Algebraicpreliminaries
Arbitrarylevel
sequences

A finite sequence (of nonnegative integers)
a0 , a1 , . . . , an is unimodal if
a0 ≤ · · · ≤ ak−1 ≤ ak ≥ ak+1 ≥ · · · ≥ an ,PureO-sequences

for some index k , 0 ≤ k ≤ n.

Pure
f -vectors

Unimodality is a central concept in combinatorics,
algebra and other fields. Many important results onlevel algebras and subsets thereof concern unimodality
issues. For a wealth of info, examples and techniques,
see the two classics: “Log-concave and unimodal
sequences in Algebra, Combinatorics and Geometry”,
by R. Stanley (1989, written in 1986); and “Same title:
an update”, by F. Brenti (1994).

Matroid
h-vectors

Unimodality

Algebraicpreliminaries
Arbitrarylevel
sequences

A finite sequence (of nonnegative integers)
a0 , a1 , . . . , an is unimodal if
a0 ≤ · · · ≤ ak−1 ≤ ak ≥ ak+1 ≥ · · · ≥ an ,PureO-sequences

for some index k , 0 ≤ k ≤ n.

Pure
f -vectors

Unimodality is a central concept in combinatorics,
algebra and other fields. Many important results onlevel algebras and subsets thereof concern unimodality
issues. For a wealth of info, examples and techniques,
see the two classics: “Log-concave and unimodal
sequences in Algebra, Combinatorics and Geometry”,
by R. Stanley (1989, written in 1986); and “Same title:
an update”, by F. Brenti (1994).

Matroid
h-vectors

Arbitrary level and Gorenstein HFs

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In codimension r = 2, level and Gorenstein HFs arecompletely characterized. (In fact, so are the HFs
associated to any given socle-vector; see e.g. A.
Iarrobino, TAMS 1984.)
In codimension r = 3, trouble starts. Essentially the
only nice characterization is still that of Gorenstein HFs:
They are exactly the SI-sequences, i.e., they are
symmetric and their first half is differentiable. (Asequence is differentiable if its first difference is an
O-sequence.) See Stanley, Adv. Math. 1978, and thenFZ, PAMS 2006 for a combinatorial proof.
In codimension 3, Geramita et al. (AMS Memoir 2007)
characterized all level HFs of socle degree 5, and thoseof socle degree 6 and type 2, by a variety of techniques.

Arbitrary level and Gorenstein HFs

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In codimension r = 2, level and Gorenstein HFs arecompletely characterized. (In fact, so are the HFs
associated to any given socle-vector; see e.g. A.
Iarrobino, TAMS 1984.)
In codimension r = 3, trouble starts. Essentially the
only nice characterization is still that of Gorenstein HFs:
They are exactly the SI-sequences, i.e., they are
symmetric and their first half is differentiable. (Asequence is differentiable if its first difference is an
O-sequence.) See Stanley, Adv. Math. 1978, and thenFZ, PAMS 2006 for a combinatorial proof.
In codimension 3, Geramita et al. (AMS Memoir 2007)
characterized all level HFs of socle degree 5, and thoseof socle degree 6 and type 2, by a variety of techniques.

Arbitrary level and Gorenstein HFs

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In codimension r = 2, level and Gorenstein HFs arecompletely characterized. (In fact, so are the HFs
associated to any given socle-vector; see e.g. A.
Iarrobino, TAMS 1984.)
In codimension r = 3, trouble starts. Essentially the
only nice characterization is still that of Gorenstein HFs:
They are exactly the SI-sequences, i.e., they are
symmetric and their first half is differentiable. (Asequence is differentiable if its first difference is an
O-sequence.) See Stanley, Adv. Math. 1978, and thenFZ, PAMS 2006 for a combinatorial proof.
In codimension 3, Geramita et al. (AMS Memoir 2007)
characterized all level HFs of socle degree 5, and thoseof socle degree 6 and type 2, by a variety of techniques.

Nonunimodal examples

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For for any r ≥ 3, level HFs can be nonunimodal in the“worst possible way”, i.e., with arbitrarily many peaks(FZ, J. Algebra 2006). In particular, a classification ofsuch HFs seems entirely out of reach.
The above result answered a question from theGeramita et al. 2007 Memoir, and was later improved inseveral directions, both combinatorial and geometric(see e.g. Migliore, Canad. J. Math. 2008).
The state of the art on nonunimodal level HFs for r = 3
and r = 4 is A. Weiss’ PhD thesis (Tufts, 2007).
Weiss constructed classes of nonunimodal HFs of any
type t ≥ 5 for r = 3, and of any type t ≥ 3 for r = 4, bynicely extending some inverse system ideas of
Iarrobino and myself.

Nonunimodal examples

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For for any r ≥ 3, level HFs can be nonunimodal in the“worst possible way”, i.e., with arbitrarily many peaks(FZ, J. Algebra 2006). In particular, a classification ofsuch HFs seems entirely out of reach.
The above result answered a question from theGeramita et al. 2007 Memoir, and was later improved inseveral directions, both combinatorial and geometric(see e.g. Migliore, Canad. J. Math. 2008).
The state of the art on nonunimodal level HFs for r = 3
and r = 4 is A. Weiss’ PhD thesis (Tufts, 2007).
Weiss constructed classes of nonunimodal HFs of any
type t ≥ 5 for r = 3, and of any type t ≥ 3 for r = 4, bynicely extending some inverse system ideas of
Iarrobino and myself.

Nonunimodal examples

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For for any r ≥ 3, level HFs can be nonunimodal in the“worst possible way”, i.e., with arbitrarily many peaks(FZ, J. Algebra 2006). In particular, a classification ofsuch HFs seems entirely out of reach.
The above result answered a question from theGeramita et al. 2007 Memoir, and was later improved inseveral directions, both combinatorial and geometric(see e.g. Migliore, Canad. J. Math. 2008).
The state of the art on nonunimodal level HFs for r = 3
and r = 4 is A. Weiss’ PhD thesis (Tufts, 2007).
Weiss constructed classes of nonunimodal HFs of any
type t ≥ 5 for r = 3, and of any type t ≥ 3 for r = 4, bynicely extending some inverse system ideas of
Iarrobino and myself.

Nonunimodal examples

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For for any r ≥ 3, level HFs can be nonunimodal in the“worst possible way”, i.e., with arbitrarily many peaks(FZ, J. Algebra 2006). In particular, a classification ofsuch HFs seems entirely out of reach.
The above result answered a question from theGeramita et al. 2007 Memoir, and was later improved inseveral directions, both combinatorial and geometric(see e.g. Migliore, Canad. J. Math. 2008).
The state of the art on nonunimodal level HFs for r = 3
and r = 4 is A. Weiss’ PhD thesis (Tufts, 2007).
Weiss constructed classes of nonunimodal HFs of any
type t ≥ 5 for r = 3, and of any type t ≥ 3 for r = 4, bynicely extending some inverse system ideas of
Iarrobino and myself.

Codimension 3 ad 4

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Unimodality in codimension r = 3 is open for types 2, 3
and 4. For r = 4, unimodality is open for types 1 and 2.
In particular, proving the unimodality of Gorenstein HFs
of codimension 4 is an important open problem in thisarea. See Iarrobino-Srinivasan (JPAA 2005),
Migliore-Nagel-FZ (MRL 2008) and Seo-Srinivasan(Comm. Algebra 2012) for some progress, mostly incharacteristic zero.
Another important open question: Are all level HFs ofcodimension 3 flawless? (Recall that h = (1, h1 , . . . , he )is flawless if hi ≤ he−i for all i ≤ e/2.)
If the answer is “no”, this will have striking algebraic
consequences on the Lefschetz properties forGorenstein algebras (BMMNZ2, J. Algebra 2014).

Codimension 3 ad 4

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Unimodality in codimension r = 3 is open for types 2, 3
and 4. For r = 4, unimodality is open for types 1 and 2.
In particular, proving the unimodality of Gorenstein HFs
of codimension 4 is an important open problem in thisarea. See Iarrobino-Srinivasan (JPAA 2005),
Migliore-Nagel-FZ (MRL 2008) and Seo-Srinivasan(Comm. Algebra 2012) for some progress, mostly incharacteristic zero.
Another important open question: Are all level HFs ofcodimension 3 flawless? (Recall that h = (1, h1 , . . . , he )is flawless if hi ≤ he−i for all i ≤ e/2.)
If the answer is “no”, this will have striking algebraic
consequences on the Lefschetz properties forGorenstein algebras (BMMNZ2, J. Algebra 2014).

Codimension 3 ad 4

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Unimodality in codimension r = 3 is open for types 2, 3
and 4. For r = 4, unimodality is open for types 1 and 2.
In particular, proving the unimodality of Gorenstein HFs
of codimension 4 is an important open problem in thisarea. See Iarrobino-Srinivasan (JPAA 2005),
Migliore-Nagel-FZ (MRL 2008) and Seo-Srinivasan(Comm. Algebra 2012) for some progress, mostly incharacteristic zero.
Another important open question: Are all level HFs ofcodimension 3 flawless? (Recall that h = (1, h1 , . . . , he )is flawless if hi ≤ he−i for all i ≤ e/2.)
If the answer is “no”, this will have striking algebraic
consequences on the Lefschetz properties forGorenstein algebras (BMMNZ2, J. Algebra 2014).

Higher codimension

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For r ≥ 5, very little is known. Stanley (Adv. Math.
1978), Bernstein-Iarrobino (Comm. Algebra 1992),
Boij-Laksov (PAMS 1994), etc., proved the existence of
nonunimodal level and Gorenstein HFs for all types
t ≥ 1 and codimensions r ≥ 5. In particular, these HFs
are extremely hard to characterize.
For instance, given r , in general we don’t even know
the minimum f (r ) for a Gorenstein HF (1, r , f (r ), r , 1).
Migliore and I recently proved (arXiv:1512.01433;
PAMS, to appear) that Stanley’s 1978 nonunimodal
example, (1, 13, 12, 13, 1), is the smallest possible.
Asymptotically: f (r ) ∼r −→∞ (6r )2/3 , as conjectured by
Stanley, 1983 (see Migliore-Nagel-FZ, PAMS 2008).
However, regardless(1, r , c, r , 1) is Gorenstein
h of f (r ),i
r +1
if and only if c ∈ f (r ), 2
(FZ, J. Algebra 2009).

Higher codimension

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For r ≥ 5, very little is known. Stanley (Adv. Math.
1978), Bernstein-Iarrobino (Comm. Algebra 1992),
Boij-Laksov (PAMS 1994), etc., proved the existence of
nonunimodal level and Gorenstein HFs for all types
t ≥ 1 and codimensions r ≥ 5. In particular, these HFs
are extremely hard to characterize.
For instance, given r , in general we don’t even know
the minimum f (r ) for a Gorenstein HF (1, r , f (r ), r , 1).
Migliore and I recently proved (arXiv:1512.01433;
PAMS, to appear) that Stanley’s 1978 nonunimodal
example, (1, 13, 12, 13, 1), is the smallest possible.
Asymptotically: f (r ) ∼r −→∞ (6r )2/3 , as conjectured by
Stanley, 1983 (see Migliore-Nagel-FZ, PAMS 2008).
However, regardless(1, r , c, r , 1) is Gorenstein
h of f (r ),i
r +1
if and only if c ∈ f (r ), 2
(FZ, J. Algebra 2009).

Higher codimension

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For r ≥ 5, very little is known. Stanley (Adv. Math.
1978), Bernstein-Iarrobino (Comm. Algebra 1992),
Boij-Laksov (PAMS 1994), etc., proved the existence of
nonunimodal level and Gorenstein HFs for all types
t ≥ 1 and codimensions r ≥ 5. In particular, these HFs
are extremely hard to characterize.
For instance, given r , in general we don’t even know
the minimum f (r ) for a Gorenstein HF (1, r , f (r ), r , 1).
Migliore and I recently proved (arXiv:1512.01433;
PAMS, to appear) that Stanley’s 1978 nonunimodal
example, (1, 13, 12, 13, 1), is the smallest possible.
Asymptotically: f (r ) ∼r −→∞ (6r )2/3 , as conjectured by
Stanley, 1983 (see Migliore-Nagel-FZ, PAMS 2008).
However, regardless(1, r , c, r , 1) is Gorenstein
h of f (r ),i
r +1
if and only if c ∈ f (r ), 2
(FZ, J. Algebra 2009).

Higher codimension

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For r ≥ 5, very little is known. Stanley (Adv. Math.
1978), Bernstein-Iarrobino (Comm. Algebra 1992),
Boij-Laksov (PAMS 1994), etc., proved the existence of
nonunimodal level and Gorenstein HFs for all types
t ≥ 1 and codimensions r ≥ 5. In particular, these HFs
are extremely hard to characterize.
For instance, given r , in general we don’t even know
the minimum f (r ) for a Gorenstein HF (1, r , f (r ), r , 1).
Migliore and I recently proved (arXiv:1512.01433;
PAMS, to appear) that Stanley’s 1978 nonunimodal
example, (1, 13, 12, 13, 1), is the smallest possible.
Asymptotically: f (r ) ∼r −→∞ (6r )2/3 , as conjectured by
Stanley, 1983 (see Migliore-Nagel-FZ, PAMS 2008).
However, regardless(1, r , c, r , 1) is Gorenstein
h of f (r ),i
r +1
if and only if c ∈ f (r ), 2
(FZ, J. Algebra 2009).

Higher codimension

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

For r ≥ 5, very little is known. Stanley (Adv. Math.
1978), Bernstein-Iarrobino (Comm. Algebra 1992),
Boij-Laksov (PAMS 1994), etc., proved the existence of
nonunimodal level and Gorenstein HFs for all types
t ≥ 1 and codimensions r ≥ 5. In particular, these HFs
are extremely hard to characterize.
For instance, given r , in general we don’t even know
the minimum f (r ) for a Gorenstein HF (1, r , f (r ), r , 1).
Migliore and I recently proved (arXiv:1512.01433;
PAMS, to appear) that Stanley’s 1978 nonunimodal
example, (1, 13, 12, 13, 1), is the smallest possible.
Asymptotically: f (r ) ∼r −→∞ (6r )2/3 , as conjectured by
Stanley, 1983 (see Migliore-Nagel-FZ, PAMS 2008).
However, regardless(1, r , c, r , 1) is Gorenstein
h of f (r ),i
r +1
if and only if c ∈ f (r ), 2
(FZ, J. Algebra 2009).

The Interval Property

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Interval Conjecture (IC) (FZ, J. Algebra 2009).
Suppose that for some α ≥ 0, (1, h1 , . . . , hi , . . . , he ) and(1, h1 , . . . , hi + α, . . . , he ) are both level HFs.
Then (1, h1 , . . . , hi + β, . . . , he ) is also level, for each
integer β in the interval [0, α].
Gorenstein Interval Conjecture (GIC).
Suppose that for some α ≥ 0,
(1, h1 , . . . , hi , . . . , he−i , . . . , 1) and(1, h1 , . . . , hi + α, . . . , he−i + α, . . . , 1) are both
Gorenstein HFs. Then(1, h1 , . . . , hi + β, . . . , he−i + β, . . . , 1)
is also Gorenstein, for each β in the interval [0, α].

The Interval Property

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

Interval Conjecture (IC) (FZ, J. Algebra 2009).
Suppose that for some α ≥ 0, (1, h1 , . . . , hi , . . . , he ) and(1, h1 , . . . , hi + α, . . . , he ) are both level HFs.
Then (1, h1 , . . . , hi + β, . . . , he ) is also level, for each
integer β in the interval [0, α].
Gorenstein Interval Conjecture (GIC).
Suppose that for some α ≥ 0,
(1, h1 , . . . , hi , . . . , he−i , . . . , 1) and(1, h1 , . . . , hi + α, . . . , he−i + α, . . . , 1) are both
Gorenstein HFs. Then(1, h1 , . . . , hi + β, . . . , he−i + β, . . . , 1)
is also Gorenstein, for each β in the interval [0, α].

How much does the characteristic matter?

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The IC and GIC seem consistent with all existing
techniques for level and Gorenstein HFs, and would
finally provide a strong structural result on their sets.
The Interval Property is true in many instances;
however, it’s false in general for pure O-sequences(conjectured in BMMNZ, and disproved by
Varbaro-Constantinescu), but true (and consequential)
in socle degree 3; false also for pure f -vectors(Pastine-FZ, PAMS 2015); false in general forr -differential posets (Stanley-FZ, E-JC 2012), but open
for 1-differential posets, which is the main class.
Finally, another question I’d love to see answered on
arbitrary level and Gorenstein HFs:
Does there exist a HF which is level in somecharacteristic but not in another??
If yes, extremely interesting, though no big ideas so far.

How much does the characteristic matter?

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The IC and GIC seem consistent with all existing
techniques for level and Gorenstein HFs, and would
finally provide a strong structural result on their sets.
The Interval Property is true in many instances;
however, it’s false in general for pure O-sequences(conjectured in BMMNZ, and disproved by
Varbaro-Constantinescu), but true (and consequential)
in socle degree 3; false also for pure f -vectors(Pastine-FZ, PAMS 2015); false in general forr -differential posets (Stanley-FZ, E-JC 2012), but open
for 1-differential posets, which is the main class.
Finally, another question I’d love to see answered on
arbitrary level and Gorenstein HFs:
Does there exist a HF which is level in somecharacteristic but not in another??
If yes, extremely interesting, though no big ideas so far.

How much does the characteristic matter?

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The IC and GIC seem consistent with all existing
techniques for level and Gorenstein HFs, and would
finally provide a strong structural result on their sets.
The Interval Property is true in many instances;
however, it’s false in general for pure O-sequences(conjectured in BMMNZ, and disproved by
Varbaro-Constantinescu), but true (and consequential)
in socle degree 3; false also for pure f -vectors(Pastine-FZ, PAMS 2015); false in general forr -differential posets (Stanley-FZ, E-JC 2012), but open
for 1-differential posets, which is the main class.
Finally, another question I’d love to see answered on
arbitrary level and Gorenstein HFs:
Does there exist a HF which is level in somecharacteristic but not in another??
If yes, extremely interesting, though no big ideas so far.

Pure O-sequences

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A monomial order ideal is a nonempty, finite set X of(monic) monomials, such that for M ∈ X and N dividing
M, we have N ∈ X . (It’s a special case of an order ideal
in a poset.)
The h-vector of X is its degree vector,
h = (1, h1 , . . . , he ), counting the monomials of X ineach degree.
A pure O-sequence is the h-vector of a monomial orderideal X whose maximal (by divisibility) monomials all
have the same degree.
Equivalently, via Macaulay’s inverse systems, pureO-sequences coincide with HFs of artinian monomial
level algebras.

Pure O-sequences

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A monomial order ideal is a nonempty, finite set X of(monic) monomials, such that for M ∈ X and N dividing
M, we have N ∈ X . (It’s a special case of an order ideal
in a poset.)
The h-vector of X is its degree vector,
h = (1, h1 , . . . , he ), counting the monomials of X ineach degree.
A pure O-sequence is the h-vector of a monomial orderideal X whose maximal (by divisibility) monomials all
have the same degree.
Equivalently, via Macaulay’s inverse systems, pureO-sequences coincide with HFs of artinian monomial
level algebras.

Pure O-sequences

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A monomial order ideal is a nonempty, finite set X of(monic) monomials, such that for M ∈ X and N dividing
M, we have N ∈ X . (It’s a special case of an order ideal
in a poset.)
The h-vector of X is its degree vector,
h = (1, h1 , . . . , he ), counting the monomials of X ineach degree.
A pure O-sequence is the h-vector of a monomial orderideal X whose maximal (by divisibility) monomials all
have the same degree.
Equivalently, via Macaulay’s inverse systems, pureO-sequences coincide with HFs of artinian monomial
level algebras.

Pure O-sequences

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A monomial order ideal is a nonempty, finite set X of(monic) monomials, such that for M ∈ X and N dividing
M, we have N ∈ X . (It’s a special case of an order ideal
in a poset.)
The h-vector of X is its degree vector,
h = (1, h1 , . . . , he ), counting the monomials of X ineach degree.
A pure O-sequence is the h-vector of a monomial orderideal X whose maximal (by divisibility) monomials all
have the same degree.
Equivalently, via Macaulay’s inverse systems, pureO-sequences coincide with HFs of artinian monomial
level algebras.

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Example
Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The pure monomial order ideal (inside k[x, y, z]) withmaximal monomials xy 3 z and x 2 z 3 is:
X = {xy3 z, x2 z3 ;y 3 z, xy 2 z, xy 3 , xz 3 , x 2 z 2 ;y 2 z, y 3 , xyz, xy 2 , xz 2 , z 3 , x 2 z;z, y, x;yz, y 2 , xz, xy, z 2 , x 2 ;1}Hence the h-vector of X is the pure O-sequenceh = (1, 3, 6, 7, 5, 2).

Some old and new results

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, AMS Memoir 2012, we attempted todevelop a systematic theory of pure O-sequences.
Very few results were known before, and many are still
unknown now. Among the main previous results:
THEOREM (Hibi). Let h = (1, h1 , . . . , he ) be a pureO-sequence. Then hi ≤ hj for all 0 ≤ i ≤ j ≤ e − i.
THEOREM (Hausel). The first half of h = (1, h1 , . . . , he )is differentiable. (A converse to Hausel’s result wasproved in BMMNZ, thus making this a characterization.)
NOTE 1. Differentiability holds characteristic free,though Hausel’s proof was of a “g-theorem” incharacteristic zero.
NOTE 2. This fact connects with the Lefschetz
Properties, an interesting topic of combinatorial
algebra, which we recently related to the combinatorics
of plane partitions. We won’t discuss it here.

Some old and new results

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, AMS Memoir 2012, we attempted todevelop a systematic theory of pure O-sequences.
Very few results were known before, and many are still
unknown now. Among the main previous results:
THEOREM (Hibi). Let h = (1, h1 , . . . , he ) be a pureO-sequence. Then hi ≤ hj for all 0 ≤ i ≤ j ≤ e − i.
THEOREM (Hausel). The first half of h = (1, h1 , . . . , he )is differentiable. (A converse to Hausel’s result wasproved in BMMNZ, thus making this a characterization.)
NOTE 1. Differentiability holds characteristic free,though Hausel’s proof was of a “g-theorem” incharacteristic zero.
NOTE 2. This fact connects with the Lefschetz
Properties, an interesting topic of combinatorial
algebra, which we recently related to the combinatorics
of plane partitions. We won’t discuss it here.

Some old and new results

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, AMS Memoir 2012, we attempted todevelop a systematic theory of pure O-sequences.
Very few results were known before, and many are still
unknown now. Among the main previous results:
THEOREM (Hibi). Let h = (1, h1 , . . . , he ) be a pureO-sequence. Then hi ≤ hj for all 0 ≤ i ≤ j ≤ e − i.
THEOREM (Hausel). The first half of h = (1, h1 , . . . , he )is differentiable. (A converse to Hausel’s result wasproved in BMMNZ, thus making this a characterization.)
NOTE 1. Differentiability holds characteristic free,though Hausel’s proof was of a “g-theorem” incharacteristic zero.
NOTE 2. This fact connects with the Lefschetz
Properties, an interesting topic of combinatorial
algebra, which we recently related to the combinatorics
of plane partitions. We won’t discuss it here.

Some old and new results

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, AMS Memoir 2012, we attempted todevelop a systematic theory of pure O-sequences.
Very few results were known before, and many are still
unknown now. Among the main previous results:
THEOREM (Hibi). Let h = (1, h1 , . . . , he ) be a pureO-sequence. Then hi ≤ hj for all 0 ≤ i ≤ j ≤ e − i.
THEOREM (Hausel). The first half of h = (1, h1 , . . . , he )is differentiable. (A converse to Hausel’s result wasproved in BMMNZ, thus making this a characterization.)
NOTE 1. Differentiability holds characteristic free,though Hausel’s proof was of a “g-theorem” incharacteristic zero.
NOTE 2. This fact connects with the Lefschetz
Properties, an interesting topic of combinatorial
algebra, which we recently related to the combinatorics
of plane partitions. We won’t discuss it here.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THEOREM (Stanley). There exist nonunimodal pureO-sequences.
Using his own characterization of the h-vectors of
Cohen-Macaulay simplicial complexes, Stanley (1977)
constructed this nonunimodal example of an f -vector ofa CM complex (hence a pure O-sequence):h = (1, 505, 2065, 3395, 3325, 3493).
Other nonunimodal examples then followed withdifferent techniques (see Björner, Bull. AMS 1981;
Michael-Traves, Graphs and Comb. 2003; Boij-FZ,
PAMS 2007; Pastine-FZ, PAMS 2015; etc.).
A shortest nonunimodal pure O-sequence ish = (1, 49, 81, 79, 81), obtained by adding(1, 5, 15, 35, 70) (once) and (1, 4, 6, 4, 1) (11 times).
In general, unimodality fails in any socle degree e ≥ 4(BMMNZ), but not for e ≤ 3, because of Hibi’s theorem.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THEOREM (Stanley). There exist nonunimodal pureO-sequences.
Using his own characterization of the h-vectors of
Cohen-Macaulay simplicial complexes, Stanley (1977)
constructed this nonunimodal example of an f -vector ofa CM complex (hence a pure O-sequence):h = (1, 505, 2065, 3395, 3325, 3493).
Other nonunimodal examples then followed withdifferent techniques (see Björner, Bull. AMS 1981;
Michael-Traves, Graphs and Comb. 2003; Boij-FZ,
PAMS 2007; Pastine-FZ, PAMS 2015; etc.).
A shortest nonunimodal pure O-sequence ish = (1, 49, 81, 79, 81), obtained by adding(1, 5, 15, 35, 70) (once) and (1, 4, 6, 4, 1) (11 times).
In general, unimodality fails in any socle degree e ≥ 4(BMMNZ), but not for e ≤ 3, because of Hibi’s theorem.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THEOREM (Stanley). There exist nonunimodal pureO-sequences.
Using his own characterization of the h-vectors of
Cohen-Macaulay simplicial complexes, Stanley (1977)
constructed this nonunimodal example of an f -vector ofa CM complex (hence a pure O-sequence):h = (1, 505, 2065, 3395, 3325, 3493).
Other nonunimodal examples then followed withdifferent techniques (see Björner, Bull. AMS 1981;
Michael-Traves, Graphs and Comb. 2003; Boij-FZ,
PAMS 2007; Pastine-FZ, PAMS 2015; etc.).
A shortest nonunimodal pure O-sequence ish = (1, 49, 81, 79, 81), obtained by adding(1, 5, 15, 35, 70) (once) and (1, 4, 6, 4, 1) (11 times).
In general, unimodality fails in any socle degree e ≥ 4(BMMNZ), but not for e ≤ 3, because of Hibi’s theorem.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THEOREM (Stanley). There exist nonunimodal pureO-sequences.
Using his own characterization of the h-vectors of
Cohen-Macaulay simplicial complexes, Stanley (1977)
constructed this nonunimodal example of an f -vector ofa CM complex (hence a pure O-sequence):h = (1, 505, 2065, 3395, 3325, 3493).
Other nonunimodal examples then followed withdifferent techniques (see Björner, Bull. AMS 1981;
Michael-Traves, Graphs and Comb. 2003; Boij-FZ,
PAMS 2007; Pastine-FZ, PAMS 2015; etc.).
A shortest nonunimodal pure O-sequence ish = (1, 49, 81, 79, 81), obtained by adding(1, 5, 15, 35, 70) (once) and (1, 4, 6, 4, 1) (11 times).
In general, unimodality fails in any socle degree e ≥ 4(BMMNZ), but not for e ≤ 3, because of Hibi’s theorem.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THEOREM (Stanley). There exist nonunimodal pureO-sequences.
Using his own characterization of the h-vectors of
Cohen-Macaulay simplicial complexes, Stanley (1977)
constructed this nonunimodal example of an f -vector ofa CM complex (hence a pure O-sequence):h = (1, 505, 2065, 3395, 3325, 3493).
Other nonunimodal examples then followed withdifferent techniques (see Björner, Bull. AMS 1981;
Michael-Traves, Graphs and Comb. 2003; Boij-FZ,
PAMS 2007; Pastine-FZ, PAMS 2015; etc.).
A shortest nonunimodal pure O-sequence ish = (1, 49, 81, 79, 81), obtained by adding(1, 5, 15, 35, 70) (once) and (1, 4, 6, 4, 1) (11 times).
In general, unimodality fails in any socle degree e ≥ 4(BMMNZ), but not for e ≤ 3, because of Hibi’s theorem.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, we proved that unimodality in fact may fail,with arbitrarily many peaks, for pure O-sequences ofany codimension r ≥ 3.
Results further generalized by Pastine-FZ (PAMS2015), proving nonunimodality with arbitrarily manypeaks for CM f -vectors. In particular, we gave anentirely combinatorial proof for pure f -vectors.
However, for the further subset of matroid f -vectors,
unimodality conjecturally holds. (See Stanley’s
Twenty-Fifth Problem, from his ICM 2000 essay.)
For a recent proof of this unimodality conjecture forrepresentable matroids, see the impressive works of
Huh (JAMS 2012, in characteristic zero), and Huh-Katz(Math. Ann. 2013, in arbitrary characteristic).

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, we proved that unimodality in fact may fail,with arbitrarily many peaks, for pure O-sequences ofany codimension r ≥ 3.
Results further generalized by Pastine-FZ (PAMS2015), proving nonunimodality with arbitrarily manypeaks for CM f -vectors. In particular, we gave anentirely combinatorial proof for pure f -vectors.
However, for the further subset of matroid f -vectors,
unimodality conjecturally holds. (See Stanley’s
Twenty-Fifth Problem, from his ICM 2000 essay.)
For a recent proof of this unimodality conjecture forrepresentable matroids, see the impressive works of
Huh (JAMS 2012, in characteristic zero), and Huh-Katz(Math. Ann. 2013, in arbitrary characteristic).

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, we proved that unimodality in fact may fail,with arbitrarily many peaks, for pure O-sequences ofany codimension r ≥ 3.
Results further generalized by Pastine-FZ (PAMS2015), proving nonunimodality with arbitrarily manypeaks for CM f -vectors. In particular, we gave anentirely combinatorial proof for pure f -vectors.
However, for the further subset of matroid f -vectors,
unimodality conjecturally holds. (See Stanley’s
Twenty-Fifth Problem, from his ICM 2000 essay.)
For a recent proof of this unimodality conjecture forrepresentable matroids, see the impressive works of
Huh (JAMS 2012, in characteristic zero), and Huh-Katz(Math. Ann. 2013, in arbitrary characteristic).

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

In BMMNZ, we proved that unimodality in fact may fail,with arbitrarily many peaks, for pure O-sequences ofany codimension r ≥ 3.
Results further generalized by Pastine-FZ (PAMS2015), proving nonunimodality with arbitrarily manypeaks for CM f -vectors. In particular, we gave anentirely combinatorial proof for pure f -vectors.
However, for the further subset of matroid f -vectors,
unimodality conjecturally holds. (See Stanley’s
Twenty-Fifth Problem, from his ICM 2000 essay.)
For a recent proof of this unimodality conjecture forrepresentable matroids, see the impressive works of
Huh (JAMS 2012, in characteristic zero), and Huh-Katz(Math. Ann. 2013, in arbitrary characteristic).

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

So far it’s known that all pure O-sequences of type 1
are unimodal, in any number of variables (trivial to see,e.g. using generating functions).
Are all pure O-sequences of type 2 unimodal??
The answer is known to be positive only for r = 3(BMMNZ), and r = 4 (Boyle, PhD Thesis, Notre Dame
2012). Note that, in both instances, unimodality is still
unknown for arbitrary level algebras of type 2.
Also, what is the largest type t forcing unimodality for all
pure O-sequences of codimension 3? Boyle recently
proved that t ≥ 3.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

So far it’s known that all pure O-sequences of type 1
are unimodal, in any number of variables (trivial to see,e.g. using generating functions).
Are all pure O-sequences of type 2 unimodal??
The answer is known to be positive only for r = 3(BMMNZ), and r = 4 (Boyle, PhD Thesis, Notre Dame
2012). Note that, in both instances, unimodality is still
unknown for arbitrary level algebras of type 2.
Also, what is the largest type t forcing unimodality for all
pure O-sequences of codimension 3? Boyle recently
proved that t ≥ 3.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

So far it’s known that all pure O-sequences of type 1
are unimodal, in any number of variables (trivial to see,e.g. using generating functions).
Are all pure O-sequences of type 2 unimodal??
The answer is known to be positive only for r = 3(BMMNZ), and r = 4 (Boyle, PhD Thesis, Notre Dame
2012). Note that, in both instances, unimodality is still
unknown for arbitrary level algebras of type 2.
Also, what is the largest type t forcing unimodality for all
pure O-sequences of codimension 3? Boyle recently
proved that t ≥ 3.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

So far it’s known that all pure O-sequences of type 1
are unimodal, in any number of variables (trivial to see,e.g. using generating functions).
Are all pure O-sequences of type 2 unimodal??
The answer is known to be positive only for r = 3(BMMNZ), and r = 4 (Boyle, PhD Thesis, Notre Dame
2012). Note that, in both instances, unimodality is still
unknown for arbitrary level algebras of type 2.
Also, what is the largest type t forcing unimodality for all
pure O-sequences of codimension 3? Boyle recently
proved that t ≥ 3.

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Pure f -vectors

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

A collection ∆ of subsets of V = {v1 , . . . , vn } is a
simplicial complex if, for each F ∈ ∆ and G ⊆ F , wehave G ∈ ∆. (Yet another order ideal in a poset!)
The elements of ∆ are dubbed faces, and the maximal
faces (under inclusion) are the facets of ∆.
The f -vector of ∆ is the vector f (∆) = (1, f0 , . . . , fd−1 ),where fi is the number of cardinality i + 1 faces of ∆.
A pure f -vector is the f -vector of a pure simplicial
complex, i.e., one with all facets of the same cardinality.
Equivalently, a pure f -vector is a squarefree pureO-sequence, hence an even smaller class of level HFs!
Little is known about pure f -vectors. See
Colbourn-Keranen-Kreher, Discrete Math. 2014, for acharacterization in socle degree 3. Virtually any result
might be interesting. Numerous applications, including
to finite geometries and combinatorial designs...

Matroid simplicial complexes

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors

Let k be a field and ∆ any simplicial complex overV = {v1 , . . . , vn }. We can associate to ∆ a squarefree
monomial ideal in S = k [x1 , . . . , xn ], namelyYI∆ = 
xi | F 6∈ ∆ ⊆ S.
vi ∈F

Matroid
h-vectorsI∆ is the Stanley-Reisner ideal of ∆, and the quotient
algebra k [∆] = S/I∆ is its Stanley-Reisner ring.
∆ is a matroid complex if, for every subset W ⊆ V , therestriction ∆|W is a pure simplicial complex.

Matroid simplicial complexes

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors

Let k be a field and ∆ any simplicial complex overV = {v1 , . . . , vn }. We can associate to ∆ a squarefree
monomial ideal in S = k [x1 , . . . , xn ], namelyYI∆ = 
xi | F 6∈ ∆ ⊆ S.
vi ∈F

Matroid
h-vectorsI∆ is the Stanley-Reisner ideal of ∆, and the quotient
algebra k [∆] = S/I∆ is its Stanley-Reisner ring.
∆ is a matroid complex if, for every subset W ⊆ V , therestriction ∆|W is a pure simplicial complex.

Matroid simplicial complexes

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors

Let k be a field and ∆ any simplicial complex overV = {v1 , . . . , vn }. We can associate to ∆ a squarefree
monomial ideal in S = k [x1 , . . . , xn ], namelyYI∆ = 
xi | F 6∈ ∆ ⊆ S.
vi ∈F

Matroid
h-vectorsI∆ is the Stanley-Reisner ideal of ∆, and the quotient
algebra k [∆] = S/I∆ is its Stanley-Reisner ring.
∆ is a matroid complex if, for every subset W ⊆ V , therestriction ∆|W is a pure simplicial complex.

The big, unproved connection

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The h-vector of a simplicial complex ∆ is defined as theh-vector of its Stanley-Reisner ring k[∆].
It’s a standard combinatorial algebra fact that theh-vector of a matroid is level (though in general ∆ has
positive Krull-dimension). In fact, more is conjectured:
CONJECTURE (Stanley 1977). A matroid h-vector is apure O-sequence!
A huge amount of work over the past 40 years, though
only solutions of special cases. With Hà and Stokes(Annals Comb. 2013), I introduced a first possible
general approach, using the theory of pureO-sequences. However, in the paper we completely
solved only the Krull-dimension 2 case.
See also, among the many interesting recent works:
Constantinescu-Varbaro, De Loera-Kemper-Klee,
Merino-Noble-Ramı́rez-Villarroel, Oh, Schweig.

The big, unproved connection

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The h-vector of a simplicial complex ∆ is defined as theh-vector of its Stanley-Reisner ring k[∆].
It’s a standard combinatorial algebra fact that theh-vector of a matroid is level (though in general ∆ has
positive Krull-dimension). In fact, more is conjectured:
CONJECTURE (Stanley 1977). A matroid h-vector is apure O-sequence!
A huge amount of work over the past 40 years, though
only solutions of special cases. With Hà and Stokes(Annals Comb. 2013), I introduced a first possible
general approach, using the theory of pureO-sequences. However, in the paper we completely
solved only the Krull-dimension 2 case.
See also, among the many interesting recent works:
Constantinescu-Varbaro, De Loera-Kemper-Klee,
Merino-Noble-Ramı́rez-Villarroel, Oh, Schweig.

The big, unproved connection

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The h-vector of a simplicial complex ∆ is defined as theh-vector of its Stanley-Reisner ring k[∆].
It’s a standard combinatorial algebra fact that theh-vector of a matroid is level (though in general ∆ has
positive Krull-dimension). In fact, more is conjectured:
CONJECTURE (Stanley 1977). A matroid h-vector is apure O-sequence!
A huge amount of work over the past 40 years, though
only solutions of special cases. With Hà and Stokes(Annals Comb. 2013), I introduced a first possible
general approach, using the theory of pureO-sequences. However, in the paper we completely
solved only the Krull-dimension 2 case.
See also, among the many interesting recent works:
Constantinescu-Varbaro, De Loera-Kemper-Klee,
Merino-Noble-Ramı́rez-Villarroel, Oh, Schweig.

The big, unproved connection

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The h-vector of a simplicial complex ∆ is defined as theh-vector of its Stanley-Reisner ring k[∆].
It’s a standard combinatorial algebra fact that theh-vector of a matroid is level (though in general ∆ has
positive Krull-dimension). In fact, more is conjectured:
CONJECTURE (Stanley 1977). A matroid h-vector is apure O-sequence!
A huge amount of work over the past 40 years, though
only solutions of special cases. With Hà and Stokes(Annals Comb. 2013), I introduced a first possible
general approach, using the theory of pureO-sequences. However, in the paper we completely
solved only the Krull-dimension 2 case.
See also, among the many interesting recent works:
Constantinescu-Varbaro, De Loera-Kemper-Klee,
Merino-Noble-Ramı́rez-Villarroel, Oh, Schweig.

The big, unproved connection

Algebraicpreliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

The h-vector of a simplicial complex ∆ is defined as theh-vector of its Stanley-Reisner ring k[∆].
It’s a standard combinatorial algebra fact that theh-vector of a matroid is level (though in general ∆ has
positive Krull-dimension). In fact, more is conjectured:
CONJECTURE (Stanley 1977). A matroid h-vector is apure O-sequence!
A huge amount of work over the past 40 years, though
only solutions of special cases. With Hà and Stokes(Annals Comb. 2013), I introduced a first possible
general approach, using the theory of pureO-sequences. However, in the paper we completely
solved only the Krull-dimension 2 case.
See also, among the many interesting recent works:
Constantinescu-Varbaro, De Loera-Kemper-Klee,
Merino-Noble-Ramı́rez-Villarroel, Oh, Schweig.

Algebraic
preliminaries
Arbitrarylevel
sequences
Pure
O-sequencesPure
f -vectors
Matroid
h-vectors

THANK YOU!!!