Algebraic Combinatorics
Math. 629, Sec. 001, Spring 2017
Announcements and Notes
Urgent stuff
The 3rd (and last) HW
has been posted here below.
NB!
(i) A crash-course in duality and the proof
of the Dehn-Sommerville Equations, and (ii)
solutions to the first HW assignment have been
posted here below.
Please Note!
There will weekly talks at our
Combinatorics, Algebra and Geometry Seminar (CAGS)
on Fridays (Exploratory Hall, room 4106, at 12:30 pm).
Many of the talks there are relevant to
what we are covering in class. -- For more info check
the following CAGS website.
General Info
Here below is a pdf version of the promotional
poster for the course, it contains a rough outline
of our plan of attack:
NB!
All handouts for this class will be listed with an itemized icon
as with the Syllabus.
For general more detailed information regarding this course, the Syllbus
will be posted here below shortly. Please read it carefully as it
outlines the overall plan for this course.
It should answer all practical questions.
In the above figure we see a regular soccer ball, which is actually a
convex polyhedron or a 3-dimensional convex polytope if each pentagonal or hexagonal
face is not curved but rather a flat two-dimensional surface. This polytope
is called the
Truncated icosahedron,
and is one of the 13 Archimedean solids, which are semi-regular convex polyhedrons composed
of regular polygons.
A convex polytope is basically a multidimensional convex polyhedron with vertices,
edges and faces of various dimensions. A 2-dimensional polytope is a convex
polygon and trivially each polygon has equally as many edges as vertices,
which can be of any order, and just one 2-dimensional face, namely the
internal of the polygon itself. In 3-dimensions it gets a tad more complicated
but still there are some very nicely stated conditions on the number of vertices, edges,
and 2-dimensional faces due to Stenitz from 1922. The interesting (and annoying!)
thing is that in dimensions 4 and higher, it is unknown how to describe
the conditions the number of vertices, edges and i-dimensional
faces must satisfy in order for them to count (or represent) the number
of vertices, edges and faces of a polytope. Finding such a description
has been, and still is, an active area of research in algebraic combinatorics.
The reason this is a branch in algebraic combinatorics is
that it commutative algebra has turned out to be indispensable tool
in obtaining important and descriptive results about polytopes
and related combinatorial topics.
This topics course will serve as an introduction
to the application of abstract algebra in combinatorics,
and the hope is that, after this course, it should be clear
that a firm knowledge in commutative algebra is necessary for
the working combinatorist. -- I will assume basic knowledge
in algebra (Math 621, Algebra, first year graduate course, or equivalent),
and some combinatorics (Math 325, Discrete Mathematics II, or equivalent)
is preferred. However, the most important thing is to have mathematical
maturity and an open mind.
There is no required text for the course. The lectures will be self contained
and be based in part of some of the following books, which also will serve
as a list of references.
- Takayuki Hibi,
Algebraic Combinatorics; on Convex Polytopes, Carslaw Publications, (1992). (Out of print).
- Richard P. Stanley,
Combinatorics and Commutative Algebra, Second Edition,
Progress in Mathematics (PM-41), Birkhauser, Boston, (1996).
- Ezra Miller; Bernd Sturmfels,
Combinatorial Commutative Algebra, Graduate Texts in Mathematics (GTM-227), Springer Verlag, New York, (2005).
- Arne Brøndsted,
An Introduction to Convex Polytopes,
Graduate Texts in Mathematics (GTM-90), Springer Verlag, New York, (1983).
In addition, and if time permits, we might discuss some results
from selected research articles.
Weekly recap of Lectures and Problems
Here below are some notes to recap the lecture each week.
Also, occasional problems and/or their solutions might
be posted here as well.
- First week, January 23.
- Part I, Convex polytopes and simplicial complexes
In this first lecture we introduced a convex polytope,
one of the fundamental object of algebraic combinatorics. We
defined a convex polytope as the convex hull of finitely many
points in the N-dimensional Euclidean space, with
its usual topology. Each polytope has faces which
are precisely the non-empty intersections of the polytope
and its supporting hyperplanes. Each face is itself
a polytope and an i-face is a face of dimension
i. By convention for a polytope of dimension d
we call its 0-faces vertices,
its 1-faces edges and its
(d-1)-faces facets. We showed that each polytope
is precisely the convex hull of its vertices, and that each face
is the precisely the convex hull of those vertices contained in
the face. This shows that the polytope can be presented in a finite
way, so to speak, as a set-system or hypergraph on a finite set of vertices,
i.e. a collection of subsets of a given finite set we call a vertex set. -- Next
week we will continue to discuss the properties of polytopes and introduce
some parameters attached to them.
- Second week, January 30.
In this second lecture we used two fundamental properties
of a polytope to define a polyhedral complex,
as a finite collection of convex polytopes such that
(i) every face of a polytope in the complex is also contained in
the complex, and (ii) the intersection of two polytopes in the complex
is also in the complex. We have seen that if we know the vertices
of a polytope, then every face is uniquely determined by a subset
of vertices of the polytope. However, not every subset will yield
a face. -- A simplex is precisely a polytope where every
subset of its vertices will yield a face. Simplices are the fundamental
polytopes partily for this very reason. A polyhedral complex
is simplicial if every polytope in it is a simplex
of some dimension. Likewise, a polytope is simplicial
if every face is a simplex. Two polyhedral complexes are
of the same combinatorial type if their face posets
are order isomorphic.
- Third week, February 6.
The f-vector
of a complex of dimension d-1 is the
d-tuple where the i-th coordinate denotes
the number of faces of dimension i for each i
from 0 to d-1. We also defined the corresponding
h-vector which is a (d+1)-tuple.
Two complexes of the same combinatorial type have the same
f- and h-vectors, but the converse
is far from true, as easy examples show us. Characterizing
the vectors that are f-vectors or h-vectors
is a daunting task and only small parts have been solved.
The f-vector and the h-vector of a polyhedral
complex are equivalent, in that one can be obtained from the other.
Since the f-vector directly describes the number of faces
of each dimension, one might ask why bother with the h-vector?
One reason for the h-vector is that each simplex of
dimension d has (1,1,1,...,1) as its
h-vector. Another reason is that there are certain equations
or condictions easier to state using the h-vector (we will
see later that the Dehn-Sommervill Equations have an elegant representation
using the h-vector.)
- Part II, The Katona-Kruskal-Shutzenberger theorem
The information of the f-vector or the h-vector for a polyhedral
complex is by no means enough to characterize the complex, as we
have seen with simple examples, but it provides the first
tool to distinguish complexes from one another. Some integer
vectors are f-vectors for polyhedral complexes others
are not. Hence, it is of great interest to characterize
which vectors are indeed f-vectors for polyhedral
complexes. In general this is unknown, but the characterization
of f-vectors for simplicial complexes is known
and has a nice solution. The theorem describing this characterization
is due to Katona, Kruskal and Schutzenberger (KKS Thm), all who obtained
the same characterization independently.
- Forth week, February 13.
To describe the characterization in the KKS theorem,
one must use the i-binomial representation of a positive
integer, and the reverse lexicographical ordering of subsets
of [n] of cardinality i.
The proof of the KKS Thm is unusual from most mathematical theorems,
in that when proving a characterization, the direction to prove
that the given conditions in (iii) of the f-vector imply the existence of a
concrete simplicial complex with the given vector as its f-vector as in (i)
is realitively easy. The hard part is actually to show that the
f-vector of every simplicial complex in (i) actually does satisfies
the given conditions in (iii). To do that one must rely on a fundamental
theorem of the combinatorics of finite sets, that the boundary of
the compression of a given collection of sets is contained in the
compression of the boundary of the same collection of sets. The proof
of this theorem, which we just used and didn't prove, can be found
in I. Anderson, Combinatorics of finite sets, Oxford University
Press, 1987.
- Part III, Simplicial polytopes and spheres
Here we will discuss (i) The Dehn-Sommerville Equations (DSE),
(ii) The Upper Bound Theorem (UBT), and (iii) The Lower Bound Theorem
(LBT) for simplicial spheres, which are simplicial complexes whose
underlying sets are homeomorphic to topological shperes. Clearly,
the boundary complex of a polytope is homeomorphic to a sphere,
so a simplicial sphere does generalize the boundary complex of a simplicial
polytope. In dimension 3 simplicial spheres are exactly boundary
complexes of simplicial polytopes, but in dimension 4 there are
simplicial spheres that are not boundary complexes of any polytope.
The DSE actually also hold for any simplicial Eulerian poset.
(i.e. a ranked poset such that any interval has the same number
of elements of even rank as elements of odd rank, and it is simplicial
if every down-set (the set of elements less than or equal to a given element)
is actually a boolean algebra (as a hyper-cube)). -- The neat think about
the DSE is that among the d entries of the f-vector,
the latter half are a linear combination of the first half (for odd
d the first "half" really means the lower integer part [d/2]
of d/2). Hence, if we know the enties of the f-vector
labelled with 0,1,...,[d/2]-1 then we can compute the rest
of the entries as linear (in fact, integer) combination of the first ones.
This greatly restricts the structure of an f-vector for a general
(d-1)-sphere, and hence the boundary complex of a polytope.
- Fifth week, February 20.
The cyclic polytope C(n,d) is a simplicial polytope generated
by n distinct points on the moment curve in the
d-dimensional space. The cyclic polytope has the obvious
maximum number (n choose i+1) of i-faces
for i among 0,1,...,[d/2]-1.
The Upper Bound Theorem (UBT)
states that among every simplicial (d-1)-sphere, the cyclic polytope
C(n,d) has the maximum number of i-faces for every
i, that is for i among 0,1,...,d-1.
The UBT for simplicial polytopes was proved by McMullen in 1070. -- NB!
The cyclic polytope C(n,d) has
n choose i+1 faces of dimension i
only for the first half of the i-faces.! -- The proof of the UBT used the fact
that each polytope has the combinatorial property of being shellable
(something we will discuss soon, intuitively it means that we can build
the boundary complex up and close it as if we are building closed brick-tower
or an igloo). Since (d-1)-spheres are in general not shellable, this proof of
McMullen does not carry over to (d-1)-spheres. To prove the UBT
(which was done be Richard Stanley at MIT) one must use that a
(d-1)-sphere is Cohen-Macaulay, which we will
discuss in the coming weeks.
- Sixth week, February 27.
- Part IV, M-vectors and Cohen-Macaulay rings
This part is at the heart of our coverage since it shows how commutative
algebra is important in combinatorics. Here R is a graded,
commutative algebra over a given field. We further assume that R
is finitely generated (by part/summand nr. 1, as part/summand nr. 0 is the field itself).
The version of the Hilbert Basis Theorem (HBT) that we will be using
states that "If I is a graded ideal and R is finitely
generated, then I is generated by finitely many homogeneous
elements of R". The "finitely generated"-part here is indeed a
special version of the HBT which states that
"any ideal of the polynomial algebra over a field
in finitely many variables/indeterminates, is finitely generated."
Again, this is a special case of the purest form of the HBT that
states that "if R is Noetherian, then so is R[x],
the polynomial ring over R." --
The Hilbert function
of R is simply the dimension of the n-th
summand of R over the given field, and the Hilbert series
is then the generating function of this function. Although it can many
times be hard to express the dimension of each summand in a graded
algebra R, the Hilbert series might actually sum to
a nicely expressable function, something which is informative when
investigating these dimensions. We will demonstrate the usufulness
of the Hilbert functions and series in lectures to come.
- Seventh week, March 6.
We say that a graded R is standard
if all the generating variables have the same degree (of one). For
a standard R we have the Noether Normalization Lemma (NNL),
which basically states that R (being finitely generated)
always contains a copy of the free polynomial ring over the base
field k in some number of variables.
The maximum number d of these variables turns out to be unique
and is called the Krull dimension of R. In that case
the elements of R representing these free variables are then
called homogeneous system of parameters (h.s.o.p., or just "hsop") for R.
The hsop for R are called regular if every element of
R can be written uniquely as a linear combination of certain given
elements of R, where the coefficients are polynomials
in the hsop's. As we saw, to show that hsop is regular can take
quite a bit of work, when done directly.
- Eighth week, March 20.
If a standard graded algebra R has a regular hsop, then it turns
out that every hsop for R is regular. In this
case we call R a Cohen-Macaulay ring. To figure
out whether or not a given set of hsop is regular can be a daunting
task if we prefer to do that directly. Luckily, we can instead
compute the Hilbert series for R and decide from
its form whether the given set of hsop is regular or not. Note
that computing the Hilbert series can itself be involved, but it
is much more direct, and can (for that matter) be programmed
for a computer to compute. In short computing the Hilbert series
is mechanical.
- Nineth week, March 27.
An M-vector (or O-sequence) is simply
the vector corresponding to the Hilbert series of a finitely
dimensional standard algebra S. The entries of an M-vector
must all be positive and less than or equal to a binomial coefficient
describing the number or monimial of a certain total degree. This
gives us the first criterion to check if a given vector is indeed
an M-vector. A nice algebraic condition describing exactly
when a given vector is an M-vector, is given by Macaulay's
Theorem (hence the "M" in M-vector!). This theorem is very
similar to the previous KKS theorem we discussed.
- Part V, Stanley's idea of the proof of the UBC
Stanley's proof of the Upper Bound Conjecture (UBC), a purely combinatorial
result, involves both commutative algebra and algebraic topology.
The main tool is the face ring aka Stanley-Reisner ring
of a given simplicial complex. This face ring is a graded standard
algebra over a field, which is a quotient algebra of the polynomial
ring in n variables, where n is the number of
vertices of the simplicial complex. One nice property of the face
ring is that its Hilbert series is given by a rational function
in lambda, where the numerator is precisely the polynomial given
by the h-vector of the complex. This result alone serves
as a vital justification for studying the h-vector of
a simplicial complex.
- Tenth week, April 3.
What Stanley needed to show, essentially, was
that the face ring of a (d-1)-sphere is Cohen-Macaulay.
That result was provided by a seminal paper by Gerald Allen Reisner,
which gives topological criteria on when exactly the face ring
is Cohen-Macaulay. From Reisner's result if follows that any
(d-1)-sphere is Cohen-Macaulay, and hence Stanley was
able to prove the UBC.
- Part VI, Reisner's topological criteria
In a seminal paper, Gerald Allen Reisner gave necessary and sufficient
conditions for the face ring of a simplicial complex to be Cohen Macaulay (CM).
These conditions turn out to be topological, in the sense that if two
simplicial complexes are homeomorphic, then they are either both CM
or neither is CM. -- In order to fully understand and appreciate Reisner's
theorem, some algebraic topology is needed, in particular the
reduced simplicial homology spaces of a simplicial complex.
The word "group" is used if the coefficients are integers.
The word "reduced" is here due to the fact that for i = -1
then in the reduced vector space
case C_{i} is generated by one element, namely the empty
set (so its dimension is f_{-1} = 1 consistent with
the h-vector), but in the usual topological (abelian group) case,
C_{i} = (0) corresponding to dimension zero.) Having
the boundary operator and the fact that the composition of any two consecutive
bounfary operators is the zero map, justifies the definition of
the i-th reduced homology quotient space over k.
- Eleventh week, April 10.
As we noticed in class the only reduced homology groups we can spot right
away are for i = -1,0, where for i = -1
the i-th reduced homology group is zero if the complex is non-empty,
but it is homomorphic to k if it is empty, and for
i = 0 the dimension of the 0-th reduced homology group
over k equals the number of connected components of the complex
minus one. Hence we see that the 0-th reduced homology group only
depends on the topological property of its connectivity. This holds
in general, in that if two complexes are homeomorphic, then their
reduced homology groups are homomorphic as vector spaces over the
field k. We also discussed two examples that illustrated
the reduced homology groups for the (d-1) dimensional ball and
(d-1) dimensional sphere. --
To state and understand Reisner's theorem, we define the link of a face
of a given simplicial complex. The link is itself a simplicial complex of dimension
given by the dimension of the simplicial complex minus the number of vertices of the face.
Reisner's theorem states that the face ring of a simplicial complex is Cohen Macaulay (CM)
if and only if for every face F (including the empty face!)
the i-th homology of the link of F is trivial, or equal to
(0), for all i not equal to the dimension of the link of F.
- Twelfth week, April 17th.
As an important corollary of Reisner's theorem, one obtains that if a simplicial
complex is homeomorphic to either the
(d-1) dimensional ball or (d-1) dimensional sphere,
then the face ring of the simplicial complex is CM over any field. One important
point, that was pointed out by the topologist James R. Munkres in 1984, is that
Reisner's criterion is topological in the sense that if two simplicial complexes
have homeomorphic geometric realizations then the face rings of them are either
both CM or neither is CM. As an example, if a simplicial complex has a geometric
realization homeomorphic to the torus, then its face ring is not CM. Another
example shows that if a simplicial complex has a geometric
realization homeomorphic to the real projective plane, then the face ring
is CM if and only if the base field k is not of characteristic 2. -- We know
that if the face ring is CM then the h-vector of the complex is an M-vector,
but we don't, in general, have the converse. However, by Stanley's theorem from 1977,
we do have that for a given h-vector/tuple it is an h-vector
of a simplicial complex whose face ring is CM if and only if it is an M-vector.
- Part VII, Cohen Macaulay complexes
A simplicial complex is said to be Cohen Macaulay (CM) over the
field k if its face ring over k is CM.
For each simplicial complex this definition clearly does depend on
the field k (notably on the characteristic!), but for
most our examples it does not. Hence, we can just talk about a cimplicial
complex being CM. It turns out that being CM is quite geometrical
in many ways: in particular if our complex is CM, then (i) each link
of the complex (itself a simplicial complex) is also CM, (ii) if the
dimension of the complex is one or more, then its geometric realization
is connected, and (iii) the complex is pure, which means that the
complex is composed of simplices, all of the same dimension.
EOF/EOS