Geometric Representation Theory Seminar  Fall 2007
John Baez and James Dolan
This fall, our seminar is tackling geometric
representation theory — the marvelous borderland where
geometry, groupoid theory and logic merge into a single subject.
The seminar is jointly run by John Baez and James Dolan.
Besides explaining wellknown stuff, we'll report on
research we've done with Todd Trimble over the last few years.
For much more on the subject of this seminar, see:
Below you can find videos and handwritten notes of
the seminar.
As usual, the seminar will meet on Tuesdays and Thursdays, and
you can ask questions and read more
discussion at the
nCategory
Café: the classes all have blog entries to go with them,
which you can access below.
For more information on the videos and problems you may encounter
watching them, see this.

Lecture 1
(Sept. 27)  John Baez on some
of the basic ideas of geometric representation theory. Classical
versus quantum; the category of sets and functions versus the
category of vector spaces and linear operators. Group representations
from group actions. Representations of the symmetric group n!
from types of structure on nelement sets. Representations of
the general linear group GL(n,F_{q}) from types of structure
on the ndimensional vector spaces over the field with q elements,
F_{q}. Uncombed
Young diagrams D, and "Dflags" as structures either on nelement sets or
ndimensional vector spaces.
Irreducible representations of n! versus representations
coming from the actions of n! on sets of Dflags. Counting Dflags:
qfactorials and their limit as q → 1. The "field with one element".
Projective geometry.

Lecture 2 (Oct. 2) 
James Dolan on transformation groups, logic and the orbisimplex.
A "transformation group" is a group acting as transformations
of some set S. Every transformation group is the group of all
permutations preserving some structure on S, and this structure is
essentially unique. The bigger the transformation group, the less
structure: symmetry and structure are dual, just like "entropy" and
"information", or "relativity" and "invariance".
To describe
structure on sets we can use a logical theory, with types,
abstract predicates and axioms. If the theory is "complete" (i.e. all models
are isomorphic), then the essentially unique model has a group
of symmetries. In this case, how can we recover the theory from this group?
For simplicity suppose its model is finite, so we have a subgroup
G of the permutation group S! for some finite set S. Form the
simplex Δ_{S} with S as vertices, and then take
the quotient Δ_{S}/G: the "orbisimplex".
This quotient is nicely
described as a quotient of the barycentric subdivision of Δ_{S}.
A simplex in the barycentric subdivision of Δ_{S}
is the same as a Dflag on some nelement subset of S, where D is
any nbox Young diagram. We can think of this as a "Dary predicate" on S:
an nary predicate on S invariant under the "Young subgroup" corresponding to
D (that is, the subgroup of n! preserving the partition of n into rows of D).
A simplex in the barycentric subdivision of Δ_{S}/G
is the same as an atomic Ginvariant Dary predicate on S. These
are the predicates our logical theory — and we can read off the axioms
geometrically, too!

Lecture 3 (Oct. 4) 
James Dolan on the orbisimplex. Pictures of orbisimplices
for subgroups of 3!, the group of all permutations of the 3element
set. How the simplices in an orbisimplex get labelled by
Young diagrams D: a Dlabelled simplex
in the orbisimplex of a subgroup G ⊆ S! is a Gorbit
in the space of Dflags in S. Example: the Dlabelled
simplices in the orbisimplex for the 3element cyclic subgroup of
3!. How Dlabelled simplices in the orbisimplex
these correspond to atomic invariant Dary predicates, and
how to read off the axioms these predicates satisfy, recovering an
axiomatic theory whose model on S has G as symmetries.
The relation
to traditional representation theory. Theorem: let G be a subgroup
of S! for some finite set S, and let R be the corresponding
representation of G on C^{S}. Then the space of intertwining
operators from R to R has a basis given by the orbits of G on
S × S — that is, atomic Ginvariant binary relations
on S. These operators are called "Hecke operators".
Apart from the diagonal orbit {(s,s): s ∈ S},
the orbits in S × S correspond to certain edges in
the orbisimplex — namely, those labelled by this Young diagram:
X
X

Lecture 4 (Oct. 9) 
John Baez on categorifying and qdeforming the theory of binomial
coefficients  and multinomial coefficients!  using the analogy between
projective geometry and set theory. Review of uncombed Young diagrams D,
and Dflags on finite sets and finitedimensional vector spaces over the
field with q elements, F = F_{q}.
When D has n boxes, two rows, and just one box in the first
row, the set of all Dflags on F^{n}, denoted D(F^{n}),
is just the (n1)dimensional
projective
space over F, and
the number of points in D(F^{n}) is the nth qinteger:
[n]_{q} = (q^{n}  1)/(q  1)
When D has n boxes, two rows, and k boxes in the first row,
D(F^{n}) is the Grassmannian consisting
of kdimensional subspaces of F^{n}, and the number
of points in D(F^{n}) is the qbinomial coefficient
(n choose k)_{q} = [n]!_{q} / [k]!_{q} [nk]!_{q}
where the qfactorial [n]!_{q} is given by
[n]!_{q} = [1]_{q} [2]_{q} … [n]_{q}
For a general uncombed Young diagram
D, D(F^{n}) is a partial flag variety, and its number of points
is a "qmultinomial coefficient". Young subgroups versus parabolic
subgroups. Decomposing projective spaces into Schubert cells.

Lecture 5 (Oct. 11) 
James Dolan on Hecke operators. Examples of the big theorem from
Lecture 3:
for any finite group G and finite Gsets X and Y, there's
a basis of intertwining operators from C^{X} to C^{Y}
coming from Gorbits in X × Y. These intertwining operators are
examples of "Hecke operators", and when X = Y they span an algebra,
called a "Hecke algebra". Gorbits in X × Y are "atomic
geometricological relations between types of geometric figures".
Example 1: G is the isometry group of a cube. X is the set of corners
of the cube. Y is the set of edges. Example 2: G = GL(4,F) for some field
F. X is the set of "points" in projective 3space, that is,
Dflags where D is the Young diagram
XX
XX
Y is the set of "complete flags", that is,
Eflags where E is the Young diagram
X
X
X
X

Lecture 6 (Oct. 16) 
John Baez on categorifying and qdeforming the theory of multinomial
coefficients.
A surprising fact: qmultinomial coefficients are actually polynomials
in q with natural number coefficients. It suffices to prove this
for qbinomial coefficients, since every qmultinomial coefficient is
a product of qbinomial coefficients. So, it's enough to decompose any
Grassmannian into "Bruhat classes", and show that each of these
is isomorphic (as a set) to F_{q}^{k} for some k.
For this, we show that each Bruhat class corresponds to a set of
matrices in
reduced
row echelon form. Example: the Young diagram
D = XX
XX
for which Dflags on F^{4} correspond to 2d subspaces of
F^{4}, or equivalently, lines in projective 3space.
Another surprising fact: any qmultinomial is actually a "palindromic"
polynomial in q. The closure of a Bruhat class is called a "Schubert
cell", and this palindromic property follows from Poincaré
duality, since Schubert cells are a basis for the cohomology of the
Grassmannian over C.

Lecture 7 (Oct. 18) 
James Dolan on applications of Hecke operators. Theorem: if
a finite group G acts in a doubly transitive way on a finite
set X, then the resulting permutation representation of G on C^{X}
is the direct sum of two irreducible representations, one being
the trivial representation. Proof: every permutation representation
contains the trivial representation, and
there are only two Hecke operators from C^{X} to itself.
Lemma: if G is a finite group, Rep(G) is a 2Hilbert space with
the irreducible representations of G as an orthonormal basis.
(This is a combination of Schur's Lemma and Maschke's Theorem.)
Another application: using
GramSchmidt
orthonormalization to take
the permutation representations of G = n! coming from nbox Young diagrams
and turn them into an "orthonormal basis" of Rep(G): that is, a
complete collection of irreducible representations. Beginning of an
explicit calculation for n = 4.

Lecture 8
(Oct. 23)  John Baez on the qdeformed Pascal's triangle.
Categorifying and qdeforming the recursion relation for binomial
coefficients. If (n choose k)_{F} is the set of kdimensional
subspaces of the vector space F^{n}, we have:
(n choose k)_{F} ≅
(n1 choose k)_{F} +
F^{nk} × (n1 choose k1)_{F}
so in particular, taking F to be the field with q elements, we
obtain this relation for qbinomial coefficients:
(n choose k)_{q} =
(n1 choose k)_{q} +
q^{nk} (n1 choose k1)_{q}
Using this to compute the qdeformed Pascal's triangle.
Symmetries of the qdeformed Pascal's triangle. Why the
binomial coefficient (n choose k) is the number of combed
Young diagrams with ≤ k columns and ≤ nk rows.
Why the qbinomial coefficient (n choose)_{q} is the
sum over such Young diagrams D of q^{# of boxes of D}.
Why each term in this sum corresponds to a specific Bruhat
class in the Grassmannian of kdimensional subspaces of F^{n}.
The relation between Young diagrams and matrices in
reduced rowechelon form.

Lecture 9
(Oct. 25)  James Dolan on Hecke operators for the groups
n!. Each nbox Young diagram D gives an action of n! on the set of Dflags
on the nelement set. These actions give permutation representations
of n! called "flag representations". Flag representations are usually
reducible, but we can extract a complete set of irreducible representations
using Hecke operators, via a categorified
version of GramSchmidt orthonormalization. So, we obtain one irreducible
representation of n! for each nbox Young diagram. The example of 4!,
continued.
Given an nbox Young diagram D with d rows and an nbox Young diagram E with
e rows, we can use "crackpot matrices" —
d × e matrices of natural numbers with specified row and column sums
— to give explicit descriptions of all the
Hecke operators from one flag representation to another.

Lecture 10
(Oct. 30)  John Baez on the qdeformed Pascal's triangle
and the quantum group GL_{q}(2,k). Putting Pascal's triangle
in a magnetic field, we obtain the qdeformed Pascal's triangle. Now
the operation of moving down and to right (called x) and the operation
of moving down and to the left (called y) no longer commute, but
instead satisfy:
xy = qyx
This relation implies the qdeformed binomial formula:
(x + y)^{n} = Σ_{k = 0}^{n}
(n choose k)_{q} y^{k} x^{nk}
Picking a field k, the "algebra of functions on the
quantum
plane",
k_{q}[x,y], is the associative algebra over k generated
by variables x and y satisfying the relation xy = qyx. The
symmetries of the quantum plane form the
quantum group GL_{q}(2,k)
The basic philosophy of algebraic geometry. The functor from
geometry to algebra. Noncommutative geometry as a mutant version
of algebraic geometry. Hopf algebras, and how they "coact" on algebras.
A sketch of how we'll simultaneously
qdeform and categorify the following structures:

binomial coefficients (to obtain Grassmanians)

the variables x and y showing up in the binomial theorem
(to obtain certain Hecke operators)

the group GL(2,k) (to obtain a categorified version of the
quantum group GL_{q}(2,k))

Lecture
11 (Nov. 1)  James Dolan on Hecke operators between
flag representations. Describing these Hecke operators using
matrices with specified row and column sums.
The problem of composing these operators: the composite
of two such operators is not a single operator but a "superposition"
of many. However, in the limit where we rescale our Young diagrams
by making the rows longer and longer, this superposition is
sharply peaked at some definite answer. The result looks like
an imitation of ordinary matrix multiplication, with a certain
"correction factor" thrown in.

Lecture 12
(Nov. 6)  John Baez on matrix mechanics and its generalizations.
Heisenberg's original matrix mechanics, where a quantum process from a
set X of states to a set Y of states is described by a matrix of
complex "amplitudes":
F: X × Y → C
We can generalize this
by replacing the complex numbers with any rig R, obtaining a category Mat(R)
where the objects are finite sets, and the morphisms from X to Y are
Rvalued matrices
F: X × Y → R
Mat(R) is equivalent to the category of finitely generated free Rmodules.
For example, Mat(C) is equivalent to the category
of finitedimensional complex vector spaces, FinVect_{C}.
If {0,1} is the rig of
truth values with "or" as addition and "and" as multiplication,
Mat({0,1}) is equivalent to the category with finite sets as
objects and relations as morphisms, FinRel.
There's
an obvious map from Mat({0,1}) to Mat(C), which lets us reinterpret
invariant relations as Hecke operators. But this map is not a functor,
so we don't get a functor
FinRel → FinVect_{C}. To fix this, we can consider Mat(N),
where N is the rig of natural numbers. This is equivalent to FinSpan,
the category where morphisms are isomorphism class of spans between finite
sets. The theory of spans as categorified matrix mechanics.

Lecture 13
(Nov. 8)  James Dolan on Hecke operators between
flag representations of n!. Comparing two notations for such Hecke operators:
crackpot matrices and braid diagrams.
Preview of the qdeformed
case, where the braid diagrams will allow us to categorify the Jones
polynomial (thought of as an invariant of positive braids).
Seeing a Young diagram in the braid describing a Hecke operator coming
from a Schubert cell of a Grassmannian.

Lecture 14 (Nov. 13)  John Baez on matrix mechanics and Hecke operators.
Any rig R gives a category Mat(R), whose objects are finite sets and whose
morphisms are Rvalued matrices. Any rig homomorphism from R to R' gives
a functor from Mat(R) to Mat(R). The homomorphism from N to C lets us
turn spans of finite sets into linear operators between finitedimensional
vector spaces. We can thus turn Ginvariant spans between Gsets into
intertwining operators between finitedimensional representations of G.
These are "Hecke operators".
A flawed attempt to formally state the "Fundamental Theorem of Hecke
Operators" in terms of this functor.

Lecture 15
(Nov. 15)  James Dolan on the fundamental theorem
of Hecke operators and various forms of decategorification.
The problem with the statement from last time.
Decategorification processes. Turning a category into a set: its set of
isomorphism classes of objects. Turning a finite set into a natural number:
its cardinality. Turning a finitedimensional vector space into a natural
number: its dimension. Another way to turn a category into a set: its set of
components. π_{0} turns a topological space into a set: its set
of components. π_{1} turns a space into a truth value: the
empty space become 'false', while nonempty spaces become 'true'. The
Grothendieck group construction turns an abelian category into an
abelian group. Degroupoidification turns finite groupoids into
finitedimensional vector spaces, and spans into linear operators.

Lecture 16
(Nov. 20)  John Baez on Hecke operators and groupoidification.
Correcting the mistake from last time: a quick
fix is easy, but the real solution requires "groupoidification". For
starters, this means replacing a group G acting on a set S by a groupoid
S//G, the "weak quotient" or "action groupoid". Object of S//G are just
elements of S, while morphisms are of the form (g,s): s → gs.
Examples: suppose D is an nbox uncombed
Young diagram. Then the group G = n! acts on the set S of Dflags on the
nelement set, and S//G is equivalent to the groupoid of "Dflagged
sets". Similarly, for any field F, the group G = GL(n,F) acts on the
set S of Dflags
on the vector space F^{n}, and S//G is equivalent to
the groupoid of "Dflagged vector spaces".

Lecture 17
(Nov. 27)  James Dolan on degroupoidification.
The 0th homology of a groupoid.
Why groupoids don't get enough respect.
Why 0th homology doesn't get enough respect.
Transfer maps for 0th homology.

Lecture 18 (Nov. 29)  John Baez on degroupoidification.
Turning a group G acting on a set S into a groupoid, the
weak quotient S//G. Turning
a map between groups acting on sets into a functor between
groupoids. Degroupoidification as a 2functor from the
bicategory
[finite groupoids, spans of finite groupoids, equivalences between
spans ]
to the bicategory
[finitedimensional vector spaces, linear operators, equations
between linear operators]
Turning a groupoid X into a vector space, namely the "zeroth homology"
of X with coefficients in the field k, denoted H_{0}(X,k).
This is the free vector space on the set of isomorphism classes of objects
of X. Cohomology as dual to homology. Example: the homology of
the groupoid of finite sets is the polynomial ring k[z], while
its cohomology is the ring of formal power series, k[[z]].
Turning a span of finite groupoids into a linear operator using
the concept of "groupoid cardinality". Heuristic introduction
to groupoid cardinality. The "cardinality" of a groupoid X is the
sum over objects x, one from each isomorphism class, of the
fractions 1/Aut(x), where Aut(x) is the automorphism group of x.
A puzzle: what's the cardinality of the groupoid of finite sets?

Lecture 19 (December 4)  James Dolan on the Fundamental
Theorem of Hecke Operators. Answer to last week's puzzle.
A new puzzle: find an interesting groupoid with cardinality e^{e}.
A harder one: find an interesting groupoid with cardinality π.
Degroupoidification turns a finite groupoid
G into a finitedimensional vector space, its zeroth homology
H_{0}(G). It turns a span of finite groupoids
j k
G < S > H
into the linear operator defined as the composite
j_{*} k_{*}
H_{0}(G) > H_{0}(S) > H_{0}(H)
where k_{*} is the pushforward (defined in an obvious way)
and j^{!} is the transfer (defined in a clever way using
groupoid cardinality, as explained here).
Degroupoidification is a weak monoidal 2functor
D: FinSpan → FinVect
where
FinSpan = [finite groupoids, spans of finite groupoids, equivalences between
spans ]
and
FinVect = [finitedimensional vector spaces, linear operators, equations
between linear operators]
The latter is really just a category in disguise. So, we can use
degroupoidification to obtain a weak 3functor
D^{–}: [bicategories enriched over FinSpan] → [categories enriched over FinVect]
For us, the key example of a bicategory enriched over FinSpan is
the "Hecke bicategory" of a finite group G, Hecke(G). This has finite
Gsets as objects, and for any pair of finite Gsets A and B it has
hom(A,B) = (A × B)//G
Composition in the Hecke bicategory involves a "trispan".
Future directions: following the plan outlined on page 400
of Daniel Bump's book on Lie Groups, in the chapter "The
Philosophy of Cusp Forms".

Lecture 20 (December 6)  John Baez on the Fundamental
Theorem of Hecke Operators. This theorem says that for any
finite group G, if we take the bicategory Hecke(G) and
degroupoidify it using
D^{–}: [bicategories enriched over FinSpan] → [categories enriched over FinVect]
the result is equivalent to the category of (finitedimensional)
permutation representations of G.
In short: the Hecke bicategory is a groupoidification of the
the category of permutation representations.
Future directions: groupoidifying the qdeformed
Pascal's triangle, the action of the quantum group GL_{q}(2) on the
quantum plane, and more generally the action of GL_{q}(n) on
"quantum nspace". (Final words cut off
as the power cable to the video camera is accidentally unplugged!)
For a more precise and thorough statement of the Fundamental
Theorem, read this:
This seminar is continuing into the winter quarter 
check it out!
We're offering the above videos in streaming and/or
downloadable form,
both as .mov files. Downloading them takes a long time, but you may
need to do this, since the streaming videos seem to work well only if
you have a good internet connection.
.mov files can best be played using a free program
called QuickTime.
If you have QuickTime and your web browser has .mov files associated to
this program, you should be able to click on the "Streaming Video" links
above and watch the videos. An alternate method is to launch the
QuickTime player on your computer, click on "File" and then "Open URL",
and type in the URLs provided above. This has the advantage that you
can easily make the picture bigger.
If you can handle URL's that begin with
rtsp, you can instead go the corresponding URL of that form, for example:
This may also have advantages, but at present my computer gags on such URL's,
so I don't know.
Errata
If you catch mistakes, let me know and I'll add them to the
list of errata.
There may also eventually be some
LaTeX, encapsulated
PostScript and xfig files to download
if for some bizarre reason you want them.
However, we reserve all rights to this work.
© 2007 John Baez and James Dolan
baez@math.removethis.ucr.andthis.edu