Geometric Representation Theory Seminar: Fall 2007

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 well-known 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:

John Baez, Groupoidification.

Below you can find handwritten notes of the seminar. There are also videos of every class on my YouTube channel. As usual, the seminar meets on Tuesdays and Thursdays, and you can ask questions and read more discussion at the n-Category Café: the classes all have blog entries to go with them, which you can access below.

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 n-element sets. Representations of the general linear group GL(n,F_q) from types of structure on the n-dimensional vector spaces over the field with q elements, F_q. Uncombed Young diagrams D, and "D-flags" as structures either on n-element sets or n-dimensional vector spaces. Irreducible representations of n! versus representations coming from the actions of n! on sets of D-flags. Counting D-flags: q-factorials 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 orbi-simplex. 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 "orbi-simplex". 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 D-flag on some n-element subset of S, where D is any n-box Young diagram. We can think of this as a "D-ary predicate" on S: an n-ary 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 G-invariant D-ary predicate on S. These are the predicates our logical theory — and we can read off the axioms geometrically, too!
- Supplementary reading:
  Todd Trimble, Concrete groups and axiomatic theories I.
  Todd Trimble, Concrete groups and axiomatic theories II.

Lecture 3 (Oct. 4) - James Dolan on the orbi-simplex. Pictures of orbi-simplices for subgroups of 3!, the group of all permutations of the 3-element set. How the simplices in an orbi-simplex get labelled by Young diagrams D: a D-labelled simplex in the orbi-simplex of a subgroup G ⊆ S! is a G-orbit in the space of D-flags in S. Example: the D-labelled simplices in the orbi-simplex for the 3-element cyclic subgroup of 3!. How D-labelled simplices in the orbi-simplex these correspond to atomic invariant D-ary 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 G-invariant 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 orbi-simplex — namely, those labelled by this Young diagram:
```
X
X
```

Lecture 4 (Oct. 9) - John Baez on categorifying and q-deforming the theory of binomial coefficients - and multinomial coefficients! - using the analogy between projective geometry and set theory. Review of uncombed Young diagrams D, and D-flags on finite sets and finite-dimensional 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 D-flags on Fⁿ, denoted D(Fⁿ), is just the (n-1)-dimensional projective space over F, and the number of points in D(Fⁿ) is the nth q-integer:

[n]_q = (qⁿ - 1)/(q - 1)

When D has n boxes, two rows, and k boxes in the first row, D(Fⁿ) is the Grassmannian consisting of k-dimensional subspaces of Fⁿ, and the number of points in D(Fⁿ) is the q-binomial coefficient

(n choose k)_q = [n]!_q / [k]!_q [n-k]!_q

where the q-factorial [n]!_q is given by

[n]!_q = [1]_q [2]_q … [n]_q

For a general uncombed Young diagram D, D(Fⁿ) is a partial flag variety, and its number of points is a "q-multinomial 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 G-sets X and Y, there's a basis of intertwining operators from C^X to C^Y coming from G-orbits in X × Y. These intertwining operators are examples of "Hecke operators", and when X = Y they span an algebra, called a "Hecke algebra". G-orbits in X × Y are "atomic geometrico-logical 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 3-space, that is, D-flags where D is the Young diagram
```
XX
XX
```
Y is the set of "complete flags", that is, E-flags where E is the Young diagram
```
X
X
X
X
```

Lecture 6 (Oct. 16) - John Baez on categorifying and q-deforming the theory of multinomial coefficients. A surprising fact: q-multinomial coefficients are actually polynomials in q with natural number coefficients. It suffices to prove this for q-binomial coefficients, since every q-multinomial coefficient is a product of q-binomial 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 D-flags on F⁴ correspond to 2d subspaces of F⁴, or equivalently, lines in projective 3-space.
Another surprising fact: any q-multinomial 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 2-Hilbert 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 Gram-Schmidt orthonormalization to take the permutation representations of G = n! coming from n-box 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.
- Supplementary reading: John Baez, Higher-dimensional algebra II: 2-Hilbert spaces.

Lecture 8 (Oct. 23) - John Baez on the q-deformed Pascal's triangle. Categorifying and q-deforming the recursion relation for binomial coefficients. If (n choose k)_F is the set of k-dimensional subspaces of the vector space Fⁿ, we have:

(n choose k)_F ≅ (n-1 choose k)_F + F^n-k × (n-1 choose k-1)_F

so in particular, taking F to be the field with q elements, we obtain this relation for q-binomial coefficients:

(n choose k)_q = (n-1 choose k)_q + q^n-k (n-1 choose k-1)_q

Using this to compute the q-deformed Pascal's triangle. Symmetries of the q-deformed Pascal's triangle. Why the binomial coefficient (n choose k) is the number of combed Young diagrams with ≤ k columns and ≤ n-k rows. Why the q-binomial 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 k-dimensional subspaces of Fⁿ. The relation between Young diagrams and matrices in reduced row-echelon form.
- Supplementary reading: John Baez, The Quantum Pascal's Triangle

Lecture 9 (Oct. 25) - James Dolan on Hecke operators for the groups n!. Each n-box Young diagram D gives an action of n! on the set of D-flags on the n-element 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 Gram-Schmidt orthonormalization. So, we obtain one irreducible representation of n! for each n-box Young diagram. The example of 4!, continued. Given an n-box Young diagram D with d rows and an n-box 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 q-deformed Pascal's triangle and the quantum group GL_q(2,k). Putting Pascal's triangle in a magnetic field, we obtain the q-deformed 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 q-deformed binomial formula:

(x + y)ⁿ = Σ_{k = 0}ⁿ (n choose k)_q y^k x^n-k

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 q-deform 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 R-valued matrices

F: X × Y → R

Mat(R) is equivalent to the category of finitely generated free R-modules. For example, Mat(C) is equivalent to the category of finite-dimensional 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 q-deformed 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 R-valued 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 finite-dimensional vector spaces. We can thus turn G-invariant spans between G-sets into intertwining operators between finite-dimensional 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 finite-dimensional vector space into a natural number: its dimension. Another way to turn a category into a set: its set of components. π₀ 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 finite-dimensional 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 n-box uncombed Young diagram. Then the group G = n! acts on the set S of D-flags on the n-element set, and S//G is equivalent to the groupoid of "D-flagged sets". Similarly, for any field F, the group G = GL(n,F) acts on the set S of D-flags on the vector space Fⁿ, and S//G is equivalent to the groupoid of "D-flagged 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 2-functor from the bicategory
[finite groupoids, spans of finite groupoids, equivalences between spans]
to the bicategory
[finite-dimensional 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 H₀(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 (Dec. 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 finite-dimensional vector space, its zeroth homology H₀(G). It turns a span of finite groupoids
```
       j     k
   G <--- S ---> H 
```
into the linear operator defined as the composite
```
       j_*         k_*
H₀(G) ---> H₀(S) ---> H₀(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 2-functor
D: FinSpan → FinVect
where
FinSpan = [finite groupoids, spans of finite groupoids, equivalences between spans]
and
FinVect = [finite-dimensional 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 3-functor
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 G-sets as objects, and for any pair of finite G-sets 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".
- Supplementary reading: John Baez and Alex Hoffnung, Higher-Dimensional Algebra VIII: The Hecke Bicategory (draft version).

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 (finite-dimensional) permutation representations of G. In short: the Hecke bicategory is a groupoidification of the the category of permutation representations.
Future directions: groupoidifying the q-deformed 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 n-space". (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:
- Supplementary reading: John Baez, Alex Hoffnung and Christopher Walker, Higher-Dimensional Algebra VII: Groupoidification.

This seminar is continuing into the winter quarter - check it out!

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.