For much more on the subject of this seminar, see:
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.
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!
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
When D has n boxes, two rows, and k boxes in the first row, D(F^{n}) is the Grassmannian consisting of k-dimensional subspaces of F^{n}, and the number of points in D(F^{n}) is the q-binomial coefficient
where the q-factorial [n]!_{q} is given by
For a general uncombed Young diagram D, D(F^{n}) 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.
XX XXY is the set of "complete flags", that is, E-flags where E is the Young diagram
X X X X
D = XX XXfor which D-flags on F^{4} correspond to 2d subspaces of F^{4}, 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.
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.
so in particular, taking F to be the field with q elements, we obtain this relation for q-binomial coefficients:
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 F^{n}. The relation between Young diagrams and matrices in reduced row-echelon form.
This relation implies the q-deformed binomial formula:
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:
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
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.
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 F^{n}, and S//G is equivalent to the groupoid of "D-flagged vector spaces".
[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 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?
Degroupoidification turns a finite groupoid G into a finite-dimensional vector space, its zeroth homology H_{0}(G). It turns a span of finite groupoids
j k G <--- S ---> Hinto 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 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".
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:
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.