Ottawa-Carleton Joint Algebra Seminar

Let \(G\) be a finite group. The faithful dimension of \(G\) is defined to be the smallest possible dimension for a faithful complex representation of \(G\). Aside from its intrinsic interest, the problem of determining the faithful dimension of \(p\)-groups is motivated by its connection to the theory of essential dimension. In this paper, we will address this problem for groups of the form \(\mathbf{G}_p:=\exp(\mathfrak{g} \otimes_{\mathbb{Z}}\mathbb{F}_p)\), where \(\mathfrak{g}\) is a nilpotent \(\mathbb{Z}\)-Lie algebra of finite rank, and \(\mathbf{G}_p\) is the \(p\)-group associated to \(\mathfrak{g} \otimes_{\mathbb{Z}}\mathbb{F}_p\) in the Lazard correspondence. We will show that in general the faithful dimension of \(\mathbf{G}_p\) is given by a finite set of polynomials associated to a partition of the set of prime numbers into Frobenius sets. At the same time, we will show that for many naturally arising groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single polynomial. The arguments are reliant on various tools from number theory, model theory, combinatorics and Lie theory.

Starting from a graded Frobenius superalgebra \(B\), we consider a graphical calculus of \(B\)-decorated string diagrams. From this calculus we produce algebras consisting of closed planar diagrams and of closed annular diagrams. The action of annular diagrams on planar diagrams can be used to make clockwise (or counterclockwise) annular diagrams into an inner product space. In turns out that this space can be identified with the space of symmetric functions equipped with the Jack inner product at Jack parameter \(\dim B_\textrm{even} - \dim B_\textrm{odd}\). In this way, we obtain a graphical realization of that inner product space. This is joint work with Anthony Licata and Daniele Rosso.

To a finite directed graph, we can associate functorially a \(\mathbf{Z}\)-action on a zero-dimensional compact space, and take either the (pre-)ordered (first) Čech cohomology group, or the (pre-)ordered \(\mathrm{K}_0\) group of the crossed product \(C^*\)-algebra. The resulting pre-ordered group is easy to describe, and is closely related to the usual first edge cohomology group of the graph, but equipped with a natural partial ordering. This yields two classes of invariants; one easy to calculate with. The more complicated invariant is surprisingly effective at distinguishing graphs. Many examples will be given.

Let \(G/B\) be a flag variety over \(\mathbb{C}\), where \(G\) is a simple algebraic group with a simply laced Dynkin diagram, and \(B\) is a Borel subgroup. The Bruhat decomposition of \(G\) defines subvarieties of \(G/B\) called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. The Chow ring of \(G/B\) is generated as a ring by the classes of Schubert divisors, and it is generated as an abelian group by the classes of all Schubert varieties. In particular, each product of Schubert divisors is a linear combination of Schubert varieties.

I am going to discuss the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination equals 1. Also, I am going to talk about possible applications of my result to the computation of so-called canonical dimension of flag varieties over non-algebraically-closed fields.

I will show that the space \(\Omega^{\leq 1}\) of formal differential \(\leq 1\)-forms on \(\mathbb{R}^n\) has an (induced) SAYD module structure on the Connes-Moscovici Hopf algebra \(\mathcal{H}_n\). Then we will see that the Hopf-cyclic cohomology \(\mathcal{H}_n\) with coefficients in formal differential forms is identified with the Gelfand-Fuks cohomology of the Lie algebra \(W_n\) of formal vector fields on \(\mathbb{R}^n\). Furthermore, I will introduce a multiplicative structure on the Hopf-cyclic bicomplex, and we will see that this van-Est type isomorphism is multiplicative. I finally show the whole machinery in the case \(n=1\); by pulling back the multiplicative generators of \(H^\ast(W_1,\Omega^1_{1})\) to \(H^\ast(\mathcal{H}_1, \Omega^1_{1 \delta})\).

It has long been known that the numbers of solutions to certain equations over finite fields can be described in terms of modular forms. The most famous results of this type are that of Eichler-Shimura, showing that every rational Hecke eigenform of weight 2 gives rise to an elliptic curve over Q whose point counts are expressed in terms of the coefficients of the form, and Wiles, showing that conversely all elliptic curves over Q come from a modular form in this way. In dimension 2, the situation is understood by work of Elkies and Schütt together with some analytic number theory. In dimension 3, there are many examples, but no general theory, and almost nothing is known beyond that. I will describe three examples of varieties of dimension 5 whose point counts appear to be explained by the Hecke eigenform of weight 6 and level 8. For one of these the relation can be proved by connecting it to work of Frechette-Ono-Papanikolas on products of families of elliptic curves; for a second, the large degree of symmetry may make it practical to construct and analyze a resolution of singularities explicitly; and the third appears to give a new relation for hypergeometric functions over finite fields.

The Robinson-Schensted-Knuth correspondence is a many-faceted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of n and pairs of standard Young tableaux with the same shape, which is a partition of n. In another (more general) version, it provides a bijection between fillings of a partition lambda by arbitrary non-negative integers and fillings of the same shape lambda by non-negative integers which weakly increase along rows and down columns (i.e., reverse plane partitions of shape lambda). I will discuss an interpretation of RSK in terms of the representation theory of type A quivers (i.e., directed graphs obtained by orienting a path graph). This allows us to generalize RSK to other Dynkin types (plus a choice of minuscule weight), and is related to periodicity results for piecewise-linear toggling. I will not assume familiarity with either RSK or with quiver representations. This is joint work with Al Garver and Becky Patrias.

Koszul duality is a beautiful and wide-reaching set of ideas occuring in algebra, topology and representation theory. In its simplest form it exchanges pairs of associative \(\mathbb{C}\)-algebras with presentations by ``good'' quadratic relations and relates their representation theory, e.g. the symmetric algebra \(\mathrm{Sym} (V)\) and antisymmetric algebra \(\bigwedge (V^*)\). This talk will be a short introduction to the ideas behind Koszul duality. Time willing, we will present recent joint results with Ben Briggs on presenting the Koszul dual to arbitrary commutative complete intersection algebras, whose relations are neither "good" nor even quadratic.

It is well known that a cocomplete abelian category A with a compact projective generator has to be a module category, and vice-versa. What happens if we just assume that our category A has a generator M? Does it have to have a projective generator (we do not know whether A has enough projective objects)? We will give a positive answer to this question in case where the endomorphism ring of M is nice enough. Finally, we will consider some applications in representation theory.

A frieze is an array of positive integers with a finite number of infinite rows and whose entries satisfy a diamond rule: the \(2 \times 2\) matrix formed by the neighboring entries is 1. Introduced in the 1970's by Conway and Coxeter, the interest in friezes gained fresh momentum in the last decade, when strong relations to cluster theory were discovered. In particular, there exists a bijection between friezes and cluster algebras of type A. In cluster theory, the key concept is that of mutation. We introduce mutations of Conway-Coxeter friezes that are compatible with these cluster mutations, and describe the resulting entries using combinatorics of quiver representations. We will also consider a generalization of the Conway-Coxeter friezes, so-called \(SL_k\)-friezes, where the determinant of every \(k \times k\) determinant is 1. Here we obtain \(SL_k\)-friezes from cluster categories associated to certain Grassmannians \(Gr(k,n)\). This is joint work with K. Baur, S. Gratz, K. Serhiyenko, and G. Todorov.