## Time and location

The Ottawa-Carleton joint algebra seminar meets during the academic year approximately once per week at either The University of Ottawa or Carleton University. Unless otherwise indicated, time and location are as follows:

• University of Ottawa:  Friday, 2:30pm–3:30pm, KED B015
• Carleton University:  Friday, 2:30pm–3:30pm, HP 4325

## Talks

Date Speaker Title (click for abstract)
Jan 12 (O) Mohammad Bardestani (Cambridge)
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.
Jan 19 (O) Alistair Savage (Ottawa)
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.
Jan 26 (O) David Handelman (Ottawa)
To a finite directed graph, we can associate functorially a $$\mathbb{Z}$$-action on a zero-dimensional compact space, and take either the (pre-)ordered (first) Čech cohomology group, or the (pre-)ordered $$K_0$$ 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.
Feb 2
Feb 9
Feb 16
Mar 2
Mar 9
Mar 16
Mar 23
Apr 6
(O) = uOttawa, (C) = Carleton