Given a finite group $\text{G}$ and a field $K$, the faithful dimension of $\text{G}$ over $K$ is defined to be the smallest integer $n$ such that $\text{G}$ embeds into $\operatorname{GL}_{n}(K)$. We address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_{q}:=\exp (\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q})$ associated to $\mathfrak{g}_{q}:=\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q}$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_{p}$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such that on each part the faithful dimension of $\mathscr{G}_{q}$ for $q:=p^{f}$ is equal to $fg(p^{f})$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

This is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.

Standard codeterminants for Donkin's symplectic Schur algebra $\bar{S}(n,r)$ are defined. It is shown that they form a basis of $\bar{S}(n,r)$. They are then used to give a purely combinatorial proof that $\bar{S}(n,r)$ is quasi-hereditary.

The determination of the generalized Springer correspondence for disconnected reductive groups is completed here by being achieved in the case of disconnected groups of types $D_4$ in characteristic 3 and $E_6$ in characteristic 2. It is expected that this correspondence will provide an important ingredient for the computation of character values of finite groups of Lie type. Some evidence is provided in support of the latter statement.