28 problems

Let $U_n(q)$ be the group of upper-triangular $n\times n$ matrices over $\mathbb{F}_q$ with units on the diagonal; equivalently, it is a maximal unipotent subgroup of…

Let $U_n(q)$ be the group of upper-triangular $n\times n$ matrices over $\mathbb{F}_q$ with units on the diagonal, and let $O_e(q)$ denote the number of its irreducible characters…

Unique representatives conjecture. For every coadjoint orbit, there exists a unique linear form $\lambda\in S$ lying on that orbit.

Assume that $Q$ is of finite Dynkin type and let $N_K$ be the unipotent radical associated with the complementary subset $K$ of vertices. Let $R_K$ be the subalgebra of…

Let $G_0$ be any connected unipotent group over ${\mathbb F}_q$, let $G=G_0\otimes_{{\mathbb F}_q}{\mathbb F}$ with Frobenius $\operatorname{Fr}$, and let…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, and let its functional dimension be $d_e=(\dim G-n_e)/2$. The integrality…

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, let ${\mathcal M}_e^{perv}$ be the full subcategory of perverse objects in…

The character-sheaf duality and perversity conjecture. For every minimal idempotent $e$, there exists $n_e\in\\{0,1,\dotsc,\dim G\\}$ such that

For an integer $n$, let $\bbH$ be the affine group scheme of formal series $P(X)\in 1+XA[[X]]$ satisfying $P(X)P(Y)=P(X+Y+XY)$. The assignment $m\mapsto(1+X)^m$ defines a morphism…

Boyarchenko–Drinfeld's conjecture. $G$ is easy if and only if its $cL$-packets of character sheaves are trivial.

Let $N$ be the maximal unipotent subgroup under consideration, and let ${\rm C}[N]$ have the biperfect bases introduced in the paper, including the MV basis and the theta basis. MV…

Group-theoretic Frattini-resistance conjecture. If there exist homomorphisms of pro-$p$ groups $\tau,\tau':G\to\mathbb{U}_3$ such that

Platonov–Potapchik's conjecture. If all primitive elements of $G$ are unipotent, then the group $G$ generated by $S$ is unipotent.

Let $S_n$ be the symmetric group with identity element $e$ and Bruhat order, and let $U_-^{\geq 0}$ be the totally nonnegative lower unipotent group, decomposed into unipotent cell…

The dual canonical basis conjecture. For every $T\in {\rm SSYT}(k-1,[k])$,

Cluster compatibility conjecture. For any decorated reduced word for $w_0$, the set ${\bf X}_{\mathbb{I}}$ is a cluster for $U$ mutation-equivalent to an initial cluster attached t…

Tossici's conjecture.

Verschiebung lower-bound conjecture. The essential dimension of $G$ over $k$ satisfies

Let $k$ be a field of positive characteristic, and let $G$ be a finite commutative unipotent group scheme over $k$. Let…

Let $\mathcal{O}$ be the ring of integers of a number field $K$, and let $\mathbf{G}$ be a unipotent group scheme arising from a finitely generated free torsion-free $\mathcal{O}$-…

Let $G$ be a connected unipotent group over ${\mathbb F}_q$, let $\overline{G}=G\otimes_{{\mathbb F}_q}\overline{{\mathbb F}}_q$, and let…

Conjecture. (1) ${\mathcal M}^{perv}_e$ is a semisimple abelian category with finitely many simple objects. (2) There exists $n_e\in{\mathbb Z}$ such that…

Let $\widetilde w=(i_1,\ldots,i_l)$ be a reduced expression of a Weyl group element $w$, and let ${\mathscr A}(\Gamma_{\widetilde w})$ be the associated classical cluster algebra.…

Lusztig's conjecture. There should exist an interesting theory of character sheaves for unipotent groups in positive characteristic, including nontrivial $L$-packets.