10 problems

The conjecture. For every $\lambda\in\mathfrak{u}_n(q)^*$,

Let $n\in\mathbb{N}^{*}$, let $q$ be a prime power, and let $U_n(\mathbb{F}_q)$ be the group of $n\times n$ upper triangular matrices over $\mathbb{F}_q$ with ones on the diagonal.…

Let $k\geq 11$ and let $\mathcal{H}\subset \mathsf{UT}(n,\mathbb{Q})$ be any finite set of matrices. Write $\mathcal{I}=\{1,2,\ldots,k+1\}$, let $\operatorname{PI}^{\mathcal{I}}$ b…

Let $\mathbb{F}_q$ be a finite field, let $n\geq 1$, and let $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$ matrices over $\mathbb{F}_q$. Thompson's $q$-power degree…

For $n\geq 1$, let $U_n(q)$ be the group of upper triangular $n\times n$ matrices over $\mathbb F_q$ with ones on the diagonal, and let $k(U_n(q))$ denote its number of conjugacy c…

Let $U_n(q)$ be the group of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$ with ones on the diagonal, and let $k(U_n(q))$ denote its number of conjugacy…

Let $l_{n,c}(t)\in\mathbb Z[t]$ be the polynomial from Lehrer's conjecture, counting irreducible complex characters of $U_n(q)$ of degree $q^c$. Isaacs's positivity conjecture. The…

Exponential Kirillov-function conjecture. The irreducible characters and exponential Kirillov functions of $\mathrm{UT}_n(q)$ coincide if and only if $n\leq 6p$.

Lehrer–Isaacs conjecture. For fixed $n$ and $e$, $N_{n,e}(q)$ can be expressed as a polynomial in $q$ with integer coefficients.

Unitriangular polynomiality conjecture. For every positive integer $m$, $\rho_{(1^m)}(q)$ is a polynomial in $q$ with rational coefficients. This is the special case corresponding…