15 problems

Kirillov's conjecture. These Kirillov functions comprise all the irreducible characters of $\mathrm{UT}_n(q)$ if and only if $n\leq 12$. The paper notes that the assertion is known…

Let $p$ be a prime, let $q$ be a power of $p$, let $U=U_n(q)$ be the group of lower unitriangular $(n\times n)$-matrices over $\mathbb F_q$, and fix $c\in\mathbb Z_{\geqslant 0}$.…

Let $UT_n(\mathbb{F})$ be the group of upper unitriangular matrices over a field $\mathbb{F}$, and let $T_n(\mathbb{F})$ be the group of invertible upper triangular matrices, conta…

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 $\mathbb F_q$ with ones on the diagonal, and let $k(U_n)$ denote its conjugacy-class-counting polynomial. Ar…

Fix a prime power $q$, 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 con…

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 $G=\operatorname{UT}(n,F)$ be a group of upper unitriangular matrices over a field $F$, where $n\in\mathbb N\cup\{\infty\}$, and let $e$ denote the identity matrix. A bijection…

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 $n\leq 6p$.

Let $p$ be the characteristic of the finite field $\mathbb{F}_q$, let $\mathrm{UT}_n(q)$ be the unitriangular group, and let $\mathfrak{u}_n(q)^*$ be the dual of its strictly upper…

Let $U_{13}(2)$ be the group of upper triangular $13\times13$ matrices over ${\mathbb F}_2$ with all diagonal entries equal to $1$. An irreducible character is irrational if some o…