169 problems

Let $\lambda \vdash n$ be a partition. Write $\operatorname{Ev}(\lambda)$ for the set of partitions obtained from $\lambda$ in the construction used in the paper, let…

Picky Conjecture. There exists a bijection

Let $b_1(n)$ be the largest irreducible character degree of the symmetric group $S_n$, and for each $m\geq 2$ let $b_m(n)$ be the $m$-th largest irreducible character degree of…

Let $G$ be a finite group and let $\chi$ be an orthogonally stable rational character of $G$. Its orthogonal discriminant is the square class represented by a unique square-free in…

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

For positive integers $k_1,\ldots,k_l$, let $\Sigma_{k_1,\ldots,k_l}$ denote the normalized character and let $\hat{B}_i$ denote the twisted Boolean cumulants. Positivity conjectur…

Let $\operatorname{\bf{G}}_0$ be a simple group of type $F_4$, and let $F_0$ be the generalized Frobenius map defining a Ree group of type $F_4$. Write…

Let $\lambda$ be the partition of $n$ whose diagram is the union of $m$ rectangles described above. Let $\mu\vdash k$ and fix a permutation $w_\mu\in\mathfrak S_k$ of cycle type…

Let $R(x)=1+\sum_{i\geq 2}R_ix^i$, and define $C_m$ by $C_0=1$, $C_1=0$, and … For $n\geq 0$, let $\Sigma_{k,2n}$ be the sum of the terms of weight $k+1-2n$ in $\Sigma_k$. A polyno…

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of…

Let $G$ be a solvable group, and let $\psi,\varphi\in\operatorname{Irr}(G)$ be faithful, where $\operatorname{Irr}(G)$ denotes the irreducible characters of $G$. Suppose that … whe…

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, and let $\chi, \psi \in \operatorname{Irr}(G)$ be faithful characters. Let $\eta(\chi,\psi)$ denote the number of d…

Let $V$ be a rational vertex operator algebra of CFT type, let $M$ be an irreducible $V$-module, and let ${\mathfrak h}$ and $L^{\circ}$ be the spaces defined in the source, with…

Commutator abelianness conjecture. If $\Mcd G N=\{1,f\}$ for some positive integer $f$, then the commutator subgroup

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Let $n_r$ be the least i…

Let $G$ be a finite group and let $\ell$ be a prime number. For an $\ell$-element $x\in G$, define the subnormaliser … Let $\operatorname{Irr}^x(G)$ denote the set of complex irred…

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a $p$-element. Write $\operatorname{Irr}^x(G)$ for the irreducible complex characters of $G$ that are nonzero at…

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a picky $p$-element, meaning that $x$ lies in a unique Sylow $p$-subgroup. Let $P\in\operatorname{Syl}_p(G)$ cont…

Let $G$ be a finite group, let $\pi$ be a set of primes, and let $H$ be a nilpotent Hall $\pi$-subgroup of $G$. Let $\mathcal H$ be the set of elements of $H$ that are $H$-picky, m…

Let $G$ be a finite group and let $x\in G$. Write $x=\prod_p x_p$ for the decomposition of $x$ into its $p$-parts. Assume that, for every prime $p$ dividing $o(x)$, the pair…

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, let $H=\mathbf{N}_G(P)$, and let $\mathcal P$ be the set of picky elements of $P$. Define…

Let $G$ be a finite group with abelian Sylow $p$-subgroups. Let $P\in\operatorname{Syl}_p(G)$ and let $x\in P$ be picky. Let $S_p(x)$ be a set of representatives for the conjugacy…

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, and let $x\in P$. Let $R_p(x)$ be a set of representatives for the conjugacy classes in the reduce…

Let $H$ be a finite group such that $L=[H,H]$ is quasi-simple and $\mathbf Z(L)=\mathbf Z(H)$, and let $\chi\in\operatorname{Irr}(H)$ be faithful. Wilde's quasi-simple commutator c…

Let $G$ be a finite group such that $S\lhd G/\mathbf Z(G)\leq\operatorname{Aut}(S)$ for some non-abelian simple group $S$, and let $\chi\in\operatorname{Irr}(G)$ be faithful. Suppo…