325 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…

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 $k$ be a positive integer, set $n=k(k+1)/2$, and let $\lambda=(k,k-1,\ldots,2,1)$ be the staircase partition of $n$. Let $\sigma_\lambda$ be the irreducible representation of…

Non-distributivity conjecture. If $r\geq 10$, then $S_2^\lambda$ is not distributive.

Let $n\geq 7$ be odd and let $I$ satisfy … For a partition $\lambda$ of $n$, write $\lambda_\lambda^I$ for the eigenvalue associated with the corresponding irreducible representati…

Hecke-algebra extension conjecture. Theorem should hold in $\mathcal{H}$ when each $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

Let $K\in\mathbb{N}$, let $\lambda\vdash 4K$, and write $\lambda^{(n-3K)}$ for the partition obtained by adding a first row of length $n-3K$ to $\lambda$. Let…

Polynomial growth conjecture. For any fixed $p\geq 0$, $m_{p,\{\lambda\}_n}(V_n(I))$ is eventually given by a polynomial in $n$.

Shift-invariance conjecture. If $\\{\alpha,\beta\\}$ is not an inversion of $w$, then the three probability ratios satisfy

Let $n\geq 2$, and let $e\leq \sigma_1\leq \sigma_2\leq \Omega_n$. For the Kazhdan–Lusztig $R$-polynomial $\widetilde{R}_{\sigma_1,\sigma_2}(q)$, write $F_j(q)$ for the $q$-Fibonac…

Let $W$ be a Coxeter group of type $B_n$ or $D_{2n+1}$. Let $V$ be its reflection representation, and let $U$ be the non-faithful reflection representation of dimension $n-1$ descr…

Character bound conjecture. There is a constant $A$ such that, for sufficiently large $n$, every big $\lambda$ satisfies

Let $m$ be a positive integer, let $p$ be an odd prime, and let ${\rm DP}(m)$ denote the relevant set of strict partitions of $m$. Let ${\rm DPR}_p(m)$ denote the indexing set for…

Let $[m]=\{1,2,\dots,m\}$, let $\mu\vdash k$, and let $\omega_\mu$ be a fixed element of the conjugacy class $C_\mu$ in $\mathfrak{S}_k$. Let $\mathfrak{S}^{(m)}_k$ be the set of p…

Let $r,s$ be positive integers, let $\Phi(x,j)$ be the generating-function expression used in the preceding propositions, and write $[x^{r+1}]$ and $[y^{s+1}]$ for coefficient extr…

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 $C>0$. A Young diagram $\lambda$ with $n$ boxes has at most $C\sqrt{n}$ rows and columns, and let $\pi\in S_n$ be a permutation. Moore and Russell's character bound. There exis…

Trivial-source conjecture. Indecomposable, self-dual $k\Sigma_d$-modules with Specht filtrations are trivial source modules. This is a proposed weaker replacement for the earlier c…

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…

Dipper–James conjecture. The centre of ${\mathcal H}$ is the set of symmetric polynomials in the Jucys–Murphy operators $L_1,\ldots,L_n$.

Let $k$ be the underlying field, let $\N\in\operatorname{mod} k\Sigma_d$ be indecomposable and self-dual, and suppose that $N$ has a Specht, and hence also a dual Specht, filtratio…

Let $q_n$ denote the polynomial equation problem associated with the unordered configuration space of $n$ points, let $g(q_n)$ be its Schwartz genus, and let $L_n$ be the relevant…

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 $G^\bullet_{\mu^+,\mu^-}(\lambda;\tau)$ be the disconnected generating function of two-partition Hodge integrals for partitions $\mu^+$ and $\mu^-$. Let…