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…

For each partition $\lambda$, let $t_\lambda$ denote a formal variable corresponding to the normalized character value $\chi(\lambda)/\chi(1)$, and let $T_\lambda$ be the specified ratio…

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 $\EuScript{E}_m$ be the Fomin–Kirillov algebra over a field $\mathbf{k}$ associated with the symmetric group $\mathbb{S}_m$. Fomin–Kirillov's conjecture. The algebra…

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…

Let $S_{\bullet}=\{S_N\}_{N\ge 1}$, and let $\chi^{\mu[\bullet]}=\{\chi^{\mu[N]}\}$ denote the stable irreducible character associated with a partition $\mu$. For a word $w$ in the…

Let $\rho_m$ be the staircase partition of size $n=\binom{m+1}{2}$, and let $g(\lambda,\lambda,\mu)$ denote the Kronecker coefficient. Saxl Conjecture. For every partition…

Hanany–Puder conjecture. For any $k\in\mathbb{Z}_{\geq2}$, any $\lambda\vdash k$, and any $w\in F_m$,

Let $n$ be a positive integer, and let a partition be self-conjugate if it equals its conjugate. Its Durfee size is the side length of its largest square contained in its Young dia…

Let $n\in\mathbb N$ and let $\lambda\vdash n$. A character is multiplicity-free when every irreducible constituent occurs with multiplicity at most one. The partitions listed in pa…

Basis conjecture. One has

Parity-unimodality conjecture. The fake-degree polynomials $f^{clambda}(q)$ are parity-unimodal for all partitions $clambda$.

Let $\Omega_n$ be the superspace ring with diagonal $\mathfrak{S}_n$-action, let $SR_n=\Omega_n/SI_n$ be its superspace coinvariant ring, and let $\mathbb{F}[\mathcal{OP}_n]$ be th…

Equivariant Gordan's lemma. The monoid $M$ is a $\operatorname{Sym}$-equivariantly finitely generated normal monoid.

For each $n$, let $S_n$ be the symmetric group, and let $z_I(S_n)$, $z_{I\!I}(S_n)$, and $z(S_n)$ denote respectively the proportions of character-table zeros of type I, of type II…

Higher-order asymptotic conjecture. There exists a constant $c>0$ such that

Shift-strategy characterization conjecture. Every optimal strategy is a shift strategy.

Concavity conjecture. For any fixed $n$, the function $f(m)=F(n,m)$ is concave.

Let $n$ be a prime number, let $\operatorname{sgn}_n$ be the partition of $n$ corresponding to the sign representation of $S_n$, and let $r$ be a term in the $n$-th Farey sequence.…

Let $S_n$ be the symmetric group, let $\mathcal{I}(S_n)$ denote its set of involutions, and for $z\in\mathcal{I}(S_n)$ let $\mathcal{A}(z)$ be the corresponding set of atoms, equip…

Trace formula conjecture. One has

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let…

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by … For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's…

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…

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