65 problems

Kirillov–Klyachko's conjecture. The reduced Kronecker coefficients satisfy saturation:

Let $n$ be a positive integer, and let $\mathfrak{S}_n$ be the symmetric group. A complex irreducible character of $\mathfrak{S}_n$ has a square containing all irreducible characte…

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 $g(\lambda,n^d,n^d)$ be a rectangular Kronecker coefficient, and let $\gamma_{\lambda,d,n,m}$ be the multiplicity of the irreducible representation indexed by $\lambda$ in the…

Suppose that $\alpha, \beta, \gamma, \lambda, \mu, \nu$ are partitions such that $|\alpha|=|\beta|=|\gamma|$ and $|\lambda|=|\mu|=|\nu|$. For $n\in\mathbb{Z}_{\geq 0}$, consider th…

Let $d$ be odd, let $k$ be even, and define the rectangular Kronecker coefficients … where $g$ is the Kronecker coefficient. The sequence is known to satisfy…

Crystal-lowering conjecture. The operation $$ can be completed to a lowering operation $f^1$ that picks out which Kronecker tableaux should be considered lowest weight.

Bitableaux conjecture. The Kronecker coefficient with the corresponding partitions is counted by lexicographic bitableaux of the given shape and weights whose pair of reading words…

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathbf t\in\mathcal P(n)^3$, and let $g(\mathbf t)$ denote the corresponding Kronecker coefficient. Define … Let $F_n$ be…

Let $\mathcal P(n)$ be the set of partitions of $n$, and let $\mathbf t=(\lambda,\mu,\nu)\in\mathcal P(n)^3$. If $b_\lambda$, $b_\mu$, and $b_\nu$ are the $b$-loadings of the three…

Let $\mathcal P(n)$ be the set of partitions of $n$, and let $\mathbf t=(\lambda,\mu,\nu)\in\mathcal P(n)^3$. The $a$-loading of a triple is $a(\mathbf t)=a_\lambda+a_\mu+a_\nu$. A…

Pak–Panova's conjecture. With high probability,

Pak–Panova's conjecture. With high probability,

Let $\pmb{\lambda}$ and $k$ be as in the generalized Kronecker-coefficient conjecture, and let $\omega\in\bigwedge^k V$ be the Cayley form. Define $L(v)=\omega\wedge v$. The proper…

Let $g(\lambda^{(1)},\ldots,\lambda^{(d)})$ be the generalized Kronecker coefficient, let $\rho_k(\lambda)=(k,\lambda)$, and apply $\rho_k$ componentwise to tuples of partitions. L…

Let $g(\lambda^{(1)},\ldots,\lambda^{(d)})$ denote the generalized Kronecker coefficient, where the arguments are partitions of the same integer, and write…

Maximizer conjecture for $b_\star$. For $n\ge 6$, the values $b_\star$ are attained by triples such that

Maximizer conjecture for $r_\star$. When $n=4k$ with $k\ge 2$, the values $r_\star$ are attained by

Normal-limit conjecture. The distribution of the $r$-loadings of $\mathbf t$ converges to a normal distribution as $n\to\infty$.

For $m\ge 7$, let $(m^m)$ be the square partition of $m^2$, let $\mu'$ denote the conjugate partition of $\mu$, and define … Write $g(\lambda,\lambda,\mu)$ for the Kronecker coeffi…

Let $S_n$ be the symmetric group, and let $V$ be an irreducible representation of $S_n$. For partitions $tau, uvdash n$, write $g(tau,tau,nu)$ for the Kronecker coefficient giving…

For a partition $\lambda$, let $g(\lambda,\mu,\nu)$ be the Kronecker coefficient and define the symmetric Kronecker coefficient by … A counting function is concise if every positiv…

Let $\textsc{ComputeKron}$ be the problem of computing $g(\lambda,\mu,\nu)$ from partitions $\lambda,\mu,\nu$, and let $\ell(\lambda)$ denote the number of parts of $\lambda$. Pano…

Let $g(\lambda,\lambda,\mu)$ be a Kronecker coefficient, and let $\mathbb{S}_\lambda$ denote the irreducible $S_n$-module indexed by the partition $\lambda$. Tensor square conjectu…

Let $\alpha$, $\beta$, and $\gamma$ be partitions, and let $\overline{\operatorname{k}}(\alpha,\beta,\gamma)$ denote the reduced Kronecker coefficient. Saturation conjecture. If ……