35 problems

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 $g(\lambda^{(1)},\ldots,\lambda^{(d)})$ 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

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 $n$ 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 $\rho_m$ 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…

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

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…

Saxl's conjecture.

Multipartition sorting conjecture.

Sorting conjecture.

Representation-valued concavity conjecture. The function $V$ is concave under tensor multiplication: whenever $(\lambda+\mu)/2$ is a partition,

Kirillov–Klyachko conjecture. The reduced Kronecker coefficients satisfy

Let $\rho_k$ denote the staircase partition and let $\delta_\ell$ denote the square partition. Their sizes are $|\rho_k|=\binom{k}{2}$ and $|\delta_\ell|=\ell^2$. For a self-conjug…

Fully symmetric Kronecker asymptotic conjecture.

Unimodality conjecture. For every $1\leq i\leq\binom{n}{2}-1$, the sequence $d_{\nu,i}$ is symmetric and unimodal. This is proposed as a stronger conjecture implying equivariant lo…

For $n\geq3$, let $\nu\vdash n$, let $g_{\lambda\mu}^{\nu}$ be the Kronecker coefficient in $V^\lambda\otimes V^\mu$, and let $H_\lambda$ denote the hook product of the partition…

For $n\in\mathbb{N}$, let ${\bf D}^{\mathbb{C}}(\lambda)$ denote the simple complex $\mathfrak{S}_n$-module labelled by the partition $\lambda$. A partition is a symmetric $p$-core…

Let $\Bbbk$ be a field of characteristic $2$, let $n=k(k+1)/2$, and let $\rho(k)=(k,k-1,\dots,2,1)$. A partition is 2-regular if no positive part occurs at least twice, and…

Let $n$ be a positive integer, and let $\lambda\vdash n$ be a staircase-like partition, meaning a self-conjugate partition satisfying one of the three conditions described in the s…