24 problems

Antenna-vertex conjecture. For every $n\geq 4$, the only vertices of local simplex dimension $1$ are the two antenna vertices:

Global chain decomposition conjecture. There exist collections of partitions $\mathcal{C}_{\mu}$, indexed by deficit partitions $\mu$, and a size-preserving involution…

Let $\lambda$ be a strict partition, and let $Q_{\lambda}$ be the Schur Q-function. Give each power sum symmetric function degree $1$, so that $\deg(p_{\nu})=\ell(\nu)$. Let…

Maximizer conjecture. At most one part of $\lambda$ is the sum of a part in $\mu$ and a part in $\nu$, while the other parts of $\lambda$ are distributed in $\mu$ and $\nu$.

Rank-factorization conjecture. For any partition $\lambda$ of rank $r$, there is a polynomial $Q_\lambda(z;q,t)\in\mathbb{Z}_{\geq 0}[z,q,t]$ such that

For a partition $\mu$ of $n$, let … be its lower interval in the immersion poset, and let $\lessdot_I$ denote a cover relation. The $(n-2,2)$ interval conjecture states that, for…

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…

Balanced-partition product conjecture. If $\lambda$ is balanced, then

Converse CDE conjecture for shifted diagrams. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced or trapezoidal.

Converse CDE conjecture. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced.

Let $a$ and $b$ be positive integers. Let $\lambda$ and $\mu$ be partitions of the same size, with $\lambda$ $\operatorname{Reg}_{a,b}$-valid and $\mu$ $(a,b)$-regular. Write…

Let $a$ and $b$ be positive integers with $2a<b$, and let $u$ be a partition that is both $\operatorname{Reg}{a,b}$-valid and $\operatorname{Colreg}{a,b}$-valid. For each hook…

Let the cyclic group $\mathbb{Z}_3$ act diagonally on $\mathbb{C}^2$, and let $b_k(-)$ denote the $k$th topological Betti number. The equivariant Hilbert scheme…

Let rigid semisimple operators be labeled by pairs of partitions $(\lambda^{'},\lambda^{”})$. Fix a fingerprint invariant, and consider operators satisfying different conditions in…

Let $G(n)$ denote the largest integer $k$ such that every partition of $n$ is uniquely determined by its multiset of $k$-minors, with the precise definition of $G(n)$ as in the pap…

Let $\lambda$ and $\mu$ be partitions of the same size. Write $a(\lambda,\mu)$ and $b(\lambda,\mu)$ for the associated invariants introduced in the paper, with $b(\lambda,\mu)$ the…

Weighted self-conjugate core-average conjecture.

Let $k$ be a positive integer, let $u$ be a $k$-bounded partition, and let $R( u)$ be its residue table. The theory also involves $k$-bounded partitions and $k$-Schur functions, i…

Bivariate symmetry conjecture.

Let $a<b$ be coprime. For an $(a,b)$-core partition $\lambda$, let $\ell(\lambda)$ be its number of nonzero rows. Define its skew length $s\ell(\lambda)$ to be the number of boxes…

Let $a,b\in\mathbb{N}$ be coprime. An $(a,b)$-core is a partition with no hook length divisible by $a$ or $b$; a self-conjugate $(a,b)$-core is one equal to its conjugate. The aver…

Simion's conjecture. For each integer $\ell$ and each partition $\lambda$, the sequence