53 problems

Morales–Pak–Panova's conjecture. Under these assumptions,

Let $s_{\lambda/\mu}$ and $s_{\nu/\rho}$ be skew Schur functions, and write $f\leqslant_s g$ when $g-f$ is Schur-positive. For skew shapes, the LPP inequality is … The complexity c…

Let $\lambda$ be a partition with $n$ outside corners. Choose a corner $k$ and corresponding weight $\omega_\ell$, and retain the notions of Theorem … 1leq i1<cdots<irleq kleq jr<c…

Let $r$ be rank conditions satisfying the paper's rank-condition hypotheses, let $R_{ij}$ be the associated array of rectangles, and choose a fixed tableau $T_{ij}$ on each…

Let $n$ be a positive integer, let $partition \lambda\vdash n$ have length $\ell(\lambda)$ and rank $\operatorname{rank}(\lambda)$, and let $\hat{s}^2_\lambda$ denote its 2-bottom…

Let $(e_0,e_1,\dots,e_n)$ be a dimension vector for equioriented quiver representations, let $r$ be an orbit characterized by rank conditions, and write its Thom polynomial as … He…

Given an ordered pair $(\lambda,\mu)$ of partitions with the same number of parts, define … and … For any partition $\nu$, the starred-transformation conjecture. … This is a combin…

Witness-elimination conjecture. With

Linear-independence conjecture. Fix a rectangle $R$. The products

Let $X$ be a nonsingular variety, let $E_0\to E_1\to\dots\to E_n$ be a sequence of vector bundles and bundle maps, and let $r=\{r_{ij}\}$ be occurring rank conditions. The associat…

Kleber's conjecture. The set of products

Let $\lambda=(2^{e_2},\ldots,m^{e_m})$ be a partition without parts equal to $1$, set $N:=|\lambda|$ and $M:=|\widetilde\lambda|$, and write … For each admissible pair…

Shifted-binomial log-concavity conjecture. For every partition $\lambda$ without parts equal to $1$, and every admissible index $i$, the sequence…

The CII maximal-involution conjecture.

Alexandersson–Haglund–Wang conjecture. The coefficients $a_k(\lambda,\mu)$ and $b_{d-k}(\lambda,\mu)$ are nonnegative integers. Moreover, the polynomials

Let $G_w$ be the Stanley symmetric function of a permutation $w$. Call $G_w$ multiplicity-free when, in its Schur expansion, every coefficient is either $0$ or $1$. Billey–Pawlowsk…

Purity and exterior-power basis conjecture. For every $w\in Z_{r+1}$, the cohomology of $(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})$ is pure,

Let $c^\lambda_{\mu,\nu}$ be the Littlewood–Richardson coefficients defined by … Writing $\operatorname{sc}_{LR}$ for the counting problem that computes these coefficients, Unary L…

Regular factorization conjecture. For certain factorizations $\kappa$, there exist factorization solutions for which

Mizukawa–Nakajima–Yamada's conjecture. The following equalities hold:

Let $G(k,m)$ be the symmetric function whose Schur expansion is studied in the paper, and let $p_2$ denote the power sum symmetric function of degree $2$. An expansion in the Schur…

Let $\tau$ be a triangular partition of size $n$, and let $\mathcal{A}_{\tau}(q,t)$ be the associated polynomial. For each partition $\lambda$ of length at most $2$, write…

Fix a rectangle $a\times b$. For each partition $\lambda\subseteq a\times b$, let $\lambda^{\vee}$ be the partition obtained by rotating the complement of $\lambda$ in the rectangl…

Let $G(k,m)\in\Lambda$ be the Petrie symmetric function, let $p_2$ be the second power-sum symmetric function, let $\operatorname{Par}$ denote the set of partitions, and let…

Let $m,k\in\mathbb{Z}_{>0}$, let $\mu=(m^k)$, and let $\nu=(m+k-1,m+k-3,\dots,|m-k|+1)$. For partitions $\lambda$ and $\mu$ with $\mu\subseteq\lambda$, let…