32 problems

Let $n$ be a positive integer, let $clambda$, $cmu$, and $cnu$ be partitions of length at most $n$, and let $T_n$ consist of the triples for which $V_\nu$ occurs in…

Let $c_{\mu, \nu}^{\lambda}$ denote the Littlewood–Richardson coefficient associated to partitions $\lambda$, $\mu$, and $\nu$. For a partition $\alpha$ and a positive integer $N$,…

Let $G_{\Bbb R}/K_{\Bbb R}$ be an irreducible Hermitian symmetric space of rank $r$, let $X=\frak p^+$, and let $\Lambda$ be the set of partitions of length at most $r$. For…

Let $\lambda$, $\mu$ and $u$ be partitions, and let $c{lambdamu}^{nu}$ denote the Littlewood–Richardson coefficient. Assume that $c{lambdamu}^{nu}0$. For each positive integer $N$…

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…

Let $\lambda,\mu,\nu \in ({\mathbb Z}^n)^3$ be a triple of dominant weights for $\operatorname{GL}_n({\mathbb C})$, and let $V_\lambda,V_\mu,V_\nu$ be the corresponding irreducible…

Let $\lambda,\mu,\nu$ be partitions with $|\lambda|+|\mu|=|\nu|$, and let $c^{\nu}_{\lambda\mu}$ be the Littlewood–Richardson coefficient, the multiplicity of the Schur function…

Let shifted Jack Littlewood–Richardson coefficients be defined for partitions with $|\lambda|\leq|\mu|+|\nu|$. Shifted Jack congruence conjecture. The adjacent-triple congruence co…

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be adjacent triples of partitions corresponding to a pivot at $b$. Let $\mathcal W'$ and $\mathcal W$ be the associated…

Let $T$ be a triple of partitions and let $W$ be a window for $T$. A rule is a Stanley sum whose evaluation gives the corresponding Jack Littlewood–Richardson coefficient. Jack win…

For partitions $\mu,\nu,\lambda$, let $\mathsf{SD}_{\mu\nu}^{\lambda}$ be the set of Stanley diagrams and let a Stanley sum be an integer linear combination of these diagrams. Stan…

Let $c_{\mu\nu}^{\lambda}$ be the ordinary Littlewood–Richardson coefficient. For a triple of partitions $(\mu,\nu,\lambda)$, let $\mathsf{SD}_{\mu\nu}^{\lambda}$ denote the set of…

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be triples of partitions that differ by a single box move in one pair, with pivot boxes $b_1\in\sigma_1$ and…

Fix partitions $\mu$ and $\nu$. For each family $\mathcal F$, let … and let $M_{\mu\nu}$ be the set of partitions $\lambda\supseteq\mu,\nu$ for which there exists a Molev tableau o…

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

Factorization conjecture. The Jack Littlewood–Richardson coefficient has an expansion

Kernel-exhaustion conjecture. The relations of Lemma exhaust the kernel of the channel equation. The claim concerns the completeness of the listed linear relations among channel so…

Channel conjugation symmetry. The transformed coefficients

Window positivity conjecture. The window factor accounts for all boxes outside $R$ in the positivity expression, namely

Rule formulation of the general structure conjecture. There exists an integer rule $\bm g_{\mu\nu}^{\lambda}\in\mathcal H_{\mu,\nu}^{\lambda}(\mathbb Z)$ of the form

Strong Stanley conjecture. If $c_{\mu\nu}^\lambda=1$, then

Let $\lambda$, $\mu$, and $\nu$ be partitions with at most $r$ parts, and let ${\sf LR}_r^{\mathbb Z}$ be the semigroup of triples $(\lambda,\mu,\nu)$ with positive Littlewood-Rich…

Stronger Newell-Littlewood saturation conjecture. If $N_{k\mu,k\nu,k\lambda}>0$, then there exist partitions $\alpha,\beta,\gamma$ such that

The combinatorial-model conjecture. The restriction of this map to $\overline{\mathcal{T}(\mu^t,\mu,s(\mathcal{T}_\lambda))}$ is a bijection onto…

Let $\lambda\vdash n$, and let $\mu$ and $\nu$ be partitions with $c^\lambda_{\mu,\nu}={\rm C}(n)$, where ${\rm C}(n)$ denotes the largest Littlewood–Richardson coefficient at size…