59 problems

Let $k$ be a positive integer, set $n=k(k+1)/2$, and let $\lambda=(k,k-1,\ldots,2,1)$ be the staircase partition of $n$. Let $\sigma_\lambda$ be the irreducible representation of…

Tropical tensor-product conjecture. The tropicalization of $F$ is an isomorphism of torsion-free Kashiwara crystals

Boundary-stratum conjecture. There exists a subcone $S+\sigma\subset S$ such that

Singular-triple criterion. There exists a triple $\sigma\in {\mathcal P}(G)\cap L^3$ with $Tr(\sigma)\in Q(R)$ and $\sigma\notin Tens(G)$ if and only if there exists a singular tri…

The structure conjecture. (1) If $R$ is simply-laced, then

Let $\widehat{G}$ be a connected semisimple complex group, let $\widehat{T}$ be a maximal torus of $\widehat{G}$, and let $\mu_1,\mu_2,\lambda$ be weights of $\widehat{T}$. Suppose…

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…

Saturation conjecture. One can take $N=1$ in the above theorem. Equivalently, for integer $\lambda$, $\mu$, and $u$, the relations $lambdaboxplusmusimcnu$ and…

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody Lie algebra, let $\delta$ be its null root, and let $V(\gamma)$ denote the irreducible highest weight representation of highest…

Let $P^+$ be the set of dominant integral weights, and let $V(\gamma)$ denote the irreducible highest weight representation of highest weight $\gamma$. For pairs of dominant weight…

Let $\mathfrak{g}$ be a simple Lie algebra or an untwisted affine Kac–Moody Lie algebra, let $\rho$ denote the sum of the fundamental weights, and let $V(\gamma)$ be the irreducibl…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with weight lattice notation $P^+$, and let $\rho$ be the half-sum of its positive roots. For a dominant integral weight…

Barker's conjecture. $K_1 \otimes_* K_2 = K_1 \otimes^* K_2$ if and only if one of $K_1$ and $K_2$ is a Choquet simplex.

Let $\mathfrak{sl}(m|n)$ be the type $A$ Lie superalgebra, let $V_{(2s)}$ be its covariant representation, and write … where $\lambda$ ranges over $(m,n)$-hook Young diagrams. Deco…

Conjectural tensor-product formulae.

For each positive integer $n$, let $V_\lambda$ denote the irreducible representation of $S_n$ indexed by a partition $\lambda$ of $n$. Tensor-square conjecture. There exists a part…

Null-summand conjecture. There is some $m\ge 1$ such that $Z$ is a direct summand of $V^{\otimes m}$, and consequently

He–et al.'s conjecture. There is some $n\in\mathbb{N}$ such that $V^{\otimes n}$ contains a direct summand of the form $\operatorname{Ind}(W)$ for a $kG$-module $W$.

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, let ${\bf T}$ be its torus, and let $\operatorname{St}$ denote the Steinberg module. For a cha…

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, and let $\mathscr{X}({\bf G})$ be the stated category of $\Bbbk {\bf G}$-modules. For a…

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, and let $\mathscr{X}({\bf G})$ be the abelian category of $\Bbbk {\bf G}$-modules defined by t…

Let $(E,\top)$ be a locality vector space, and let $V$ and $W$ be locality vector subspaces of $E$. The locality tensor product $V\otimes_\top W$ is equipped with the quotient loca…

Let $\mathcal{F} \in \mathbb{C}^{m \times p \times n}$ be a signal, let $\Psi$ denote its samples under the stated tensor-product dynamical evolution, and let…

Let $J,J'$ be $k$-element subsets of $[n]$, and let $L(M_J)$ and $L(M_{J'})$ be the corresponding simple $U_q(\widehat{\mathfrak{sl}_k})$-modules. Weak-separation tensor-product co…

Let $n$ be a positive integer, let $\mu=(\mu_1,\mu_2,\dots)$ be a partition of $n$, and let $\tau$ be a partition of $n$. For triples of partitions define … where…