114 problems

Let $\mathbb{F}$ be the free group on two generators, and let $C^*(\mathbb{F})\otimes_{\mathrm{max}}C^*(\mathbb{F})$ and $C^*(\mathbb{F})\otimes_{\mathrm{min}}C^*(\mathbb{F})$ deno…

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…

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…

Product-state conjecture. For every state $\rho$, there exists a product state $\rho_1\otimes\rho_2$ with the same pair of entropies.

For $d\geq2$, let $\mathrm{C}^*\mathbb F_d$ denote the full group $C^*$-algebra of the free group on $d$ generators, and let $\otimes_{\min}$ and $\otimes_{\max}$ denote the minima…

Let $G$ and $H$ be connected bipartite graphs. Their tensor product $G\otimes H$ has exactly two components. If one of the factors has an automorphism that maps one of its bipartit…

Let $\gog$ be a simple Lie algebra with Weyl group $W$, and let $\VV_{\la}$ and $\VV_{\mu}$ be irreducible representations with highest weights $\la$ and $\mu$. For a weight $\xi$,…

Let $G$ be a complex reductive Lie group, let $V_\lambda$ and $V_\mu$ be irreducible representations with dominant highest weights $\lambda$ and $\mu$, and let $w$ and $v$ be Weyl…

Permutation-twisted module counting conjecture. The number of isomorphism classes of irreducible $g$-twisted $V^{\otimes k}$-modules is equal to the number of isomorphism classes o…

Riley's conjectures. If $R$ and $S$ satisfy nonmatrix identities, then $\gamma^2(R)$ is nil of bounded index, and the tensor product

Disjoint-union conjecture. For every two graphs $G$ and $H$,

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

Geometric tensor-product conjecture. The inverse of $F$ is an isomorphism of positive decorated geometric 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…

Tensor-product non-isomorphism conjecture. In general, the bialgebras

Parthasarathy–Ranga Rao–Varadarajan conjecture. The irreducible component with highest weight $\lambda+\alpha$ occurs with multiplicity at least one in…

Let $\mathfrak A$ and $\mathfrak B$ be type $1$ associative algebras whose resonant spectral values are precise in regular functionals. For an $\alpha$-regular functional, let…

Akutsu–Kashiwara's cyclicity conjecture. Under these conditions, the tensor product is cyclic.

Let $V(\bpi^i_{m,a})$ denote the finite-dimensional representation associated with the Drinfeld polynomial $\bpi^i_{m,a}$, and consider a tensor product of such representations wit…

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

Vila's conjecture. Complete continuity of the operator considered as a multilinear mapping does not imply complete continuity of the same operator considered as a linear mapping fr…

Tensor-power conjecture. For every positive integer $m$, the tensor powers