27 problems

Let $\Lambda$ be the algebra of universal Lie-theoretic diagrams introduced in the paper, and let $t$ and $x_n$ denote its special diagrams. Vogel's generation conjecture. The alge…

Let $\Lambda$ be the graded algebra of fixed diagrams defined by the antisymmetry condition under $S_3$ and the relation $\phi_1(u)=\phi_2(u)$, and let $t,x_3,x_5,\ldots$ be the el…

Let a round diagram be a diagram with a Wilson loop, and let a symmetrized diagram be its corresponding diagram in the symmetrized-diagram basis. For a positive integer $l$, consid…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space with filtration quotients ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m+2f}^{\,u}$, where $f,o,e$ are the filtration parameters. Vanis…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space. A caterpillar diagram is a diagram of the illustrated type in the source, with an even number of legs on each body segment. Cate…

Let ${\cal B}_{m}^{\,0}$ be the space of connected graph diagrams with zero univalent vertices, and call a diagram planar if it admits a planar realization. Planar spanning conject…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space equipped with the filtration whose quotient at $(f,o,e)$ is ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m}^{\,u}$. Let $T(m,u)$ be the…

Let … , let … and … such that … The conjecture proposes that these algebras are isomorphic for a suitable choice of parameter. The supplied text gives no proof or resolution.

Let $k$ be a positive integer, let $\mathcal{A}_{2k}^{\text{aff}}$ be the affine partition algebra, and let $z_l$ and $p_n$ denote the central elements introduced in the paper for…

Let $A(n)$ be the space defined by the two-dimensional topological theory, and let $\mathscr{A}^m(n)$ be the set of surface diagrams whose connected components satisfy the genus bo…

Let $\Theta$ be the coupled Young graph, let $\mathcal{P}(\Theta)$ be its pascalization, and let $A_{(\mathcal{C}_H,\delta)}(\infty)$ be the associated inductive limit algebra at t…

Let $A_{(\mathcal{H},\delta)}(\infty)$ be the inductive limit algebra associated with the hyperoctahedral case, and let its branching graph be the pascalization of the relevant pri…

Let $A_{(\a4cH,\a4b4)}(\a4cH,\a4b4)$ denote the diagram-algebra deformation associated with the hyperoctahedral group, and let $A_{(\a4cH,\a4b4)}(\a4cH,\a4b4)(\infty)$ be its induc…

Brauer full-commutant conjecture. The full commutant of the action of ${\boldsymbol U}_{\star}'$ on

Brauer-projector decomposition conjecture. The tensor power

Brauer-centralizer conjecture. The action of ${\boldsymbol U}_{\star}$ on

Three-simple-module conjecture. The tensor product

Adjoint decomposition conjecture. The commutator action of ${\boldsymbol U}_{\star}$ on itself should decompose into irreducible representations in the same way as the adjoint acti…

Stable tensor-power decomposition conjecture. The decomposition of

Let $S_Q(G)$ denote the graph invariant obtained from the planar-diagram construction at parameter $Q$, and let $\mathsf{S}^Q([n],[0])$ be the corresponding diagram space with $n$…

Let $\mathcal T$ be the free PRO under consideration, let $\mu$ be a representation of $\mathcal T$ of dimension $N$, and let $d$ denote the parameter appearing in the trace of the…

The algebra $\mathcal{E}_n$ is generated by braid generators and ties subject to the defining relations, while $\widetilde{\mathcal{E}}_n$ is the corresponding diagram algebra gene…

The algebras $A_{n,k}$ are graded algebras arising from diagrammatic categories related to category $\mathcal{O}$. They are connected with the cohomology rings of closed attracting…

Let $m,n$ be natural numbers and fix a block ${\mathcal B}$ of atypicality $k$ in ${\mathcal F}(\operatorname{SOSP}(m|2n))$. Let $\mathbb{H}_k^\infty$ be the limiting graded diagra…

Let ${\mathcal F}(\operatorname{SOSP}(m|2n))$ be the category of finite-dimensional representations of the orthosymplectic Lie supergroup, let ${\mathcal B}$ be a block of atypical…