44 problems

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Let $\mathbb{F}$ be a field, let $d$ be a nonnegative integer, and let $(\{\theta_i\}_{i=0}^d;\{\theta_i^*\}_{i=0}^d;\{\zeta_i\}_{i=0}^d)$ be a sequence of scalars in $\mathbb{F}$,…

Let $\Phi$ be a Leonard system with primitive idempotents $E_0,E_1,\ldots,E_d$ and dual primitive idempotents $E^*_0,E^*_1,\ldots,E^*_d$ in its associated algebra $\mathcal A$. The…

Assume $\mathbb K$ is algebraically closed. Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*$ be a tridiagonal pair on $V$. Let…

Let $A,A^*$ be a tridiagonal pair on $V$ satisfying the conditions of Definition 1p1 and the stated eigenvalue-form lemma. Let $K,R,L$ denote the operators in the associated…

Let $Phi$ be a generic tridiagonal-pair parameter array with diameter $d=3$, acting through $A,A^*$ on $V$, and let $w_1,w_2,w_3$ be its weights. Invariant-form conjecture. The fol…

Let $w_1,w_2,\ldots,w_d$ be the weights associated with the tridiagonal-pair construction, and let $q$ be the scalar used to define a $q$-string, namely a set of the form…

Let $U_0$ be the subspace used in the paper's decomposition associated with a tridiagonal pair, and let $L$ and $R$ denote the corresponding lowering and raising maps. Commutativit…

Let $(A,A^\star)$ be a tridiagonal pair on a finite-dimensional complex vector space, with eigenspaces ordered so that its shape is $(\rho_0,\rho_1,\dots,\rho_d)$, where $d$ is the…

Multiplicity-one conjecture. For every irreducible $\mathbb T$-module, the common eigenspaces for each of these four families all have dimension one. This conjecture would give a u…

Let $A,B$ be a tridiagonal pair over $\mathbb F$ having $q$-Racah type in the sense of Bockting-Conrad, and let $V$ be its underlying vector space. Let $\square_q$ denote the quant…

Let $V$ be a finite-dimensional irreducible $\widetilde \square_q$-module, where $\widetilde \square_q$ and $\boxtimes_q$ are the algebras appearing in the paper, with generators…

Let $A,B$ be a tridiagonal pair over $\mathbb F$ of $q$-Racah type, in the sense of Bockting-Conrad. Let $V$ be its underlying vector space. Tridiagonal-pair realization conjecture…

Let $\tilde T$ and $T$ be the algebras defined in the paper, and let $\mathcal A$ be the algebra acting on finite-dimensional irreducible modules described in the surrounding resul…

Fix an integer $d\geq 0$ and a sequence $p\in\operatorname{Feas}(d,\mathbb F)$. Let $\tilde T=\tilde T(p,\mathbb F)$ be the algebra defined in the paper, and let zigzag words mean…

Let $\Lambda$ be the grading index set, let $\check T_\lambda$ be the $\lambda$-homogeneous component of the checked algebra, let $R_\lambda$ be its relator subspace, and let…

Let $d\geq 0$ and let $p=(\{\theta_i\}_{i=0}^d;\{\theta_i^*\}_{i=0}^d)$ be a feasible sequence over $\mathbb F$. Let $T=T(p,\mathbb F)$ be the associated $\mathbb F$-algebra, let…

Classification conjecture. There exists a sharp tridiagonal system $\Phi$ over $\mathbb{K}$ with this parameter array if and only if the following conditions hold: (i)…

$\mu$-conjecture. The homomorphism $\mu$ is an isomorphism.

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, and let $e_i,e_i^*$ be the standard generators of $T$. Principal-submodule rank conjecture. For $0\le i\le d$, the $R$-submodules $e_i^*Te_0^*$…

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, let $Te_0^*$ be the principal right ideal, and call a $ZZ$ word feasible when it is nontrivial, ends in $e_0^*$, and has mutually distinct indic…

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, let $Te_0^*$ be the principal right ideal with its right $R$-module structure induced by the map $\mu$, and let $\operatorname{End}_R(Te_0^*)$ b…

Let $T$ be the tridiagonal-pair algebra, let $e_0^*$ be its distinguished starred generator, and let $ZZ$ words be the zigzag words in the standard generators. Principal-ideal basi…

Let $T$ be the algebra generated by the standard generators $\{e_i\}_{i=0}^d$ and $\{e_i^*\}_{i=0}^d$. A word in these generators is zigzag (or $ZZ$) when it is alternating and sat…

Let $\mathbb{F}$ be a field, let $d$ be a nonnegative integer, and let $T$ be the algebra associated with eigenvalue and dual-eigenvalue sequences of a tridiagonal pair. Let…