69 problems

Triple-coincidence conjecture. For every integer $n\geq4$, no three distinct pieces in $\mathcal{P}'$ share the same strength on the $n\times n$ board.

Finite basis conjecture. The algebra $M_2(K)$ does not have a finite basis of its polynomial identities.

Let $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$ be a highest-weight label in the decomposition denoted by $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$, where … and … Here…

Let $K\in\mathbb{N}$, let $\lambda\vdash 4K$, and write $\lambda^{(n-3K)}$ for the partition obtained by adding a first row of length $n-3K$ to $\lambda$. Let…

Fix $K\in\mathbb{N}$. Let $W_{n,n+K}$ be the representation considered in the paper, and let $p_K(x)$ and $N_K$ be as in the preceding corollary, so that $p_K(n)=\dim(W_{n,n+K})$ f…

For each $n,K\in\mathbb{N}$, let $W_{n,n+K}$ be the representation considered in the paper, and let $\dim(W_{n,n+K})$ denote its dimension. Dimension-growth conjecture. Fix…

Let $L$ be a Lie $F$-algebra acting on a finite-dimensional algebra $A$ by derivations, so that $A$ is an $L$-algebra. Let $c_n(A)$ and $c_n^L(A)$ denote the ordinary and different…

Let $\tilde{\mathfrak M}$ be the rock-paper-scissors algebra with unit over a field $\mathbb F$, and let $p$ be a homogeneous polynomial in one variable whose sum of coefficients i…

Hartley's conjecture. If the unit group $U(FG)$ satisfies a group identity, then $FG$ satisfies a polynomial identity.

Kuzmin's conjecture. The exact value of the nilpotency class is

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Proper containment conjecture. The ideal of polynomial identities of the path algebra is properly contained in that of its principal subalgebra:

For each positive integer $n$, let $p_n(w)$ be the polynomial counting the LEGO structures made from $n$ parallel $w\times1$ tiles. Reflection-symmetry conjecture. For every…

A polynomial solution $(\mathcal{A},\mathcal{B},\mathcal{C})$ is constructed for each standard Pythagorean triple, where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are polynom…

Nonexistence conjecture. No nontrivial $G$-grading $\Gamma$ on $M_n(K)$ satisfies the primeness property for graded central polynomials.

Finite-dimensional representability conjecture. There exists a finite-dimensional $W$-(super)algebra $B$ such that

Let $W$ be a unital $F$-algebra over a field of characteristic zero, and let $A$ be a $W$-algebra. Say that the action of $W$ on $A$ is finite when the image of the action has fini…

Let $A$ be a $W$-algebra over a field of characteristic zero. A class of algebras has the Specht property if every $T$-ideal of the class is finitely generated. The Specht conjectu…

Exponent-discrepancy factorization conjecture. For every $m\geqslant6$ there exists an irreducible polynomial $r_m(t)\in\mathbb{Q}[t]$ of degree at most $m-6$ such that

For $d\geq 1$, define … Let $a,b\in\mathbb{Z}$ be nonzero, put $c:=a+b\neq 0$, and define … For $d\in\{a,b,c\}$, … If $a$ is even and $b$ is odd, also define … Then … This identity…

Identity-generation conjecture. Every multilinear identity in the mutation of a $\mathfrak{perm}$ algebra over a field of characteristic zero is a consequence of

Cocharacter support conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if one of the following holds: (i) $\lambda(3)=\lambda(4)=\emptyset$ and…

Cocharacter multiplicity conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if either $w_1\leq 2$, or $w_1\geq 3$ and…

Generating-ideal conjecture. The ideal of all weak polynomial identities for the pair $(\mathsf{A}_1,\mathsf{V})$ is equal to $\mathcal{I}$.

Polynomial solution conjecture. For each choice of the sign, there is a solution of