16 problems

Let ${\mathcal O}_q(M_{m,n})$ denote an algebra of quantum matrices, and let a torus act on it so that one can consider its torus-invariant prime ideals. A generating set is polyno…

Let $q\in\mathbb{K}^\times$ be not a root of unity, and consider the quantized nilradical for $\mathfrak{g}=\mathfrak{sl}(n)$ and $J=\{\alpha_k\}$, which is isomorphic to the algeb…

Let ${\mathcal O}_q(M_n)$ be the algebra of quantum $n\times n$ matrices. Its torus consists of the standard diagonal scaling automorphisms, and the transposition automorphism is t…

Let $A\in M_n({\mathbb Z}_+)$, and form the graph $H(A)$ whose vertices are the matrices obtained from $A$ by the prescribed $2\times2$ matrix transformations. For $B\leq A$, let a…

Goodearl–Lenagan's conjecture. There exists an explicit order-preserving bijection between

Let ${\cal O}(M_q)$ be the coordinate ring of $n\times n$ quantum matrices, let $X=(x_{ij})$ be its generator matrix, and define the nonassociative quantum powers recursively by…

Let $A$ be the quantum matrix algebra, let $K_{{\boldsymbol r},{\boldsymbol c}}$ be the ideal associated with the row and column index sequences ${\boldsymbol r}$ and…

Let $R$ be a ring. A sequence $u_1,\ldots,u_n$ in $R$ is polynormal if $u_1$ is normal in $R$ and, for each $i=2,\ldots,n$, the image of $u_i$ is normal in…

NP-hardness conjecture. Problem $(\ast)$ is NP-hard and remains NP-hard even in the flag case.

Assume that the base field $\mathbb{K}$ has characteristic zero and that $q\in\mathbb{K}^{\ast}$ is not a root of unity. Let the algebra of quantum matrices be the corresponding qu…

For an integer $N\geq 2$, let $R_q[M_N]$ be the algebra of quantum matrices of size $N\times N$, let $\mathbb{T}^{2N-1}$ be the indicated torus of automorphisms, and let the transp…

Let $\mathcal{M}_{m,p}(\mathbb{C})$ be the Poisson variety of complex $m\times p$ matrices, and let $\mathcal{O}_q(\mathcal{M}_{m,p}(\mathbb{C}))$ be its quantized coordinate ring.…

The limiting distribution conjecture. For every natural number $m$ and every $0\le i\le m$, this proportion satisfies

Primitive-proportion conjecture. As $n\rightarrow\infty$, the proportion of $\mathcal{H}$-primes that are primitive tends to

Exponential enumeration conjecture. For every positive integer $n$,

Primitive-prime enumeration conjecture. The number of primitive $\mathcal{H}$-primes in ${\mathcal O}_q(M_{3,n})$ is