6 problems

Graded Amitsur conjecture. There exists a graded PI-exponent

Let $\eta_1$ and $\eta_2$ be gradings on $UT_n^{(-)}$. Say that $\eta_1$ and $\eta_2$ correspond to a 1-kill-coarsening when passing from one grading to the other moves exactly one…

Let $G$ be a group and let $(UT_n^{(-)},\eta)$ be an elementary $G$-grading, where $UT_n^{(-)}$ denotes the Lie algebra of upper triangular $n\times n$ matrices. A tree $\mu$ is a…

Degree-$n$ generation conjecture. The graded monomial identities of $M_n(F)$ follow from the graded identities of degree at most $n$.

Maximal graded cocharacter multiplicity conjecture. For any $\mathbb{Z}_m$-grading of $UT_m(F)$ and every partition $\lambda\vdash n$, one has

Let $BT(d_1,\dots,d_m;F)$ be the block-triangular matrix algebra with diagonal block sizes $d_1,\dots,d_m$ over the field $F$, and let $n=d_1+\cdots+d_m$. A monomial identity is a…