244 problems

Let $m,n\geq 1$, let $\lambda=(n-k,1^k)$ be a hook partition of $n$ with $0\leq k\leq n-1$, and let $\mathcal{R}_{m,n}=\mathbb{C}[x_{r,j}:1\leq r\leq m,\ 1\leq j\leq n]$ carry the…

For each integer $n\ge 1$, let $V_1,V_2,V_3$ be $n$-dimensional vector spaces and let $T\in V_1\otimes V_2\otimes V_3$ be concise, meaning that each contraction map…

Let $G$ be a finite group, let $\b5$ be a field of characteristic $p>0$, let $V$ be a $\b5 G$-module, and let $P$ be a Sylow-$p$-subgroup of $G$. For each subgroup $H\leq G$, let…

Let $G=GL_n(\mathbb{F}_q)$ act on $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $Q(m,n)=S/(x_1^{q^m},\ldots,x_n^{q^m})$, and let $P_{\alpha}$ be the parabolic subgroup associated with a co…

Let $d\ge 3$ be an integer and let $1\le n<m\le d-1$ satisfy $\operatorname{gcd}(n,m,d)=1$. Let $I^d_{0,n,m}\subset k[x,y,z]$ be the associated GT-system. Minimality conjecture. Th…

Arborization conjecture. The forest form of a cluster monomial can be found from the web form via the arborization algorithm.

Invariant exterior-algebra conjecture. Except when $e=F_4(a_3)$ in type $F_4$ or $e=E_8(a_7)$ in type $E_8$, this algebra is an exterior algebra on the subspace

Homological-dimension-one conjecture. The quotient ${\mathsf V}/\!\!/G$ is either an affine space or a hypersurface; equivalently,

Mumford's conjecture. For some $d\in \mathbb{Z}_{+}$, the submodule

Least-denominator conjecture. $\bar{C}(n,k)$ is a rational function with least denominator equal to the denominator of the $(n-1)(k-1)$-st derivative of $F(t)$.

Sum's conjecture.

Let $W=(W_1,W_2,\ldots)$ be a sequence of representations, with $W_i$ an $\operatorname{SL}_{n_i}$-representation of dimension $N_i$. An arithmetic circuit succinctly encodes the g…

Auslander's conjecture. The skew group ring $A\#G$ is naturally isomorphic to

Let $V$ be a vector space and let $G\subset SL(V)$ be finite. Set $X=V/G$, and assume that $X$ admits a smooth crepant resolution $Y\to X$. For a cyclic subgroup…

The hsop existence conjecture. If $p\equiv 1\pmod h$, then $S$ contains an hsop $\mathbf{\Theta}$ carrying $V^*(-p)$; if $p\equiv -1\pmod h$, then $S$ contains an hsop…

Let $\mathfrak g$ be a finite-dimensional reductive Lie algebra of rank $l$ over an algebraically closed field of characteristic zero. For $x\in\mathfrak g$, let $\mathfrak g_x$ be…

Diagonal-gcd conjecture. For all $n$,

Denominator-relation conjecture. For all $n$,

Vanishing-numerator conjecture. For $n\ge5$,

Binomial-denominator conjecture. The denominators $D(C_{n,2};s,t)$ and $D(T_{n,2};s,t)$ can be written as products of binomials $1-s^a t^b$, where $a$ and $b$ are nonnegative integ…

Finite-group non-reflection conjecture. Then $V$ does not have the $2$-polarization property.

Let $n\ge3$, and write the Poincaré series in lowest terms as … where $A_n,B_n\in\mathbf Z[t]$ are relatively prime and $A_n(0)=B_n(0)=1$. Irreducibility conjecture. The polynomial…

Let $n\ge3$ and let $I_n=\langle p_n,q_n\rangle\subset\mathbf Z[z,t]$, with $p_n$ and $q_n$ as defined in the paper. The intersection $I_n\cap\mathbf Z[t]$ is an ideal of…

For positive integers $n$ and $k$, let $\operatorname{STab}((k^n))$ be the set of standard tableaux of rectangular shape $(k^n)$. For each $T\in\operatorname{STab}((k^n))$, let…

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