242 problems

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

Let $V$ be an $n$-dimensional vector space over $\Bbbk$, let $G=\operatorname{GL}(V)$ with Lie algebra $\mathfrak g=\mathfrak{gl}(V)$, and let $P\subset G$ be the parabolic subgrou…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $E=(B\otimes B)^{\mathfrak{g}}$, and let $p_1,\ldots,p_r$ and $\widehat p_1,\ldots,\widehat p_r$ be as above. Let…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra with dual Coxeter number $g$. Let $p_1,\ldots,p_r$ be homogeneous generators of $(S\mathfrak{g})^{\mathfrak{g}}$, and…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $B$ be the algebra used in the paper, and set $E=(B\otimes B)^{\mathfrak{g}}$. Let…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A$ be the quotient of…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A=\wedge(\mathfrak{g}\oplus\mathfrak{g})/I$, where $I$…

Sturmfels's conjecture. The Castelnuovo–Mumford regularity of $I$ is at most $d$.