502 problems

Integrality conjecture. $X$ is integral over $\Lambda$ whenever no such subset sum vanishes.

Completeness conjecture. The invariant $y_3^{(4)}$ should satisfy a polynomial relation involving the $x$ variables.

Hochster's conjecture. The invariant ring of any linearly reductive group acting on a polynomial ring over a field is Cohen–Macaulay.

Nowicki's conjecture. The algebra $K[X_n,Y_n]^\delta$ is generated by $x_1,\ldots,x_n$ and

Let $P(N)$ be the stretched Littlewood–Richardson polynomial associated with fixed partitions, so that $P(N)=c^{N\nu}_{N\lambda,N\mu}$ for positive integers $N$. Fulton's conjectur…

Let $I=\langle f_0,f_1,\ldots,f_{p-1}\rangle\cap R^{S_n}$ be the elimination ideal associated with the transfer ideal, and let $A$ be the block matrix whose maximal minors generate…

Let $p$ and $q$ be primes with $q\mid p-1$, and let $C_q\ltimes_{\varphi} C_p$ be the corresponding semidirect product. For a finite group $G$, let $\beta(G)$ denote its Noether nu…

Let $\sigma$ be a $d$-admissible generalized signature of size $n$ and dimension $d$, and let $R_\sigma=\mathbb{C}[V^\sigma]^{\operatorname{SL}(V)}$. Natural cluster structure conj…

Let $R$ be a commutative ring, let $n$ be a positive integer, and let $A$ be an $n\times n$ symmetric matrix over $R$. For each $S\subseteq [n]$, write $A_S$ for the principal mino…

Let $G={\rm SL}_n\subset {\rm GL}_n=\widetilde G$ and let … Set $\mathfrak s=\mathfrak{sl}_n\ltimes V$, and let $H_1,\ldots,H_\ell$ be the basic symmetric invariants of…

Let ${\mathcal H}(n)$ be the rank-$n$ Heisenberg vertex algebra, with orthogonal-group orbifold ${\mathcal H}(n)^{\operatorname{O}(n)}$. Its generators have a first relation at wei…

Popov–Pommerening conjecture. Any observable regular subgroup of $G$ is a Grosshans subgroup.

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…

Bandeira–et-al. degree-three invariant conjecture. If $L\geq 10$, then the generic orbit in $\mathbf{L}^2(S^2)_L$ can be determined up to a finite list of orbits from invariants of…

Let $L$ be a nonnegative integer, let $V=\left(\bigoplus_{\ell=0}^{L}V_\ell\right)^R$ be the corresponding representation, and let the third moment be the invariant obtained by int…

Let $G$ be a crystallographic complex reflection group, meaning a subgroup of $\operatorname{AGL}_n(\mathbb C)$ generated by reflections that fix a hyperplane pointwise and have co…

A universal ring of invariants for algebraic structures of a fixed type is a polynomial algebra generated by certain trace diagrams. It admits the structure of a rational positive…

Invariant-lattice classification conjecture. Every primitive, $\operatorname{GL}_3(\mathcal{O}_K)$-invariant lattice $L$ in $\mathcal{V}(\mathcal{O}_K)$ is…

Watanabe's Bold Conjecture. There is another standard graded Artinian complete intersection $B$ of the same socle degree $d$, cut out by quadrics, together with an injective algebr…

Suppose that $W_1,W_2,\dots,W_t$ are subspaces of an $n$-dimensional $\mathbb{K}$-vector space $W\cong \mathbb{K}^n$, and let…

Shank–Wehlau conjecture. The invariant ring $S^G$ is a polynomial ring if and only if $\operatorname{Im}\operatorname{Tr}^G$ is a principal ideal of $S^G$.

Chevalley restriction conjecture. The homomorphism $c$ is always an isomorphism.

Let $c=1$, so the polynomial ring has one row of variables, and let ${\mathcal I}=(I_n)_{n\ge 1}$ be a nonzero $\operatorname{Inc}$-invariant chain of proper monomial ideals. Here…

Let $K$ be a field of characteristic zero, let a reductive group act linearly on a polynomial ring over $K$, and let $R$ be the corresponding ring of invariants. The ring of $K$-li…

Let $Q_m$ be the quasi-invariant polynomial space for $n=3$, and let the ring of symmetric polynomials act on it by multiplication. Three-variable freeness conjecture. The space…