17 problems

Let $n,d\ge 3$, let $V$ be a $d$-dimensional vector space, and let $\sigma:\mathbb{Z}\to\{1,2,\dots,d-1\}$ be a generalized signature of period $n$. Set…

Let $k$ be a field of characteristic zero, let $V_d$ be a $k$-vector space of dimension $d$, and let $\Lambda^{S^2}_{V_d}$ be the graded vector space introduced as a generalization…

Let $k$ be a field of characteristic zero, let $V_d$ be a $k$-vector space of dimension $d$, and let $\Lambda^{S^r}_{V_d}$ be the graded quotient constructed from…

Let $k$ be a field of characteristic zero, let $V_d$ be a $k$-vector space of dimension $d$, and let $\Lambda^{S^3}_{V_d}$ be the graded vector space whose degree-$n$ component is…

Let $k$ be a field, let $V_d$ be a $d$-dimensional vector space, and for $n\geq 0$ define … where $\mathcal{T}^{S^2}_{V_d}[n]=V_d^{\otimes \frac{n(n-1)}{2}}$ and…

Let $V$ be a finite-dimensional vector space over $k$, and write $d=\dim_k(V)$. Let ${\Lambda}^{S^2}_V(m)$ denote the degree-$m$ component of the exterior graded Swiss-Cheese opera…

Let $W$ be a coincidental complex reflection group, let $S(V^*)$ be the symmetric algebra on the dual of its reflection representation, and choose basic invariants $f_1,\ldots,f_n$…

Let $V$ be a finite-dimensional vector space over a field $F$, let $E(V)$ be its exterior algebra, and let … be a neat element, where each $\xi_k$ is decomposable and has even degr…

Free-module conjecture. The isotypic component of $V_{\theta_s}$ in $\bigwedge V_{\theta_s}$ is a free module over $\bigwedge(T_1,\ldots,T_{r_s-1})$, with basis vectors…

The lattice-path conjecture. The Hilbert series of $E_n/(f,g)$ is

Let $E$ be an exterior algebra over $\mathbb C$ with $n$ generators, let $f,g\in E$ be generic quadratic forms, and let $S$ be the corresponding commutative polynomial ring with ge…

Exterior-algebra expected-series conjecture. For $n-d=4,5,7,8,9$ and some larger values of $n$, the Hilbert series of $\Lambda(V)/(f)$ is the expected series; the source also recor…

Fröberg–Lundqvist conjecture. If $r\ge n$, then

Crucial–Lundqvist–Nevo conjecture. The Hilbert series of $\Lambda(V)/(f,g)$ is

Let $A$ be a finite ordered set, let $\Bbbk$ be a commutative unital ring, and let $\mathbf{Pl}_A$ denote the plactic monoid on $A$. Theorem gives an injective algebra map from the…

Let $V$ be a finite-dimensional vector space of odd dimension, and let the Hochschild cochain differential graded Lie algebra of the exterior algebra $\bigwedge V$ be equipped with…

Let $G$ be a finite abelian group, and let $A=\{a_1,\ldots,a_k\}$ and $B=\{b_1,\ldots,b_k\}$ be two $k$-subsets of $G$. Let $K$ be any field containing an element of multiplicative…